research article free convection heat and mass transfer...

Hindawi Publishing CorporationJournal of FluidsVolume 2013 Article ID 297493 14 pageshttpdxdoiorg1011552013297493

Research ArticleFree Convection Heat and Mass Transfer MHD Flow ina Vertical Porous Channel in the Presence of Chemical Reaction

R N Barik1 G C Dash2 and M Kar3

1 Department of Mathematics Trident Academy of Technology Infocity Bhubaneswar Odisha 751024 India2Department of Mathematics SOA University Bhubaneswar Odisha 751030 India3 Department of Mathematics Christ College Cuttack Odisha India

Correspondence should be addressed to R N Barik rnbmath22yahoocom

Received 20 May 2013 Revised 18 September 2013 Accepted 23 September 2013

Academic Editor Hideki Tsuge

Copyright copy 2013 R N Barik et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The objective of the present study is to examine the fully developed free convective MHD flow of an electrically conducting viscousincompressible fluid in a vertical porous channel under influence of asymmetric wall temperature and concentration in the presenceof chemical reaction The heat and mass transfer coupled with diffusion-thermo effect renders the present analysis interesting andcurious The analytical solution by Laplace transform technique of partial differential equations is used to obtain the expressionsfor the velocity temperature and concentration It is observed that under the influence of dominatingmass diffusivity over thermaldiffusivity with stronger Lorentz force the velocity is reduced at all points Further low rate of thermal diffusion delays the attainmentof free stream state Flow of aqueous solution in the presence of heavier species is prone to back flow

1 Introduction

In many transport processes and industrial applicationstransfer of heat and mass occurs simultaneously as a result ofcombined buoyancy effects of thermal diffusion and diffusionof chemical species Unsteady natural convection of heat andmass transfer is of great importance in designing controlsystems for modern free convection heat exchangers Morerecently Jha and Ajibade [1] have studied the heat and masstransfer aspect of the flow of a viscous incompressible fluid ina vertical channel considering the Dufour effect

Soundalgekar and Akolkar [2] studied the effect of masstransfer and free convection currents on the flow past animpulsively started infinite vertical plate and observed thatthe presence of foreign gasses in the flow domain leads toreduce the shear stress and rate of mass transfer significantly

In nature flow occurs due to density differences causedby temperature as well as chemical composition gradientsTherefore it warrants the simultaneous consideration oftemperature difference as well as concentration differencewhen heat and mass transfer occurs simultaneously It hasbeen found that an energy flux can be created not only by tem-perature gradients but by composition gradients also This is

called Dufour effect If on the other hand mass fluxes arecreated by temperature gradients it is called the Soret effect

The Soret and the Dufour effects have been found to beuseful as the Soret effect is utilized for isotope separationand in a mixture of gases of light and medium molecularweight the Dufour effect is found to be of considerable orderof magnitude such that it cannot be neglected In view ofimportance of the diffusion-thermo effect Kafoussias andWilliams [3] have studied the effects of thermal diffusionand diffusion-thermo on mixed free and forced convectiveand mass transfer boundary layer flow with temperaturedependent viscosity Jha and Singh [4] Kafoussias [5] andAlam et al [6 7] have contributed significantly to this fieldof study Recently Beg et al [8] studied chemically reactingmixed convective heat and mass transfer along inclinedand vertical plates considering Soret and Dufour effectsMore recently Dursunkaya and Worek [9] have studied thediffusion-thermo and thermal diffusion effects in transientand steady natural convections from a vertical surface

MHD flow with thermal diffusion and chemical reactionfinds numerous applications in various areas such as thermonuclear fusion liquid metal cooling of nuclear reactionsand electromagnetic casting of metals Abreu et al [10] have

2 Journal of Fluids

studied the boundary layer flows with Dufour and Soreteffects Osalusi et al [11] have worked on mixed and freeconvective heat and mass transfer of an electrically conduct-ing fluid considering Dufour and Soret effects Anghel etal [12] and Postelnicu [13] obtained numerical solutions ofchemical reacting mixed convective heat and mass transferalong inclined and vertical plates with the Soret and theDufour effects and concluded that skin friction increaseswith a positive increase in the concentration-to-thermal-buoyancy ratio parameter

The analysis of natural convection heat and mass transfernear a moving vertical plate has received much attention inrecent times due to its wide application in engineering andtechnological processes There are applications of interest inwhich combined heat and mass transfer by natural convec-tion occurs between amovingmaterial and ambientmediumsuch as the design and operation of chemical processingequipment design of heat exchangers transpiration coolingof a surface chemical vapour deposition of solid layernuclear reactor and many manufacturing processes like hotrolling hot extrusion wire drawing continuous casting andfiber drawing Gebhart and Pera [14] studied the effects ofmass transfer on a steady free convection flow past a semi-infinite vertical plate by the similarity method and it wasassumed that the concentration level of the diffusing speciesin fluid medium was very low This assumption enabledthem to neglect the diffusion-thermo and thermo-diffusioneffects as well as the interfacial velocity at the wall dueto species diffusion Following this assumption Das et al[15] investigated the effects of simultaneous heat and masstransfer on free convection flow past an infinite vertical plateunder different physical situations

Recently the unsteady MHD heat and mass transferfree convection flow of a polar fluid past a vertical movingporous plate in a porous medium with heat generation andthermal diffusion has been studied by Saxena and Dubey[16] Raveendra Babu et al [17] studied diffusion-thermoand radiation effects on MHD free convective heat and masstransfer flow past an infinite vertical plate in the presenceof a chemical reaction of first order Sudhakar et al [18]discussed chemical reaction effect on an unsteady MHD freeconvection flow past an infinite vertical accelerated plate withconstant heat flux thermal diffusion and diffusion thermo

In the present paper the flow of an electrically conductingviscous incompressible fluid in a vertical porous channelformed by two vertical parallel porous plates in the presenceof a transverse magnetic field is studied Further the masstransfer phenomena considered in this problem is associatedwith chemically reacting species The objective of the presentstudy is to extend the work of Jha and Ajibade [1] byincorporating the permeability and the magnetic field effectas well as chemical reaction on the flow heat and masstransfer phenomena

2 Mathematical Formulation

Let us consider the free convective heat and mass transferMHD flow of a viscous incompressible fluid in a verticalporous channel formedby two infinite vertical parallel porous

plates in the presence of chemical reaction A magnetic fieldof uniform strength 119861

0is applied transversely to the plate

The induced magnetic field is neglected as the magneticReynolds number of the flow is taken to be very small Theconvection current is induced due to both the temperatureand concentration differences The flow is assumed to be inthe 1199091015840-direction which is taken to be vertically upward along

the channel walls and 1199101015840-axis is taken to be normal to the

plates that are ℎ distance apartUnder the usual Boussinesqrsquos approximations the gov-

erning equations for flow are given by

1205971199061015840

1205971199051015840= ]

12059721199061015840

12059711991010158402+ 119892120573 (119879

1015840minus 1198790)

+ 119892120573lowast(1198621015840minus 1198620) minus

1205901198612

01199061015840

120588minus

]

1198701015840119901

1199061015840

1205971198621015840

1205971199051015840= 119863

12059721198621015840

12059711991010158402minus 1198701015840

119888(1198621015840minus 1198620)

1205971198791015840

1205971199051015840= 120572

12059721198791015840

12059711991010158402+ 1198631

12059721198621015840

12059711991010158402

(1)

where ] 119892 120573 120573lowast 120590 1198610 1205881198631198701015840

119888 1205721198631 and119870

1015840

119901are kinematic

viscosity acceleration due to gravity coefficient of thermalexpansion coefficient of mass expansion electrical conduc-tivity of the fluid uniformmagnetic field density of the fluidchemical molecular diffusivity the dimensional chemicalreaction parameter in the diffusion equation thermal diffu-sivity the dimensional coefficient of the diffusion-thermoeffect and the permeability of the medium respectively

Initial and boundary conditions of the problem are

1199051015840le 0 119906

1015840(1199101015840 1199051015840) = 0 119879

1015840(1199101015840 1199051015840) = 1198790 1198621015840(1199101015840 1199051015840) = 119862

0

1199051015840gt 0

1199061015840(119900 1199051015840) = 0 119879

1015840(0 1199051015840) = 119879119908 1198621015840(0 1199051015840) = 119862

119908

1199061015840(ℎ 1199051015840) = 0 119879

1015840(ℎ 1199051015840) = 1198790 1198621015840(ℎ 1199051015840) = 119862

0

(2)

We now introduce the following dimensionless quantities

119910 =1199101015840

ℎ 119905 =

1199051015840]

ℎ2 119906 =

1199061015840]

119892120573ℎ2 (119879119908

minus 1198790)

119875119903=

]

120572 119878

119888=

]

119863

119879 =1198791015840minus 1198790

119879119908

minus 1198790

119862 =1198621015840minus 1198620

119862119908

minus 1198620

119873 =120573lowast(119862119908

minus 1198620)

120573 (119879119908

minus 1198790)

119863lowast

=1198631(119862119908

minus 1198620)

120572 (119879119908

minus 1198790)

119872 =1205901198612

0ℎ2

120588]

(3)

119870119888=

119870lowast

119888ℎ2

] 119870

119901=

1198701015840

119875

ℎ2 (4)

Journal of Fluids 3

Here119875119903 119878119888119873 119863lowast119872119870

119888 and119870

119901are the Prandtl number

the Schmidt number the Sustentation parameter the Dufournumber the magnetic field parameter the chemical reactionparameter and the Porosity parameter respectively

The nondimensional form of (1) is given by

120597119906

120597119905=

1205972119906

1205971199102+ 119873119862 + 119879 minus (119872 +

1

119870119901

)119906

120597119862

120597119905=

1

119878119888

1205972119862

1205971199102minus 119870119888119862

120597119879

120597119905=

1

119901119903

1205972119879

1205971199102+

119863lowast

119901119903

1205972119862

1205971199102

(5)

Subject to boundary conditions

119905 le 0 119906 (119910 119905) = 0 119879 (119910 119905) = 0 119862 (119910 119905) = 0

119905 le 0 119906 (0 119905) = 0 119879 (0 119905) = 1 119862 (0 119905) = 1

119906 (1 119905) = 0 119879 (1 119905) = 0 119862 (1 119905) = 0

(6)

3 Solution of the Problem

Applying Laplace transform to (5) with boundary condition(6) we get

1198892119862

1198891199102minus 119878119888(119904 + 119870

119888) 119862 = 0

1198892119879

1198891199102minus 119904119875119903119879 = 119878

119888119863lowast(119904 + 119870

119888) 119862

(7)

119904119906 =1198892119906

1198891199102+ 119873119862 + 119879 minus (119872 +

1

119870119901

)119906 (8)

The boundary condition (6) becomes

119862 (0 119904) =1

119904 119862 (1 119904) = 0

119879 (0 119904) =1

119904 119879 (1 119904) = 0

119906 (0 119904) = 0 119906 (1 119904) = 0

(9)

Applying inverse Laplace transform to (7) and (8) subject toboundary condition (9) following Debanath and Bhatta [19]and using the formula

119871minus1

(119890minus119887radic119904+119886

119904) =

1

2119890minus119887radic119886 erfc( 119887

2radic119905minus radic119886119905)

+119890119887radic119886 erfc( 119887

2radic119905+ radic119886119905)

(10)

The solutions of (5) subject to the boundary conditions (6)are obtained by Laplace transform technique

Case 1 (119875119903

= 1 119878119888

= 1) The solutions of (5) subject to bound-ary conditions (6) are given by

119862 =

infin

sum

119899=0

[119891(119870119888 119886119899radic119878119888 1 0 119905) minus 119891(119870

119888 119887119899radic119878119888 1 0 119905)]

119879 = 1198601

infin

sum

119899=0

[119891(0 119886119899radic119875119903 119875119903minus 119878119888 119878119888119870119888 119905)

minus119891(0 119887119899radic119875119903 119875119903minus 119878119888 119878119888119870119888 119905)]

minus 1198601

infin

sum

119899=0

[119891(119870119888 119886119899radic119878119888 119875119903minus 119878119888 119878119888119870119888 119905)

minus 119891(119870119888 119887119899radic119878119888 119875119903minus 119878119888 119878119888119870119888 119905)]

+ (119863lowast+ 1)

infin

sum

119899=0

[119891(0 119886119899radic119875119903 1 0 119905)

minus119891(0 119887119899radic119875119903 1 0 119905)] minus 119863

lowast119862

119906 = 1198602

infin

sum

119899=0

[119891(119872 +1

119870119901

119886119899 1 minus 119878

119888 119878119888119870119888minus 119872 minus

1

119870119901

119905)

minus119891(119872 +1

119870119901

119887119899 1 minus 119878

119888 119878119888119870119888minus 119872 minus

1

119870119901

119905)]

minus 1198602

infin

sum

119899=0

[119891(119870119888 119886119899radic119878119888 1 minus 119878

119888 119878119888119870119888minus 119872 minus

1

119870119901

119905)

minus119891(119870119888 119887119899radic119878119888 1 minus 119878

119888 119878119888119870119888minus 119872 minus

1

119870119901

119905)]

+ 1198603

infin

sum

119899=0

[119891(119872 +1

119870119901

119886119899 1 minus 119875

119903 minus119872 minus

1

119870119901

119905)

minus119891(119872 +1

119870119901

119887119899 1 minus 119875


1

119870119901

119905)]

minus 1198603

infin

sum

119899=0

[119891(119870119888 119886119899radic119875119903 1 minus 119875


1

119870119901

119905)



1

119870119901

119905)]

+ 21198604

infin

sum

119899=0

[119891(119872 +1

119870119901

119886119899 119875119903minus 119878119888 119878119888119870119888 119905)

minus119891(119872 +1

119870119901

119887119899 119875119903minus 119878119888 119878119888119870119888 119905)]

minus 1198604

infin

sum

119899=0

[119891(119870119888 119886119899radic119878119888 119875119903minus 119878119888 119878119888119870119888 119905)


4 Journal of Fluids

minus 1198604

infin

sum

119899=0

[119891(0 119886119899radic119875119903 119875119903minus 119878119888 119878119888119870119888 119905)


+ (1198605minus 1198606)

infin

sum

119899=0

[119891(119872 +1

119870119901

119886119899 1 0 119905)

minus119891(119872 +1

119870119901

119887119899 1 0 119905)]

+ 1198606

infin

sum

119899=0

[119891(0 119886119899radic119875119903 1 0 119905) minus 119891(0 119887

119899radic119875119903 1 0 119905)]

(11)where 119886

119899= 2119899 + 119910 and 119887

119899= 2119899 + 2 minus 119910

The functions 119891 and erfc are given in Appendix AThe constants 119860

1 1198602 1198603 1198604 1198605 and 119860

6are given in

Appendix B

The rate of mass transfer (Sh) the rate of heat transfer(Nu) and the skin function (120591) at the walls of the channelare obtained as follows

Sherwood Number (119878ℎ) Consider

Sh0=

120597119862

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

= minusradic119878119888

infin

sum

119899=0

[119892 (119870119888 2119899radic119878

119888 1 0 119905)

+119892 (119870119888 (2119899 + 2)radic119878

119888 1 0 119905)]

Sh1=

120597119862

120597119910

10038161003816100381610038161003816100381610038161003816119910=1

= 2radic119878119888

infin

sum

119899=0

[119892 (119870119888 (2119899 + 1)radic119878

119888 1 0 119905)]

(12)Nusselt Number (119873119906) Consider

Nu0= minus

120597119879

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

= 1198601radic119878119888

infin

sum

119899=0

[119892 (119870119888 2119899radic119878

119888 119875119903minus 119878119888 119878119888119870119888 119905)

+119892 (119870119888 (2119899 + 2)radic119878

119888 119875119903minus 119878119888 119878119888119870119888 119905)]

minus 1198601radic119875119903

infin

sum

119899=0

[119892 (0 2119899radic119875119903 119875119903minus 119878119888 119878119888119870119888 119905)

+119892 (0 (2119899 + 2)radic119875119903 119875119903minus 119878119888 119878119888119870119888 119905)]

minus (119863lowast+ 1)radic119875

119903

infin

sum

119899=0

[119892 (0 2119899radic119875119903 1 0 119905)

+119892 (0 (2119899 + 2)radic119875119903 1 0 119905)]

minus 119863lowastSh0

Nu1=

120597119879

120597119910

10038161003816100381610038161003816100381610038161003816119910=1

= 21198601radic119875119903

infin

sum

119899=0

[119892 (0 (2119899 + 1)radic119875119903 119875119903minus 119878119888 119878119888119870119888 119905)]

minus 21198601radic119878119888

infin

sum

119899=0

[119892 (119870119888 (2119899 + 1)radic119878

119888 119875119903minus 119878119888 119878119888119870119888 119905)]

+ 2 (119863lowast+ 1)radic119875

119903

infin

sum

119899=0

[119892 (0 (2119899 + 1)radic119875119903 1 0 119905)]


(13)

Skin Friction (120591) Consider

1205910= minus

120597119906

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

= 1198602

infin

sum

119899=0

[119892(119872 +1

119870119901

2119899 1 minus 119878119888 119878119888119870119888minus 119872 minus

1

119870119901

119905)

+ 119892(119872 +1

119870119901

2119899 + 2 1 minus 119878119888 119878119888119870119888

minus119872 minus1

119870119901

119905)]

minus 1198602radic119878119888

infin

sum

119899=0

[119892(119870119888 2119899radic119878

119888 1 minus 119878

119888 119878119888119870119888minus 119872 minus

1

119870119901

119905)

+ 119892(119870119888 (2119899 + 2)radic119878

119888 1 minus 119878

119888 119878119888119870119888

minus119872 minus1

119870119901

119905)]

+ 1198603

infin

sum

119899=0

[119892(119872 +1

119870119901

2119899 1 minus 119875119903 minus119872 minus

1

119870119901

119905)

+ 119892(119872 +1

119870119901

2119899 + 2 1 minus 119875119903

minus119872 minus1

119870119901

119905)]

minus 1198603radic119875119903

infin

sum

119899=0

[119892(119870119888 2119899radic119875

119903 1 minus 119875

119903minus 119872 minus

1

119870119901

119905)

+ 119892(119870119888 (2119899 + 2)radic119875

119903 1 minus 119875

119903

minus119872 minus1

119870119901

119905)]

Journal of Fluids 5

+ 21198604

infin

sum

119899=0

[119892(119872 +1

119870119901

2119899 119875119903minus 119878119888 119878119888119870119888 119905)

+ 119892(119872 +1

119870119901

2119899 + 2 119875119903minus 119878119888 119878119888119870119888 119905)]

minus 1198604radic119878119862

infin

sum

119899=0

[119892 (119870119888 2119899radic119878

119888 119875119903minus 119878119888 119878119888119870119888 119905)

+ 119892 (119870119888 (2119899 + 2)radic119878

119888 119875119903minus 119878119888 119878119888119870119888 119905)]

minus 1198604radic119875119903

infin

sum

119899=0

[119892 (0 2119899radic119875119903 119875119903minus 119878119888 119878119888119870119888 119905)

+119892 (0 (2119899 + 2)radic119875119903 119875119903minus 119878119888 119878119888119870119888 119905)]

+ (1198605minus 1198606)

infin

sum

119899=0

[119892(119872 +1

119870119901

2119899 1 0 119905)

+ 119892(119872 +1

119870119901

2119899 + 2 1 0 119905)]

+ 1198605Sh0+ 1198606radic119875119903

infin

sum

119899=0

[119892 (0 2119899radic119875119903 1 0 119905)

+ 119892 (0 (2119899 + 2)radic119875119903 1 0 119905)]

1205911= minus

120597119906

120597119910

10038161003816100381610038161003816100381610038161003816119910=1

= minus21198602

infin

sum

119899=0

[119892(119872 +1

119870119901

2119899 + 1 1 minus 119878119888 119878119888119870119888

minus119872 minus1

119870119901

119905)]

+ 21198602radic119878119888

infin

sum

119899=0

[119892 (119870119888 (2119899 + 1)radic119878

119888 1 minus 119878

119888 119878119888119870119888

minus119872 119905)]

minus 21198603

infin

sum

119899=0

[119892(119872 +1

119870119901

2119899 + 1 1 minus 119875119903 minus119872 minus

1

119870119901

119905)]

+ 21198603radic119875119903

infin

sum

119899=0

[119892(119870119888 (2119899 + 1)radic119875

119903 1 minus 119875

119903 minus119872

minus1

119870119901

119905)]

minus 41198604

infin

sum

119899=0

[119892(119872 +1

119870119901

2119899 + 1 119875119903minus 119878119888 119878119888119870119888 119905)]

+ 21198604radic119878119888

infin

sum

119899=0

[119892 (119870119888 (2119899 + 1)radic119878

119888 119875119903minus 119878119888 119878119888119870119888 119905)]

+ 21198604radic119875119903

infin

sum

119899=0

[119892 (0 (2119899 + 1)radic119875119903 119875119903minus 119878119888 119878119888119870119888 119905)]

minus 2 (1198605minus 1198606)

infin

sum

119899=0

[119892(119872 +1

119870119901

2119899 + 1 1 0 119905)]

minus 21198606radic119875119903

infin

sum

119899=0

[119892 (0 (2119899 + 1)radic119875119903 1 0 119905)] + 119860

5Sh1

(14)where the function 119892 is given in Appendix A

The mass flux (volumetric flow rate) for the problem isgiven by

119876 = int

1

0

119906 119889119910 (15)

This is computed by using the trapezoidal rule for numericalintegration

Case 2 (119875119903= 119878119888= 1) The solutions of (5) subject to boundary

conditions (6) are given by

119862 =

infin

sum

119899=0

[119891 (0 119886119899 1 0 119905) minus 119891 (0 119887

119899 1 0 119905)]

119879 =

119863lowast(1 minus 119890

minus119870119888119905)

2119870119888119905radic120587119905

infin

sum

119899=0

[119886119899exp(minus

1198862

119899

4119905) minus 119887119899exp(minus

1198872

119899

4119905)]

+ (119863lowast+ 1)

infin

sum

119899=0

[119891 (0 119886119899 1 0 119905) minus 119891 (0 119887

119899 1 0 119905)] minus 119863

lowast119862

119906 = 1198608119862 +

119863lowast

2119905radic120587119905[

exp (minus119870119888119905)

119870119888(119870119888minus 119872 minus (1119870

119901))

+1

(119872 + (1119870119901))119870119888

minusexp (minus119872119905)

(119872 + (1119870119901)) (119870119888minus 119872 minus (1119870

119901))

]

times

infin

sum

119899=0

[119886119899exp(minus

1198862

119899

4119905) minus 119887119899exp(minus

1198872

119899

4119905)]

+ 1198606

infin

sum

119899=0

[119891 (0 119886119899 1 0 119905) minus 119891 (0 119887

119899 1 0 119905)]

(16)where 119860

7and 119860

8are given in Appendix B

In this case the rate of mass transfer (Sh) the rate of heattransfer (Nu) and the skin frictions (120591) on the walls of thechannel are obtained as follows

Sherwood Number Consider

Sh0= minus

120597119862

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

= minus

infin

sum

119899=0

[119892 (119870119888 2119899 1 0 119905) minus 119892 (119870

119888 2119899 + 2 1 0 119905)]

Sh1= minus

120597119862

120597119910

10038161003816100381610038161003816100381610038161003816119910=1

= 2

infin

sum

119899=0

[119892 (119870119888 2119899 + 1 1 0 119905)]

(17)

6 Journal of Fluids

Nusselt Number Consider

Nu0= minus

120597119879

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

= minus119863lowast[1 minus exp (minus119870

119888119905)]

2119870119888119905radic120587119905

times [

infin

sum

119899=0

exp(minus1198992

119905) + exp(minus

(119899 + 1)2

119905)

minus2

119905

infin

sum

119899=0

1198992 exp(minus

1198992

119905)

+(119899 + 1)2 exp(minus

(119899 + 1)2

119905)]

minus (119863lowast+ 1)

infin

sum

119899=0

[119892 (0 2119899 1 0 119905) + 119892 (0 2119899 + 2 1 0 119905)]

+ 119863lowastSh0

Nu1= minus

120597119879

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

=119863lowast[1 minus exp (minus119870

119888119905)]

2119870119888119905radic120587119905

times [2

infin

sum

119899=0

exp(minus(2119899 + 1)

2

4119905)

minus1

119905

infin

sum

119899=0

(2119899 + 1)2 exp(minus

(2119899 + 1)2

4119905)]

+ 2 (119863lowast+ 1)

infin

sum

119899=0

[119892 (0 2119899 + 1 1 0 119905)] minus 119863lowastSh1

(18)

Skin Friction Consider

1205910= minus

120597119906

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

= 1198608Sh0+

119863lowast

2119905radic120587119905

times [exp (minus119870

119888119905)

119870119888(119870119888minus 119872 minus (1119870

119901))

+1

119870119888(119872 + (1119870

119901))


(119872 + (1119870119901)) (119870119888minus 119872 minus (1119870

119901))

]

times [

infin

sum

119899=0

exp(minus1198992

119905) + exp(minus

(119899 + 1)2

119905)

minus2

119905

infin

sum

119899=0

1198992 exp(minus

1198992

119905)

+(119899 + 1)2 exp(minus

(119899 + 1)2

119905)]

+ 1198606

infin

sum

119899=0

[119892 (0 2119899 1 0 119905) + 119892 (0 2119899 + 2 1 0 119905)]

minus 1198607

infin

sum

119899=0

[119892(119872 +1

119870119901

2119899 1 0 119905)

+119892(119872 +1

119870119901

2119899 + 2 1 0 119905)]

(19)1205911= minus

120597119906

120597119910

10038161003816100381610038161003816100381610038161003816119910=1

= 1198608Sh1+

119863lowast

119905radic120587119905

times [

exp (minus119872 minus (1119870119901)) 119905

(119872 + (1119870119901)) (119870119888minus 119872 minus (1119870

119901))

minus1

119870119888(119872 + (1119870

119901))

minusexp (minus119870

119888119905)

119870119888(119870119888minus 119872 minus (1119870

119901))

]

times

infin

sum

119899=0

[exp(minus(2119899 + 1)

2

119905)

minus2(2119899 + 1)

2

119905exp(minus

(2119899 + 1)2

119905)]

minus 21198606

infin

sum

119899=0

[119892 (0 2119899 + 1 1 0 119905)]

times 21198607

infin

sum

119899=0

[119892(119872 +1

119870119901

2119899 + 1 1 0 119905)]

(20)

Steady State Setting 120597119906120597t = 0 120597119879120597t = 0 and 120597119862120597t = 0 in(5) the steady state of the problem is obtained as

1198892119906

1198891199102+ 119873119862 + 119879 minus (119872 +

1

119870119901

)119906 = 0

1198892119862

1198891199102minus 119878119888119870119888119862 = 0

1198892119879

1198891199102+ 119863lowast 1198892119862

1198891199102= 0

(21)

The solutions of (21) subject to boundary conditions (6)are given by

119862 =sinh (radic119878

119888119870119888(1 minus 119910))

sinh (radic119878119888119870119888)

119879 = (119863lowast+ 1) (1 minus 119910) minus 119863

lowast119862

Journal of Fluids 7

Curve

I 04 05 022 2 2 18 2 100

II 02 05 022 2 2 18 2 100

III 04 2 022 2 2 18 2 100

IV 04 05 03 2 2 18 2 100

V 04 05 022 1 2 18 2 100

VI 04 05 022 2 4 18 2 100

VII 04 05 022 2 2 2 2 100

VIII 04 05 022 2 2 18 4 100

IX 04 05 022 2 2 18 2 05

X Steadystate 05 022 2 4 2 100

04

02

0

minus02

minus04

minus06

minus08

minus1

u

0 02 04 06 08 1

II

IVI

V III

IXVII

VIII

IV

X

y

t M Sc Kc N Pr KpDlowast

mdash

Figure 1 Velocity profile

119906 = 1198609

sinhradic(119872 + (1119870119901)) (1 minus 119910)

sinhradic119872 + (1119870119901)

+ 1198605119862 + 119860

6(1 minus 119910)

(22)

where 1198609is given in Appendix B

In this case the Sherwood number (Sh) the Nusseltnumber (Nu) and the skin friction (120591) are given by thefollowing


Sh0= radic119878119888119870119888cothradic119878

119888119870119888

Sh1= radic119878119888119870119888cosechradic119878

119888119870119888

(23)

Nusselt Number (119873119906) Consider

Nu0= 119863lowast+ 1 + 119863

lowastSh0

Nu1= minus (119863

lowast+ 1) minus 119863

lowastSh1

(24)


1205910= minus1198609radic119872 +

1

119870119901

cothradic119872 +1

119870119901

+ 1198605Sh0minus 1198606

1205911=

1198609radic119872 + (1119870

119901)

sinhradic119872 + (1119870119901)

minus 1198605Sh1+ 1198606

(25)

Mass Flux Consider

119876 = 1198609radic119872 +

1

119870119901

coshradic119872 + (1119870119901) minus 1

sinhradic119872 + (1119870119901)

+1198605

radic119878119888119870119888

coshradic119878119888119870119888minus 1

sinhradic119878119888119870119888

+1198606

2

(26)

4 Results and Discussion

Computations have been carried out by assigning the valuesto the pertinent parameters characterising the fluids ofpractical interest The flow phenomenon is characterized bymagnetic parameter (119872) chemical reaction parameter (119870

119888)

sustentation parameter (119873) Dufour number (119863lowast) Schmidtnumber (119878

119888) Prandtl number (119875

119903) and porosity parameter

(119870119901) The effects of various parameters on velocity tempera-

ture and concentration profiles are shown graphically and intabulated form for Sherwood number (Sh) Nusselt number(Nu) and skin friction (120591)

The particular case without magnetic field and porosity(119872 = 0 119870

119901rarr infin) can be obtained from (8) which is in

good agreement with the work reported earlier [1]Another interesting point to note is that diffusion in

aqueous solution that is for higher values of 119878119888gives rise to

oscillations in the velocity as well as temperature field in thepresence of chemical reaction parameter (119870

119888)

Figure 1 depicts the velocity profile for various values ofpertinent parameters characterising flow fieldsThe common

8 Journal of Fluids

1000

500

0

minus500

minus1000

minus1500

minus2000

u

0 02 04 06 08 1 12y

Sc = 5 Kc = 0 Kp = 100

Sc = 150 Kc = 0 Kp = 100

Sc = 200 Kc = 0 Kp = 100

Sc = 200 Kc = 0 Kp = 05

Figure 2 Velocity profile for 119905 = 04 119872 = 05 119873 = 2 119875119903= 7 and 119863

lowast= 2

0 02 04 06 08 1 12y

035

03

025

02

015

01

005

0

u

VIII

VI

I II III VII

V

IV

Curve

I 05 100 1 2 2 100

II 05 022 2 2 2 100

III 05 152 1 2 2 100

IV 05 022 2 2 2 05

V 2 022 2 2 2 100

VI 05 022 2 2 30 100

VII 05 03 1 2 2 100VIII 05 022 2 4 2 100

M Sc Kc N KpDlowast

Figure 3 Velocity profile (steady state)

feature of the profiles is parabolicThis is evident from curvesII and X (119905 = 02 Curve II the steady state curve X) in caseof steady state as well as 119905 le 03 the velocity profiles remainpositive throughout the flow fieldThis is evident from curvesII and X (119905 = 02 Curve II the steady state curve X)Thus it isconcluded that the time span plays a vital role for engenderingback flow (Curve I and Curve II)

Further magnetic parameter (119872) chemical reactionparameter (119870

119888) sustentation parameter (119873) and porosity

parameter (119870119901) decrease the velocity |119906| at all points In

all other cases such as 119878119888 119875119903 and 119863

lowast the reverse effectis observed If the mass diffusivity becomes greater thanthe thermal diffusivity (ie 119873 the sustentation parametergt10) then the velocity decreases Moreover with the stronger

magnetic field Lorentz force also reduces the velocity fieldwhich is in conformity with the result of Cramer and Pai [20]

An increase in 119875119903leads to decrease in the velocity which

suggests that low rate of thermal diffusion leads to increase inthe velocity boundary layer thickness

Figure 2 depicts the velocity distribution in the absence ofchemical reaction It is noteworthy to observe that the backflow occurs for higher values of 119878

119888 that is 119878

119888= 150 and

119878119888

= 200 and the permeability of the medium representingthe diffusion in aqueous solution in the presence of constantmagnetic field and Dufour effect Further it is to mentionthat for 119878

119888lt 10 and 119870

119888= 00 no velocity profile could be

graphedpresented Further it is to note that heavier speciesthat is with increasing 119878

119888lead to flow reversal It is most

Journal of Fluids 9

CurveI 04 022 20 700II 04 022 00 700III

III

04 00 700IV 04 20 700V 02 20 700VI 04 022 00 071VII 04 022 02 071VII 04 00 071IX 04 02 071X 02 02 071

t Sc Kc Pr

6

5

4

3

2

1

00 02 04 06 08 1

y

T

V I II

IV

VI-X

WaterAir

030030030

030030030

Figure 4 Temperature profile for 119863lowast= 20 119875

119903= 70 (water) and 119875

119903= 071 (air)

0

02

04

06

08

1

12

0 02 04 06 08 1

T

y

IIII

II

Curve t Kc

I 04 20II 04 10III 02 20

Figure 5 Temperature profile (119875119903= 1 119878

119888= 1)

interesting to note that in the presence of destructive reactionthat is 119870

119888gt 0 higher value of 119878

119888leads to oscillatory flow

and in case of diffusion in the aqueous solution it leads toback flow Thus this suggests that the back flow is preventedby diffusing lighter species which is in conformity with theresults reported earlier by Pop [21] Hossain andMohammad[22] and Rath et al [23]

It is also remarked that when 119878119888varies from0 to 5 the back

flow is prevented in the absence of chemical reaction But inthe presence of heavier species the sharp fall of velocity is wellmarked

Figure 3 exhibits the steady state velocity profiles indicat-ing no back flow irrespective of higher or lower value of masstransfer coefficient and Dufour effect

On careful observation it is revealed that for highervalues of Dufour number119863lowast (curve VI) sustentation param-eter 119873 (Curve VIII) magnetic parameter 119872 (Curve V) andporosity parameter 119870

119901(Curve IV) steady state velocity is

effected significantly whereas variation in reaction parameter

and Schmidt number produces no change Thus it is impor-tant to note that mass transfer phenomena with chemicalreaction does not affect the steady flow significantly butincrease in119872 119873119863lowast and119870

119901 decrease the velocity increase

the velocity increase the velocity and decrease the velocityrespectively It may be inferred that magnetic parameterporosity parameter andDufour effect produce the same effectirrespective of steady or unsteady state except for sustentationparameter

Figures 4 5 and 6 present the temperature distribution Itis seen that for an increase in destructive reaction parameter(119870119888gt 0) temperature increases whereas increase in 119875

119903 119905 and

119878119888reduces the temperature at all points for 119875

119903= 70 (water)

when 119910 lt 07 It is noted that the slow rate of diffusion leavesthe heat energy spread in the fluidmass and it is enhanced forheavier species

It is observed in case of 119875119903= 071 (air) that the tempera-

ture falls sharply at all points with almost linear distributionregardless of other parameters Thus it may be pointed out

10 Journal of Fluids

002040608

1121416

0 02 04 06 08 1

T

y

IV

II

I

III

Curve Sc Kc Dlowast

I 022 2 2II 030 2 2III 022 1 2IV 022 2 30

Figure 6 Temperature profile (steady state)

CurveI 04 100 0II 04 617 0III 04 100 2IV 04 617 2V 02 100 2

t Kc

minus02

0

02

04

06

08

1

0 02 04 06 08 1

C

I

III

II IVV

y

Sc

Figure 7 Concentration profile

0010203040506070809

1

0 02 04 06 08 1

C

y

III

III

Curve t Kc

I 04 20II 04 10III 02 20

Figure 8 Concentration profile (119878119888= 1)

that thermal diffusivity property of the fluid plays a vitalrole in controlling the temperature distribution and hencecontributing to thermal boundary layer thickness

Figure 5 also exhibits the temperature distribution when119875119903

= 1 and 119878119888

= 1 which means that thermal diffusivityand mass diffusivity are at par This contributes to the uni-form variationdecrease in temperature Increase in reactionparameter gives rise to increase in temperature whereasallowing for a large span of time reduces it

minus02

0

02

04

06

08

1

0 02 04 06 08 1

C

y

Curve Sc Kc

I 100 2II 617 1III 617 2

Figure 9 Concentration profile (steady state)

Figure 6 presents the nonlinearity distribution of tem-perature due to high value of 119863

lowast that is for large Dufoureffect Thus sharp rise and fall of temperature is due to lowdiffusivity and high Dufour effect

Figures 7 8 and 9 exhibit the concentration variation inthe flow domain Sharp fall of concentration is indicated inFigures 7 and 9 along with mass absorption near the plate forvery high value of 119878

119888(ie 119878

119888= 617) This means that heavier

species give rise to sharp fall of concentration accelerating theprocess ofmass diffusionwhereas in case of lighter species thevariation is smooth

Skin friction plays an important role in flow characteris-tics It measures the frictional forces encountered at the solidsurfaces due to themotion of the fluid One striking feature ofthe entries of the skin friction in Table 1 in case of unsteadymotion is that the skin frictions bear same sign in case oflower and upper plate Further on careful observation it isrevealed that at the lower plate all are negative for 119905 = 04

except for the case when 119905 = 02 Thus it may be concludedthat time span plays an important role tomodify the frictionaldrag due to shear stress at the plates

Moreover from (18)-(19) it is clear that the Prandtl num-ber (119875

119903) has no role to play to affect the velocity temperature

and concentration fields in the steady flow and hence the skin

Journal of Fluids 11

Table 1 Skin friction (unsteady case)

Sl no 119905 119872 119878119888

119870119888

119873 119875119903

119863lowast

119870119901

1205910

1205911

1 04 05 022 2 2 18 2 100 minus137638 minus031532 02 05 022 2 2 18 2 100 0649408 04663083 04 2 022 2 2 18 2 100 minus040403 minus0122794 04 05 03 2 2 18 2 100 minus290421 minus098275 04 05 022 1 2 18 2 100 minus046631 minus0158386 04 05 022 2 4 18 2 100 minus091633 minus0286757 04 05 022 2 2 2 2 100 minus166428 minus033468 04 05 022 2 2 18 4 100 minus224604 minus0344339 04 05 022 2 2 18 2 05 minus125513 minus02425

Table 2 Skin friction (steady case)

Sl no 119872 119878119888

119870119888

119873 119863lowast

119870119901

1205910

1205911

1 05 022 2 2 2 100 1732053 09325152 2 022 2 2 2 100 166425 07954363 05 03 2 2 2 100 1732052 09325154 05 022 1 2 2 100 173205 09325155 05 022 2 4 2 100 1664279 08692876 05 022 2 2 4 100 2144507 13763827 05 022 2 2 2 05 142312 0524363

Table 3 Nusselt number

Sl no 119905 119878119888

119870119888

119875119903

119863lowast Nu

0Nu1

1 04 022 2 700 2 minus137638192 minus11503684072 02 022 2 700 2 minus2674621494 minus12130970043 04 030 2 700 2 minus1327044822 minus11106125154 04 022 0 700 2 minus1455009029 minus11917535935 04 022 2 071 2 0113935608 minus01139356086 04 022 2 700 4 minus1569685558 minus1279941632

friction In case of unsteady flow as 119875119903increases the shearing

stress increases at both the platesConsidering Tables 1 and 2 for both steady and unsteady

cases it is concluded that an increase in magnetic param-eter porosity parameter and sustentation parameter leadsto reduce the magnitude of frictional drag at both platesMoreover an increase in 119878

119888 119870119888 and 119863

lowast leads to enhance thefrictional drag at the plates for steady and unsteady flow

Physically we may interpret the above results as followsLorentz force and sustentation parameter that is ratios

of buoyancy effects due to temperature and concentrationdifferences reduce the frictional drag which agrees well withthe established result reported earlier Moreover presence ofheavier species and exothermic reaction and Dufour effectenhance the magnitude of the shearing stress at both plates

From Table 3 it is seen that an increase in 119905 119878119888 and 119870

119888

leads to decrease the Nusselt number at both plates Oneinteresting point is that rate of heat transfer remains negativefor all the parameters at both the plates for aqueous solution119875119903= 70 but in case of air that is 119875

119903= 071 Nusselt number

is positive at the lower plate and negative at the upper plate

Table 4 Sherwood number

Sl no 119905 119878119888

119870119888

Sh0

Sh1

1 04 100 2 59757644 minus06128007882 02 100 2 6313751515 0158384443 04 617 2 7026366229 04705642814 02 100 0 567128182 minus0466307658

Table 4 presents the variation of mass transfer for variousvalues of 119905 119878

119888 and 119870

119888 It is seen that an increase in chemical

reaction parameter119870119888increases themass transfer at the lower

plate and decreases it at the other Further it is to note thatSherwood number at both the plates increases as 119878

119888increases

that is for heavier speciesmass transfer increases at the platesThe effect of increase in time span is to reduce the rate ofmasstransfer at both plates

Tables 5 and 6 show the effects of various parameters onmass flux both for unsteady and steady cases respectivelyFrom Table 5 it is observed that mass flux increases with the


Table 5 Mass flux (unsteady case)

Sl no 119905 119872 119878119888

119870c 119873 119875119903

119863lowast

119870119901

119876

1 04 05 022 2 2 18 2 100 01978362 02 05 022 2 2 18 2 100 01765723 04 2 022 2 2 18 2 100 00576554 04 05 03 2 2 18 2 100 05684755 04 05 022 1 2 18 2 100 00911936 04 05 022 2 4 18 2 100 01666677 04 05 022 2 2 2 2 100 02649398 04 05 022 2 2 18 4 100 03645839 04 05 022 2 2 18 2 05 0034241

Table 6 Mass flux (steady case)

Sl no 119872 119878119888

119870119888

119873 119863lowast

119870119901

119876

1 05 022 2 2 2 100 01190522 2 022 2 2 2 100 01042073 05 03 2 2 2 100 01190524 05 022 1 2 2 100 01190525 05 022 2 4 2 100 01950776 05 022 2 2 4 100 01223927 05 022 2 2 2 05 0094632

increase in the values in 119878119888 119870119888 119875119903 and 119863

lowast but the reverseeffect is observed in case of 119905119872119873 and119870

119901when the motion

is unsteady Hence it is concluded that in case of unsteadymotion more flux fluid was experienced with heavier specieshaving low thermal diffusivity and increasing Dufour effectin the presence of constructive chemical reaction

Now if we analyse the steady casewe observe that moreflux is measured with higher value of 119873 and 119863

lowast whereas thereverse effect is observed in case of 119872 and 119870

119901 In case of 119878

119888

and 119870119888there is no significant effect on mass flux

Comparing both cases steady and unsteady motion it isobserved that an increase in 119870

119888and 119863

lowast gives rise to higherflux whereas magnetic field yields less amount of flux

5 Conclusion

(i) Under the influence of dominating mass diffusivityover thermal diffusivity with stronger Lorentz forcethe velocity is reduced at all points of the channel

(ii) Low rate of thermal diffusion leads to increase thethickness of boundary layer

(iii) Diffusion of heavier species leads to flow reversal Itis well marked in case of aqueous solution Flow ofaqueous solution in the presence of heavier species isprone to back flow

(iv) Destructive reaction in the presence of heavier speciesleads to oscillatory flow

(v) In case of steady state no back flow occurs irrespec-tive of high or low value of mass transfer coefficientand Dufour effects

(vi) Sharp rise and fall of temperature is the outcome oflow diffusivity and high Dufour effects

(vii) Dufour effect and chemical reaction rate in the pres-ence of heavier species enhance the frictional drag

(viii) Heat transfer bears same sign at both the plates in thepresence of aqueous solution but in case of air it is ofopposite sign

(ix) Mass transfer increases at both plates due to the pres-ence of heavier species but it reduces with increasingtime span

(x) In case of unsteady motion more flux of fluid isexperienced due to heavier species with low thermaldiffusivity Dufour effect enhances the flux both insteady and unsteady motion

Appendices

A

The complementary error function is given by

erfc (x) =2

radic120587int

infin

119909

119890minus1205822

119889120582

119891 (1199091 1199092 1199093 1199094 1199095)

=1

2exp(

11990941199095

1199093

)

times [exp(minus1199092radic

1199094

1199093

+ 1199091)


times erfc( 1199092

2radic1199095

minus radic(1199094

1199093

+ 1199091)1199095)

+ exp(1199092radic

1199094

1199093

+ 1199091)

times erfc( 1199092

2radic1199095

+ radic(1199094

1199093

+ 1199091)1199095)]

119892 (1199091 1199092 1199093 1199094 1199095)

=1

2radic

1199094

1199093

+ 1199091exp(

11990941199095

1199093

)

times [exp(1199092radic

1199094

1199093

+ 1199091)

times erfc( 1199092

2radic1199095

+ radic(1199094

1199093

+ 1199091)1199095)

minus exp(minus1199092radic

1199094

1199093

+ 1199091)

times erfc( 1199092

2radic1199095

minus radic(1199094

1199093

+ 1199091)1199095)]

minus2

radic120587exp(minus(

1199092

3

41199095

+ 11990911199095))

(A1)

B

1198601= minus119863lowast[

119878119888

119875119903minus 119878119888

+ 119875119903minus 119878119888]

1198602= minus

1 minus 119878119888

119878119888119870119888minus 119872 minus (1119870

119901)

times [119873 +

119878119888119863lowast(119870119888minus 119872 minus (1119870

119901))

119878119888119870119888(1 minus 119875

119903) + (119872 + (1119870

119901)) (119875119903minus 119878119888)

]

1198603=

1 minus 119875119903

119872 + (1119870119901)

times [1 +


119901))

119878119888119870119888(1 minus 119875

119903) + (119872 + (1119870

119901)) (119875119903minus 119878119888)

]

1198604=

119863lowast119875119903(119875119903minus 119878119888)

119878119888119870119888(1 minus 119875

119903) + (119872 + (1119870

119901)) (119875119903minus 119878119888)

1198605=

119863lowastminus 119873

119878119888119870119888minus 119872 minus (1119870

119901)

1198606=

119863lowast+ 1

119872 + (1119870119901)

1198607=

119870119888(119863lowast+ 1) minus (119872 + (1119870

119901)) (119873 + 1)

(119872 + (1119870119901)) (119870119888minus 119872 minus (1119870

119901))

1198608=


119870119888minus 119872 minus (1119870

119901)

1198609=

minus119878119888119870119888(119863lowast+ 1) + (119872 + (1119870

119901)) (119873 + 1)

(119872 + (1119870119901)) (119878119888119870119888minus 119872 minus (1119870

119901))

References

[1] B K Jha and A O Ajibade ldquoFree convection heat and masstransfer flow in a vertical channel with the Dufour effectrdquo Jour-nal of Process Mechanical Engineering vol 224 no 2 pp 91ndash1012010

[2] VM Soundalgekar and S P Akolkar ldquoEffects of free convectioncurrents and mass transfer on flow past a vertical oscillatingplaterdquoAstrophysics and Space Science vol 89 no 2 pp 241ndash2541983

[3] N G Kafoussias and E W Williams ldquoThermal-diffusion anddiffusion-thermo effects on mixed free-forced convective andmass transfer boundary layer flow with temperature dependentviscosityrdquo International Journal of Engineering Science vol 33no 9 pp 1369ndash1384 1995

[4] B K Jha and A K Singh ldquoSoret effects on free-convection andmass transfer flow in the stokes problem for a infinite verticalplaterdquo Astrophysics and Space Science vol 173 no 2 pp 251ndash255 1990

[5] N G Kafoussias ldquoMHD thermal-diffusion effects on free-con-vective and mass-transfer flow over an infinite vertical movingplaterdquo Astrophysics and Space Science vol 192 no 1 pp 11ndash191992

[6] M S Alam M M Rahman and M A Samad ldquoDufour andSoret effects on unsteady MHD free convection and masstransfer flow past a vertical porous plate in a porous mediumrdquoNonlinear Analysis Modelling and Control vol 11 no 3 pp 217ndash226 2006

[7] M S AlamMM RahmanM Ferdous KMaino EMureithiand A Postelnicu ldquoDiffusion-thermo and thermal-diffusioneffects on free convective heat andmass transfer flow in a porousmedium with time dependent temperature and concentrationrdquoInternational Journal of Applied Engineering Research vol 2 no1 pp 81ndash96 2007

[8] O A Beg T A Beg A Y Bakier andV R Prasad ldquoChemically-reactingmixed convective heat andmass transfer along inclinedand vertical plates with soret and Dufour effects numericalsolutionrdquo International Journal of Applied Mathematics andMechanics vol 5 no 2 pp 39ndash57 2009

[9] Z Dursunkaya and W M Worek ldquoDiffusion-thermo andthermal-diffusion effects in transient and steady natural con-vection from a vertical surfacerdquo International Journal of Heatand Mass Transfer vol 35 no 8 pp 2060ndash2065 1992

[10] C R A Abreu M F Alfradique and A S Telles ldquoBoundarylayer flows with Dufour and Soret effects I forced and naturalconvectionrdquo Chemical Engineering Science vol 61 no 13 pp4282ndash4289 2006


[11] E Osalusi J Side and R Harris ldquoThermal-diffusion and diffu-sion-thermo effects on combined heat and mass transfer of asteadyMHD convective and slip flow due to a rotating disk withviscous dissipation and Ohmic heatingrdquo International Commu-nications in Heat and Mass Transfer vol 35 no 8 pp 908ndash9152008

[12] M Anghel H S Takhar and I Pop ldquoDufour and Soret effectson free convection boundary layer over a vertical surfaceembedded in a porous mediumrdquo Studia Universitatis Babes-Bolyai Matematica vol 11 no 4 pp 11ndash21 2000

[13] A Postelnicu ldquoInfluence of a magnetic field on heat and masstransfer by natural convection from vertical surfaces in porousmedia considering Soret and Dufour effectsrdquo InternationalJournal of Heat andMass Transfer vol 47 no 6-7 pp 1467ndash14722004

[14] B Gebhart and L Pera ldquoThe nature of vertical natural con-vection flows resulting from the combined buoyancy effects ofthermal and mass diffusionrdquo International Journal of Heat andMass Transfer vol 14 no 12 pp 2025ndash2050 1971

[15] U N Das R Deka and V M Soundalgekar ldquoEffects of masstransfer on flowpast an impulsively started infinite vertical platewith constant heat flux and chemical reactionrdquo Forschung imIngenieurwesen vol 60 no 10 pp 284ndash287 1994

[16] S S Saxena and G K Dubey ldquoUnsteady MHD heat and masstransfer free convection flow of a polar fluid past a verticalmoving porous plate in a porous medium with heat generationand thermal diffusionrdquo Advances in Applied Science Researchvol 2 no 4 pp 259ndash278 2011

[17] K Raveendra Babu A G Vijaya Kumar and S V K VarmaldquoDiffusion-thermo and radiation effects on MHD free convec-tive heat and mass transfer flow past an infinite vertical plate inthe presence of a chemical reaction of first orderrdquo Advances inApplied Science Research vol 3 no 4 pp 2446ndash2462 2012

[18] K Sudhakar R Srinibasa Raju and M Rangamma ldquoChemicalreaction effect on an unsteady MHD free convection flow pastan infinite vertical accelerated plate with constant heat fluxthermal diffusion and diffusion thermordquo International Journalof Modern Engineering Research vol 2 no 5 pp 3329ndash33392012

[19] L Debanath andD Bhatta Integral Transforms andTheir Appli-cations Applications of Laplace Transforms Chapman andHallCRC Taylor and Francies Group London UK 2007

[20] K R Cramer and S I PaiMagnetofluid Dynamics for Engineersand Applied Physics McGraw-Hill New York NY USA 1973

[21] I Pop ldquoEffect of Hall current on hydromagnetic flow near anaccelerated platerdquo International Journal of Physics and Mathe-matical Sciences vol 5 p 375 1971

[22] M A Hossain and K Mohammad ldquoEffect of hall current onhydromagnetic free convection flow near an accelerated porousplaterdquo Japanese Journal of Applied Physics vol 27 no 8 pp 1531ndash1535 1988

[23] P K Rath G C Dash and A K Patra ldquoEffect of Hallcurrent and chemical reaction on MHD flow along an expo-nentially accelerated porous flat plate with internal heat absorp-tiongenerationrdquo Proceedings of the National Academy of Sci-ences India A vol 80 no 4 pp 295ndash308 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



AstronomyAdvances in

International Journal of


Superconductivity


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AerodynamicsJournal of

Volume 2014


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Biophysics


ThermodynamicsJournal of

2 Journal of Fluids

studied the boundary layer flows with Dufour and Soreteffects Osalusi et al [11] have worked on mixed and freeconvective heat and mass transfer of an electrically conduct-ing fluid considering Dufour and Soret effects Anghel etal [12] and Postelnicu [13] obtained numerical solutions ofchemical reacting mixed convective heat and mass transferalong inclined and vertical plates with the Soret and theDufour effects and concluded that skin friction increaseswith a positive increase in the concentration-to-thermal-buoyancy ratio parameter

The analysis of natural convection heat and mass transfernear a moving vertical plate has received much attention inrecent times due to its wide application in engineering andtechnological processes There are applications of interest inwhich combined heat and mass transfer by natural convec-tion occurs between amovingmaterial and ambientmediumsuch as the design and operation of chemical processingequipment design of heat exchangers transpiration coolingof a surface chemical vapour deposition of solid layernuclear reactor and many manufacturing processes like hotrolling hot extrusion wire drawing continuous casting andfiber drawing Gebhart and Pera [14] studied the effects ofmass transfer on a steady free convection flow past a semi-infinite vertical plate by the similarity method and it wasassumed that the concentration level of the diffusing speciesin fluid medium was very low This assumption enabledthem to neglect the diffusion-thermo and thermo-diffusioneffects as well as the interfacial velocity at the wall dueto species diffusion Following this assumption Das et al[15] investigated the effects of simultaneous heat and masstransfer on free convection flow past an infinite vertical plateunder different physical situations

Recently the unsteady MHD heat and mass transferfree convection flow of a polar fluid past a vertical movingporous plate in a porous medium with heat generation andthermal diffusion has been studied by Saxena and Dubey[16] Raveendra Babu et al [17] studied diffusion-thermoand radiation effects on MHD free convective heat and masstransfer flow past an infinite vertical plate in the presenceof a chemical reaction of first order Sudhakar et al [18]discussed chemical reaction effect on an unsteady MHD freeconvection flow past an infinite vertical accelerated plate withconstant heat flux thermal diffusion and diffusion thermo

In the present paper the flow of an electrically conductingviscous incompressible fluid in a vertical porous channelformed by two vertical parallel porous plates in the presenceof a transverse magnetic field is studied Further the masstransfer phenomena considered in this problem is associatedwith chemically reacting species The objective of the presentstudy is to extend the work of Jha and Ajibade [1] byincorporating the permeability and the magnetic field effectas well as chemical reaction on the flow heat and masstransfer phenomena

2 Mathematical Formulation

Let us consider the free convective heat and mass transferMHD flow of a viscous incompressible fluid in a verticalporous channel formedby two infinite vertical parallel porous

plates in the presence of chemical reaction A magnetic fieldof uniform strength 119861

0is applied transversely to the plate

The induced magnetic field is neglected as the magneticReynolds number of the flow is taken to be very small Theconvection current is induced due to both the temperatureand concentration differences The flow is assumed to be inthe 1199091015840-direction which is taken to be vertically upward along

the channel walls and 1199101015840-axis is taken to be normal to the

plates that are ℎ distance apartUnder the usual Boussinesqrsquos approximations the gov-

erning equations for flow are given by

1205971199061015840

1205971199051015840= ]

12059721199061015840

12059711991010158402+ 119892120573 (119879

1015840minus 1198790)

+ 119892120573lowast(1198621015840minus 1198620) minus

1205901198612

01199061015840

120588minus

]

1198701015840119901

1199061015840

1205971198621015840

1205971199051015840= 119863

12059721198621015840

12059711991010158402minus 1198701015840

119888(1198621015840minus 1198620)

1205971198791015840

1205971199051015840= 120572

12059721198791015840

12059711991010158402+ 1198631

12059721198621015840

12059711991010158402

(1)

where ] 119892 120573 120573lowast 120590 1198610 1205881198631198701015840

119888 1205721198631 and119870

1015840

119901are kinematic

viscosity acceleration due to gravity coefficient of thermalexpansion coefficient of mass expansion electrical conduc-tivity of the fluid uniformmagnetic field density of the fluidchemical molecular diffusivity the dimensional chemicalreaction parameter in the diffusion equation thermal diffu-sivity the dimensional coefficient of the diffusion-thermoeffect and the permeability of the medium respectively

Initial and boundary conditions of the problem are

1199051015840le 0 119906

1015840(1199101015840 1199051015840) = 0 119879

1015840(1199101015840 1199051015840) = 1198790 1198621015840(1199101015840 1199051015840) = 119862

0

1199051015840gt 0

1199061015840(119900 1199051015840) = 0 119879

1015840(0 1199051015840) = 119879119908 1198621015840(0 1199051015840) = 119862

119908

1199061015840(ℎ 1199051015840) = 0 119879

1015840(ℎ 1199051015840) = 1198790 1198621015840(ℎ 1199051015840) = 119862

0

(2)

We now introduce the following dimensionless quantities

119910 =1199101015840

ℎ 119905 =

1199051015840]

ℎ2 119906 =

1199061015840]

119892120573ℎ2 (119879119908

minus 1198790)

119875119903=

]

120572 119878

119888=

]

119863

119879 =1198791015840minus 1198790

119879119908

minus 1198790

119862 =1198621015840minus 1198620

119862119908

minus 1198620

119873 =120573lowast(119862119908

minus 1198620)

120573 (119879119908

minus 1198790)

119863lowast

=1198631(119862119908

minus 1198620)

120572 (119879119908

minus 1198790)

119872 =1205901198612

0ℎ2

120588]

(3)

119870119888=

119870lowast

119888ℎ2

] 119870

119901=

1198701015840

119875

ℎ2 (4)

Journal of Fluids 3

Here119875119903 119878119888119873 119863lowast119872119870

119888 and119870




120597119906

120597119905=

1205972119906

1205971199102+ 119873119862 + 119879 minus (119872 +

1

119870119901

)119906

120597119862

120597119905=

1

119878119888

1205972119862

1205971199102minus 119870119888119862

120597119879

120597119905=

1

119901119903

1205972119879

1205971199102+

119863lowast

119901119903

1205972119862

1205971199102

(5)


119905 le 0 119906 (119910 119905) = 0 119879 (119910 119905) = 0 119862 (119910 119905) = 0

119905 le 0 119906 (0 119905) = 0 119879 (0 119905) = 1 119862 (0 119905) = 1

119906 (1 119905) = 0 119879 (1 119905) = 0 119862 (1 119905) = 0

(6)



1198892119862

1198891199102minus 119878119888(119904 + 119870

119888) 119862 = 0

1198892119879

1198891199102minus 119904119875119903119879 = 119878

119888119863lowast(119904 + 119870

119888) 119862

(7)

119904119906 =1198892119906

1198891199102+ 119873119862 + 119879 minus (119872 +

1

119870119901

)119906 (8)


119862 (0 119904) =1

119904 119862 (1 119904) = 0

119879 (0 119904) =1

119904 119879 (1 119904) = 0

119906 (0 119904) = 0 119906 (1 119904) = 0

(9)


119871minus1

(119890minus119887radic119904+119886

119904) =

1

2119890minus119887radic119886 erfc( 119887


+119890119887radic119886 erfc( 119887

2radic119905+ radic119886119905)

(10)


Case 1 (119875119903

= 1 119878119888


119862 =

infin

sum

119899=0

[119891(119870119888 119886119899radic119878119888 1 0 119905) minus 119891(119870

119888 119887119899radic119878119888 1 0 119905)]

119879 = 1198601

infin

sum

119899=0

[119891(0 119886119899radic119875119903 119875119903minus 119878119888 119878119888119870119888 119905)


minus 1198601

infin

sum

119899=0

[119891(119870119888 119886119899radic119878119888 119875119903minus 119878119888 119878119888119870119888 119905)


+ (119863lowast+ 1)

infin

sum

119899=0

[119891(0 119886119899radic119875119903 1 0 119905)


lowast119862

119906 = 1198602

infin

sum

119899=0

[119891(119872 +1

119870119901

119886119899 1 minus 119878

119888 119878119888119870119888minus 119872 minus

1

119870119901

119905)

minus119891(119872 +1

119870119901

119887119899 1 minus 119878

119888 119878119888119870119888minus 119872 minus

1

119870119901

119905)]

minus 1198602

infin

sum

119899=0

[119891(119870119888 119886119899radic119878119888 1 minus 119878

119888 119878119888119870119888minus 119872 minus

1

119870119901

119905)


119888 119878119888119870119888minus 119872 minus

1

119870119901

119905)]

+ 1198603

infin

sum

119899=0

[119891(119872 +1

119870119901

119886119899 1 minus 119875


1

119870119901

119905)

minus119891(119872 +1

119870119901

119887119899 1 minus 119875


1

119870119901

119905)]

minus 1198603

infin

sum

119899=0

[119891(119870119888 119886119899radic119875119903 1 minus 119875


1

119870119901

119905)



1

119870119901

119905)]

+ 21198604

infin

sum

119899=0

[119891(119872 +1

119870119901

119886119899 119875119903minus 119878119888 119878119888119870119888 119905)

minus119891(119872 +1

119870119901

119887119899 119875119903minus 119878119888 119878119888119870119888 119905)]

minus 1198604

infin

sum

119899=0

[119891(119870119888 119886119899radic119878119888 119875119903minus 119878119888 119878119888119870119888 119905)


4 Journal of Fluids

minus 1198604

infin

sum

119899=0

[119891(0 119886119899radic119875119903 119875119903minus 119878119888 119878119888119870119888 119905)


+ (1198605minus 1198606)

infin

sum

119899=0

[119891(119872 +1

119870119901

119886119899 1 0 119905)

minus119891(119872 +1

119870119901

119887119899 1 0 119905)]

+ 1198606

infin

sum

119899=0

[119891(0 119886119899radic119875119903 1 0 119905) minus 119891(0 119887

119899radic119875119903 1 0 119905)]

(11)where 119886

119899= 2119899 + 119910 and 119887

119899= 2119899 + 2 minus 119910


1 1198602 1198603 1198604 1198605 and 119860

6are given in

Appendix B



Sh0=

120597119862

120597119910

10038161003816100381610038161003816100381610038161003816119910=0


infin

sum

119899=0

[119892 (119870119888 2119899radic119878

119888 1 0 119905)

+119892 (119870119888 (2119899 + 2)radic119878

119888 1 0 119905)]

Sh1=

120597119862

120597119910

10038161003816100381610038161003816100381610038161003816119910=1

= 2radic119878119888

infin

sum

119899=0

[119892 (119870119888 (2119899 + 1)radic119878

119888 1 0 119905)]


Nu0= minus

120597119879

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

= 1198601radic119878119888

infin

sum

119899=0

[119892 (119870119888 2119899radic119878

119888 119875119903minus 119878119888 119878119888119870119888 119905)

+119892 (119870119888 (2119899 + 2)radic119878

119888 119875119903minus 119878119888 119878119888119870119888 119905)]

minus 1198601radic119875119903

infin

sum

119899=0

[119892 (0 2119899radic119875119903 119875119903minus 119878119888 119878119888119870119888 119905)

+119892 (0 (2119899 + 2)radic119875119903 119875119903minus 119878119888 119878119888119870119888 119905)]


119903

infin

sum

119899=0

[119892 (0 2119899radic119875119903 1 0 119905)

+119892 (0 (2119899 + 2)radic119875119903 1 0 119905)]


Nu1=

120597119879

120597119910

10038161003816100381610038161003816100381610038161003816119910=1

= 21198601radic119875119903

infin

sum

119899=0

[119892 (0 (2119899 + 1)radic119875119903 119875119903minus 119878119888 119878119888119870119888 119905)]

minus 21198601radic119878119888

infin

sum

119899=0

[119892 (119870119888 (2119899 + 1)radic119878

119888 119875119903minus 119878119888 119878119888119870119888 119905)]

+ 2 (119863lowast+ 1)radic119875

119903

infin

sum

119899=0

[119892 (0 (2119899 + 1)radic119875119903 1 0 119905)]


(13)


1205910= minus

120597119906

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

= 1198602

infin

sum

119899=0

[119892(119872 +1

119870119901


1

119870119901

119905)

+ 119892(119872 +1

119870119901

2119899 + 2 1 minus 119878119888 119878119888119870119888

minus119872 minus1

119870119901

119905)]

minus 1198602radic119878119888

infin

sum

119899=0

[119892(119870119888 2119899radic119878

119888 1 minus 119878

119888 119878119888119870119888minus 119872 minus

1

119870119901

119905)

+ 119892(119870119888 (2119899 + 2)radic119878

119888 1 minus 119878

119888 119878119888119870119888

minus119872 minus1

119870119901

119905)]

+ 1198603

infin

sum

119899=0

[119892(119872 +1

119870119901


1

119870119901

119905)

+ 119892(119872 +1

119870119901

2119899 + 2 1 minus 119875119903

minus119872 minus1

119870119901

119905)]

minus 1198603radic119875119903

infin

sum

119899=0

[119892(119870119888 2119899radic119875

119903 1 minus 119875


1

119870119901

119905)

+ 119892(119870119888 (2119899 + 2)radic119875

119903 1 minus 119875

119903

minus119872 minus1

119870119901

119905)]

Journal of Fluids 5

+ 21198604

infin

sum

119899=0

[119892(119872 +1

119870119901

2119899 119875119903minus 119878119888 119878119888119870119888 119905)

+ 119892(119872 +1

119870119901

2119899 + 2 119875119903minus 119878119888 119878119888119870119888 119905)]

minus 1198604radic119878119862

infin

sum

119899=0

[119892 (119870119888 2119899radic119878

119888 119875119903minus 119878119888 119878119888119870119888 119905)

+ 119892 (119870119888 (2119899 + 2)radic119878

119888 119875119903minus 119878119888 119878119888119870119888 119905)]

minus 1198604radic119875119903

infin

sum

119899=0

[119892 (0 2119899radic119875119903 119875119903minus 119878119888 119878119888119870119888 119905)

+119892 (0 (2119899 + 2)radic119875119903 119875119903minus 119878119888 119878119888119870119888 119905)]

+ (1198605minus 1198606)

infin

sum

119899=0

[119892(119872 +1

119870119901

2119899 1 0 119905)

+ 119892(119872 +1

119870119901

2119899 + 2 1 0 119905)]

+ 1198605Sh0+ 1198606radic119875119903

infin

sum

119899=0

[119892 (0 2119899radic119875119903 1 0 119905)

+ 119892 (0 (2119899 + 2)radic119875119903 1 0 119905)]

1205911= minus

120597119906

120597119910

10038161003816100381610038161003816100381610038161003816119910=1

= minus21198602

infin

sum

119899=0

[119892(119872 +1

119870119901

2119899 + 1 1 minus 119878119888 119878119888119870119888

minus119872 minus1

119870119901

119905)]

+ 21198602radic119878119888

infin

sum

119899=0

[119892 (119870119888 (2119899 + 1)radic119878

119888 1 minus 119878

119888 119878119888119870119888

minus119872 119905)]

minus 21198603

infin

sum

119899=0

[119892(119872 +1

119870119901


1

119870119901

119905)]

+ 21198603radic119875119903

infin

sum

119899=0

[119892(119870119888 (2119899 + 1)radic119875

119903 1 minus 119875

119903 minus119872

minus1

119870119901

119905)]

minus 41198604

infin

sum

119899=0

[119892(119872 +1

119870119901

2119899 + 1 119875119903minus 119878119888 119878119888119870119888 119905)]

+ 21198604radic119878119888

infin

sum

119899=0

[119892 (119870119888 (2119899 + 1)radic119878

119888 119875119903minus 119878119888 119878119888119870119888 119905)]

+ 21198604radic119875119903

infin

sum

119899=0

[119892 (0 (2119899 + 1)radic119875119903 119875119903minus 119878119888 119878119888119870119888 119905)]

minus 2 (1198605minus 1198606)

infin

sum

119899=0

[119892(119872 +1

119870119901

2119899 + 1 1 0 119905)]

minus 21198606radic119875119903

infin

sum

119899=0

[119892 (0 (2119899 + 1)radic119875119903 1 0 119905)] + 119860

5Sh1



119876 = int

1

0

119906 119889119910 (15)




119862 =

infin

sum

119899=0

[119891 (0 119886119899 1 0 119905) minus 119891 (0 119887

119899 1 0 119905)]

119879 =


minus119870119888119905)

2119870119888119905radic120587119905

infin

sum

119899=0

[119886119899exp(minus

1198862

119899

4119905) minus 119887119899exp(minus

1198872

119899

4119905)]

+ (119863lowast+ 1)

infin

sum

119899=0

[119891 (0 119886119899 1 0 119905) minus 119891 (0 119887

119899 1 0 119905)] minus 119863

lowast119862

119906 = 1198608119862 +

119863lowast

2119905radic120587119905[

exp (minus119870119888119905)

119870119888(119870119888minus 119872 minus (1119870

119901))

+1

(119872 + (1119870119901))119870119888


(119872 + (1119870119901)) (119870119888minus 119872 minus (1119870

119901))

]

times

infin

sum

119899=0

[119886119899exp(minus

1198862

119899

4119905) minus 119887119899exp(minus

1198872

119899

4119905)]

+ 1198606

infin

sum

119899=0

[119891 (0 119886119899 1 0 119905) minus 119891 (0 119887

119899 1 0 119905)]

(16)where 119860

7and 119860




Sh0= minus

120597119862

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

= minus

infin

sum

119899=0

[119892 (119870119888 2119899 1 0 119905) minus 119892 (119870

119888 2119899 + 2 1 0 119905)]

Sh1= minus

120597119862

120597119910

10038161003816100381610038161003816100381610038161003816119910=1

= 2

infin

sum

119899=0

[119892 (119870119888 2119899 + 1 1 0 119905)]

(17)

6 Journal of Fluids


Nu0= minus

120597119879

120597119910

10038161003816100381610038161003816100381610038161003816119910=0


119888119905)]

2119870119888119905radic120587119905

times [

infin

sum

119899=0

exp(minus1198992

119905) + exp(minus

(119899 + 1)2

119905)

minus2

119905

infin

sum

119899=0

1198992 exp(minus

1198992

119905)

+(119899 + 1)2 exp(minus

(119899 + 1)2

119905)]


infin

sum

119899=0

[119892 (0 2119899 1 0 119905) + 119892 (0 2119899 + 2 1 0 119905)]

+ 119863lowastSh0

Nu1= minus

120597119879

120597119910

10038161003816100381610038161003816100381610038161003816119910=0


119888119905)]

2119870119888119905radic120587119905

times [2

infin

sum

119899=0

exp(minus(2119899 + 1)

2

4119905)

minus1

119905

infin

sum

119899=0

(2119899 + 1)2 exp(minus

(2119899 + 1)2

4119905)]

+ 2 (119863lowast+ 1)

infin

sum

119899=0

[119892 (0 2119899 + 1 1 0 119905)] minus 119863lowastSh1

(18)


1205910= minus

120597119906

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

= 1198608Sh0+

119863lowast

2119905radic120587119905


119888119905)

119870119888(119870119888minus 119872 minus (1119870

119901))

+1

119870119888(119872 + (1119870

119901))


(119872 + (1119870119901)) (119870119888minus 119872 minus (1119870

119901))

]

times [

infin

sum

119899=0

exp(minus1198992

119905) + exp(minus

(119899 + 1)2

119905)

minus2

119905

infin

sum

119899=0

1198992 exp(minus

1198992

119905)

+(119899 + 1)2 exp(minus

(119899 + 1)2

119905)]

+ 1198606

infin

sum

119899=0

[119892 (0 2119899 1 0 119905) + 119892 (0 2119899 + 2 1 0 119905)]

minus 1198607

infin

sum

119899=0

[119892(119872 +1

119870119901

2119899 1 0 119905)

+119892(119872 +1

119870119901

2119899 + 2 1 0 119905)]

(19)1205911= minus

120597119906

120597119910

10038161003816100381610038161003816100381610038161003816119910=1

= 1198608Sh1+

119863lowast

119905radic120587119905

times [

exp (minus119872 minus (1119870119901)) 119905

(119872 + (1119870119901)) (119870119888minus 119872 minus (1119870

119901))

minus1

119870119888(119872 + (1119870

119901))


119888119905)

119870119888(119870119888minus 119872 minus (1119870

119901))

]

times

infin

sum

119899=0

[exp(minus(2119899 + 1)

2

119905)

minus2(2119899 + 1)

2

119905exp(minus

(2119899 + 1)2

119905)]

minus 21198606

infin

sum

119899=0

[119892 (0 2119899 + 1 1 0 119905)]

times 21198607

infin

sum

119899=0

[119892(119872 +1

119870119901

2119899 + 1 1 0 119905)]

(20)


1198892119906

1198891199102+ 119873119862 + 119879 minus (119872 +

1

119870119901

)119906 = 0

1198892119862

1198891199102minus 119878119888119870119888119862 = 0

1198892119879

1198891199102+ 119863lowast 1198892119862

1198891199102= 0

(21)


119862 =sinh (radic119878

119888119870119888(1 minus 119910))

sinh (radic119878119888119870119888)


lowast119862

Journal of Fluids 7

Curve

I 04 05 022 2 2 18 2 100

II 02 05 022 2 2 18 2 100

III 04 2 022 2 2 18 2 100

IV 04 05 03 2 2 18 2 100

V 04 05 022 1 2 18 2 100

VI 04 05 022 2 4 18 2 100

VII 04 05 022 2 2 2 2 100

VIII 04 05 022 2 2 18 4 100

IX 04 05 022 2 2 18 2 05


04

02

0

minus02

minus04

minus06

minus08

minus1

u

0 02 04 06 08 1

II

IVI

V III

IXVII

VIII

IV

X

y


mdash


119906 = 1198609

sinhradic(119872 + (1119870119901)) (1 minus 119910)

sinhradic119872 + (1119870119901)

+ 1198605119862 + 119860

6(1 minus 119910)

(22)





119888119870119888


119888119870119888

(23)


Nu0= 119863lowast+ 1 + 119863

lowastSh0

Nu1= minus (119863


lowastSh1

(24)


1205910= minus1198609radic119872 +

1

119870119901

cothradic119872 +1

119870119901

+ 1198605Sh0minus 1198606

1205911=

1198609radic119872 + (1119870

119901)

sinhradic119872 + (1119870119901)

minus 1198605Sh1+ 1198606

(25)

Mass Flux Consider

119876 = 1198609radic119872 +

1

119870119901

coshradic119872 + (1119870119901) minus 1

sinhradic119872 + (1119870119901)

+1198605

radic119878119888119870119888

coshradic119878119888119870119888minus 1

sinhradic119878119888119870119888

+1198606

2

(26)



119888)











119888)


8 Journal of Fluids

1000

500

0

minus500

minus1000

minus1500

minus2000

u

0 02 04 06 08 1 12y

Sc = 5 Kc = 0 Kp = 100

Sc = 150 Kc = 0 Kp = 100

Sc = 200 Kc = 0 Kp = 100

Sc = 200 Kc = 0 Kp = 05


lowast= 2

0 02 04 06 08 1 12y

035

03

025

02

015

01

005

0

u

VIII

VI

I II III VII

V

IV

Curve

I 05 100 1 2 2 100

II 05 022 2 2 2 100

III 05 152 1 2 2 100

IV 05 022 2 2 2 05

V 2 022 2 2 2 100

VI 05 022 2 2 30 100

VII 05 03 1 2 2 100VIII 05 022 2 4 2 100

M Sc Kc N KpDlowast












119888 that is 119878

119888= 150 and

119878119888


119888lt 10 and 119870




Journal of Fluids 9

CurveI 04 022 20 700II 04 022 00 700III

III

04 00 700IV 04 20 700V 02 20 700VI 04 022 00 071VII 04 022 02 071VII 04 00 071IX 04 02 071X 02 02 071

t Sc Kc Pr

6

5

4

3

2

1

00 02 04 06 08 1

y

T

V I II

IV

VI-X

WaterAir

030030030

030030030


119903= 70 (water) and 119875

119903= 071 (air)

0

02

04

06

08

1

12

0 02 04 06 08 1

T

y

IIII

II

Curve t Kc

I 04 20II 04 10III 02 20


119888= 1)















119903 119905 and


119903= 70 (water)





002040608

1121416

0 02 04 06 08 1

T

y

IV

II

I

III

Curve Sc Kc Dlowast

I 022 2 2II 030 2 2III 022 1 2IV 022 2 30


CurveI 04 100 0II 04 617 0III 04 100 2IV 04 617 2V 02 100 2

t Kc

minus02

0

02

04

06

08

1

0 02 04 06 08 1

C

I

III

II IVV

y

Sc


0010203040506070809

1

0 02 04 06 08 1

C

y

III

III

Curve t Kc

I 04 20II 04 10III 02 20




= 1 and 119878119888


minus02

0

02

04

06

08

1

0 02 04 06 08 1

C

y

Curve Sc Kc

I 100 2II 617 1III 617 2





119888(ie 119878










Sl no 119905 119872 119878119888

119870119888

119873 119875119903

119863lowast

119870119901

1205910

1205911



Sl no 119872 119878119888

119870119888

119873 119863lowast

119870119901

1205910

1205911

1 05 022 2 2 2 100 1732053 09325152 2 022 2 2 2 100 166425 07954363 05 03 2 2 2 100 1732052 09325154 05 022 1 2 2 100 173205 09325155 05 022 2 4 2 100 1664279 08692876 05 022 2 2 4 100 2144507 13763827 05 022 2 2 2 05 142312 0524363


Sl no 119905 119878119888

119870119888

119875119903

119863lowast Nu

0Nu1





119888 119870119888 and 119863





119888





Sl no 119905 119878119888

119870119888

Sh0

Sh1

1 04 100 2 59757644 minus06128007882 02 100 2 6313751515 0158384443 04 617 2 7026366229 04705642814 02 100 0 567128182 minus0466307658


119888 and 119870




119888increases





Sl no 119905 119872 119878119888

119870c 119873 119875119903

119863lowast

119870119901

119876

1 04 05 022 2 2 18 2 100 01978362 02 05 022 2 2 18 2 100 01765723 04 2 022 2 2 18 2 100 00576554 04 05 03 2 2 18 2 100 05684755 04 05 022 1 2 18 2 100 00911936 04 05 022 2 4 18 2 100 01666677 04 05 022 2 2 2 2 100 02649398 04 05 022 2 2 18 4 100 03645839 04 05 022 2 2 18 2 05 0034241


Sl no 119872 119878119888

119870119888

119873 119863lowast

119870119901

119876

1 05 022 2 2 2 100 01190522 2 022 2 2 2 100 01042073 05 03 2 2 2 100 01190524 05 022 1 2 2 100 01190525 05 022 2 4 2 100 01950776 05 022 2 2 4 100 01223927 05 022 2 2 2 05 0094632







119901 In case of 119878

119888



119888and 119863


5 Conclusion











Appendices

A


erfc (x) =2

radic120587int

infin

119909

119890minus1205822

119889120582

119891 (1199091 1199092 1199093 1199094 1199095)

=1

2exp(

11990941199095

1199093

)


1199094

1199093

+ 1199091)


times erfc( 1199092

2radic1199095

minus radic(1199094

1199093

+ 1199091)1199095)

+ exp(1199092radic

1199094

1199093

+ 1199091)

times erfc( 1199092

2radic1199095

+ radic(1199094

1199093

+ 1199091)1199095)]

119892 (1199091 1199092 1199093 1199094 1199095)

=1

2radic

1199094

1199093

+ 1199091exp(

11990941199095

1199093

)


1199094

1199093

+ 1199091)

times erfc( 1199092

2radic1199095

+ radic(1199094

1199093

+ 1199091)1199095)


1199094

1199093

+ 1199091)

times erfc( 1199092

2radic1199095

minus radic(1199094

1199093

+ 1199091)1199095)]

minus2


1199092

3

41199095

+ 11990911199095))

(A1)

B


119878119888

119875119903minus 119878119888

+ 119875119903minus 119878119888]

1198602= minus

1 minus 119878119888

119878119888119870119888minus 119872 minus (1119870

119901)

times [119873 +


119901))

119878119888119870119888(1 minus 119875

119903) + (119872 + (1119870

119901)) (119875119903minus 119878119888)

]

1198603=

1 minus 119875119903

119872 + (1119870119901)

times [1 +


119901))

119878119888119870119888(1 minus 119875

119903) + (119872 + (1119870

119901)) (119875119903minus 119878119888)

]

1198604=

119863lowast119875119903(119875119903minus 119878119888)

119878119888119870119888(1 minus 119875

119903) + (119872 + (1119870

119901)) (119875119903minus 119878119888)

1198605=


119878119888119870119888minus 119872 minus (1119870

119901)

1198606=

119863lowast+ 1

119872 + (1119870119901)

1198607=

119870119888(119863lowast+ 1) minus (119872 + (1119870

119901)) (119873 + 1)

(119872 + (1119870119901)) (119870119888minus 119872 minus (1119870

119901))

1198608=


119870119888minus 119872 minus (1119870

119901)

1198609=

minus119878119888119870119888(119863lowast+ 1) + (119872 + (1119870

119901)) (119873 + 1)

(119872 + (1119870119901)) (119878119888119870119888minus 119872 minus (1119870

119901))

References






























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Journal of Fluids 3

Here119875119903 119878119888119873 119863lowast119872119870

119888 and119870




120597119906

120597119905=

1205972119906

1205971199102+ 119873119862 + 119879 minus (119872 +

1

119870119901

)119906

120597119862

120597119905=

1

119878119888

1205972119862

1205971199102minus 119870119888119862

120597119879

120597119905=

1

119901119903

1205972119879

1205971199102+

119863lowast

119901119903

1205972119862

1205971199102

(5)


119905 le 0 119906 (119910 119905) = 0 119879 (119910 119905) = 0 119862 (119910 119905) = 0

119905 le 0 119906 (0 119905) = 0 119879 (0 119905) = 1 119862 (0 119905) = 1

119906 (1 119905) = 0 119879 (1 119905) = 0 119862 (1 119905) = 0

(6)



1198892119862

1198891199102minus 119878119888(119904 + 119870

119888) 119862 = 0

1198892119879

1198891199102minus 119904119875119903119879 = 119878

119888119863lowast(119904 + 119870

119888) 119862

(7)

119904119906 =1198892119906

1198891199102+ 119873119862 + 119879 minus (119872 +

1

119870119901

)119906 (8)


119862 (0 119904) =1

119904 119862 (1 119904) = 0

119879 (0 119904) =1

119904 119879 (1 119904) = 0

119906 (0 119904) = 0 119906 (1 119904) = 0

(9)


119871minus1

(119890minus119887radic119904+119886

119904) =

1

2119890minus119887radic119886 erfc( 119887


+119890119887radic119886 erfc( 119887

2radic119905+ radic119886119905)

(10)


Case 1 (119875119903

= 1 119878119888


119862 =

infin

sum

119899=0

[119891(119870119888 119886119899radic119878119888 1 0 119905) minus 119891(119870

119888 119887119899radic119878119888 1 0 119905)]

119879 = 1198601

infin

sum

119899=0

[119891(0 119886119899radic119875119903 119875119903minus 119878119888 119878119888119870119888 119905)


minus 1198601

infin

sum

119899=0

[119891(119870119888 119886119899radic119878119888 119875119903minus 119878119888 119878119888119870119888 119905)


+ (119863lowast+ 1)

infin

sum

119899=0

[119891(0 119886119899radic119875119903 1 0 119905)


lowast119862

119906 = 1198602

infin

sum

119899=0

[119891(119872 +1

119870119901

119886119899 1 minus 119878

119888 119878119888119870119888minus 119872 minus

1

119870119901

119905)

minus119891(119872 +1

119870119901

119887119899 1 minus 119878

119888 119878119888119870119888minus 119872 minus

1

119870119901

119905)]

minus 1198602

infin

sum

119899=0

[119891(119870119888 119886119899radic119878119888 1 minus 119878

119888 119878119888119870119888minus 119872 minus

1

119870119901

119905)


119888 119878119888119870119888minus 119872 minus

1

119870119901

119905)]

+ 1198603

infin

sum

119899=0

[119891(119872 +1

119870119901

119886119899 1 minus 119875


1

119870119901

119905)

minus119891(119872 +1

119870119901

119887119899 1 minus 119875


1

119870119901

119905)]

minus 1198603

infin

sum

119899=0

[119891(119870119888 119886119899radic119875119903 1 minus 119875


1

119870119901

119905)



1

119870119901

119905)]

+ 21198604

infin

sum

119899=0

[119891(119872 +1

119870119901

119886119899 119875119903minus 119878119888 119878119888119870119888 119905)

minus119891(119872 +1

119870119901

119887119899 119875119903minus 119878119888 119878119888119870119888 119905)]

minus 1198604

infin

sum

119899=0

[119891(119870119888 119886119899radic119878119888 119875119903minus 119878119888 119878119888119870119888 119905)


4 Journal of Fluids

minus 1198604

infin

sum

119899=0

[119891(0 119886119899radic119875119903 119875119903minus 119878119888 119878119888119870119888 119905)


+ (1198605minus 1198606)

infin

sum

119899=0

[119891(119872 +1

119870119901

119886119899 1 0 119905)

minus119891(119872 +1

119870119901

119887119899 1 0 119905)]

+ 1198606

infin

sum

119899=0

[119891(0 119886119899radic119875119903 1 0 119905) minus 119891(0 119887

119899radic119875119903 1 0 119905)]

(11)where 119886

119899= 2119899 + 119910 and 119887

119899= 2119899 + 2 minus 119910


1 1198602 1198603 1198604 1198605 and 119860

6are given in

Appendix B



Sh0=

120597119862

120597119910

10038161003816100381610038161003816100381610038161003816119910=0


infin

sum

119899=0

[119892 (119870119888 2119899radic119878

119888 1 0 119905)

+119892 (119870119888 (2119899 + 2)radic119878

119888 1 0 119905)]

Sh1=

120597119862

120597119910

10038161003816100381610038161003816100381610038161003816119910=1

= 2radic119878119888

infin

sum

119899=0

[119892 (119870119888 (2119899 + 1)radic119878

119888 1 0 119905)]


Nu0= minus

120597119879

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

= 1198601radic119878119888

infin

sum

119899=0

[119892 (119870119888 2119899radic119878

119888 119875119903minus 119878119888 119878119888119870119888 119905)

+119892 (119870119888 (2119899 + 2)radic119878

119888 119875119903minus 119878119888 119878119888119870119888 119905)]

minus 1198601radic119875119903

infin

sum

119899=0

[119892 (0 2119899radic119875119903 119875119903minus 119878119888 119878119888119870119888 119905)

+119892 (0 (2119899 + 2)radic119875119903 119875119903minus 119878119888 119878119888119870119888 119905)]


119903

infin

sum

119899=0

[119892 (0 2119899radic119875119903 1 0 119905)

+119892 (0 (2119899 + 2)radic119875119903 1 0 119905)]


Nu1=

120597119879

120597119910

10038161003816100381610038161003816100381610038161003816119910=1

= 21198601radic119875119903

infin

sum

119899=0

[119892 (0 (2119899 + 1)radic119875119903 119875119903minus 119878119888 119878119888119870119888 119905)]

minus 21198601radic119878119888

infin

sum

119899=0

[119892 (119870119888 (2119899 + 1)radic119878

119888 119875119903minus 119878119888 119878119888119870119888 119905)]

+ 2 (119863lowast+ 1)radic119875

119903

infin

sum

119899=0

[119892 (0 (2119899 + 1)radic119875119903 1 0 119905)]


(13)


1205910= minus

120597119906

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

= 1198602

infin

sum

119899=0

[119892(119872 +1

119870119901


1

119870119901

119905)

+ 119892(119872 +1

119870119901

2119899 + 2 1 minus 119878119888 119878119888119870119888

minus119872 minus1

119870119901

119905)]

minus 1198602radic119878119888

infin

sum

119899=0

[119892(119870119888 2119899radic119878

119888 1 minus 119878

119888 119878119888119870119888minus 119872 minus

1

119870119901

119905)

+ 119892(119870119888 (2119899 + 2)radic119878

119888 1 minus 119878

119888 119878119888119870119888

minus119872 minus1

119870119901

119905)]

+ 1198603

infin

sum

119899=0

[119892(119872 +1

119870119901


1

119870119901

119905)

+ 119892(119872 +1

119870119901

2119899 + 2 1 minus 119875119903

minus119872 minus1

119870119901

119905)]

minus 1198603radic119875119903

infin

sum

119899=0

[119892(119870119888 2119899radic119875

119903 1 minus 119875


1

119870119901

119905)

+ 119892(119870119888 (2119899 + 2)radic119875

119903 1 minus 119875

119903

minus119872 minus1

119870119901

119905)]

Journal of Fluids 5

+ 21198604

infin

sum

119899=0

[119892(119872 +1

119870119901

2119899 119875119903minus 119878119888 119878119888119870119888 119905)

+ 119892(119872 +1

119870119901

2119899 + 2 119875119903minus 119878119888 119878119888119870119888 119905)]

minus 1198604radic119878119862

infin

sum

119899=0

[119892 (119870119888 2119899radic119878

119888 119875119903minus 119878119888 119878119888119870119888 119905)

+ 119892 (119870119888 (2119899 + 2)radic119878

119888 119875119903minus 119878119888 119878119888119870119888 119905)]

minus 1198604radic119875119903

infin

sum

119899=0

[119892 (0 2119899radic119875119903 119875119903minus 119878119888 119878119888119870119888 119905)

+119892 (0 (2119899 + 2)radic119875119903 119875119903minus 119878119888 119878119888119870119888 119905)]

+ (1198605minus 1198606)

infin

sum

119899=0

[119892(119872 +1

119870119901

2119899 1 0 119905)

+ 119892(119872 +1

119870119901

2119899 + 2 1 0 119905)]

+ 1198605Sh0+ 1198606radic119875119903

infin

sum

119899=0

[119892 (0 2119899radic119875119903 1 0 119905)

+ 119892 (0 (2119899 + 2)radic119875119903 1 0 119905)]

1205911= minus

120597119906

120597119910

10038161003816100381610038161003816100381610038161003816119910=1

= minus21198602

infin

sum

119899=0

[119892(119872 +1

119870119901

2119899 + 1 1 minus 119878119888 119878119888119870119888

minus119872 minus1

119870119901

119905)]

+ 21198602radic119878119888

infin

sum

119899=0

[119892 (119870119888 (2119899 + 1)radic119878

119888 1 minus 119878

119888 119878119888119870119888

minus119872 119905)]

minus 21198603

infin

sum

119899=0

[119892(119872 +1

119870119901


1

119870119901

119905)]

+ 21198603radic119875119903

infin

sum

119899=0

[119892(119870119888 (2119899 + 1)radic119875

119903 1 minus 119875

119903 minus119872

minus1

119870119901

119905)]

minus 41198604

infin

sum

119899=0

[119892(119872 +1

119870119901

2119899 + 1 119875119903minus 119878119888 119878119888119870119888 119905)]

+ 21198604radic119878119888

infin

sum

119899=0

[119892 (119870119888 (2119899 + 1)radic119878

119888 119875119903minus 119878119888 119878119888119870119888 119905)]

+ 21198604radic119875119903

infin

sum

119899=0

[119892 (0 (2119899 + 1)radic119875119903 119875119903minus 119878119888 119878119888119870119888 119905)]

minus 2 (1198605minus 1198606)

infin

sum

119899=0

[119892(119872 +1

119870119901

2119899 + 1 1 0 119905)]

minus 21198606radic119875119903

infin

sum

119899=0

[119892 (0 (2119899 + 1)radic119875119903 1 0 119905)] + 119860

5Sh1



119876 = int

1

0

119906 119889119910 (15)




119862 =

infin

sum

119899=0

[119891 (0 119886119899 1 0 119905) minus 119891 (0 119887

119899 1 0 119905)]

119879 =


minus119870119888119905)

2119870119888119905radic120587119905

infin

sum

119899=0

[119886119899exp(minus

1198862

119899

4119905) minus 119887119899exp(minus

1198872

119899

4119905)]

+ (119863lowast+ 1)

infin

sum

119899=0

[119891 (0 119886119899 1 0 119905) minus 119891 (0 119887

119899 1 0 119905)] minus 119863

lowast119862

119906 = 1198608119862 +

119863lowast

2119905radic120587119905[

exp (minus119870119888119905)

119870119888(119870119888minus 119872 minus (1119870

119901))

+1

(119872 + (1119870119901))119870119888


(119872 + (1119870119901)) (119870119888minus 119872 minus (1119870

119901))

]

times

infin

sum

119899=0

[119886119899exp(minus

1198862

119899

4119905) minus 119887119899exp(minus

1198872

119899

4119905)]

+ 1198606

infin

sum

119899=0

[119891 (0 119886119899 1 0 119905) minus 119891 (0 119887

119899 1 0 119905)]

(16)where 119860

7and 119860




Sh0= minus

120597119862

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

= minus

infin

sum

119899=0

[119892 (119870119888 2119899 1 0 119905) minus 119892 (119870

119888 2119899 + 2 1 0 119905)]

Sh1= minus

120597119862

120597119910

10038161003816100381610038161003816100381610038161003816119910=1

= 2

infin

sum

119899=0

[119892 (119870119888 2119899 + 1 1 0 119905)]

(17)

6 Journal of Fluids


Nu0= minus

120597119879

120597119910

10038161003816100381610038161003816100381610038161003816119910=0


119888119905)]

2119870119888119905radic120587119905

times [

infin

sum

119899=0

exp(minus1198992

119905) + exp(minus

(119899 + 1)2

119905)

minus2

119905

infin

sum

119899=0

1198992 exp(minus

1198992

119905)

+(119899 + 1)2 exp(minus

(119899 + 1)2

119905)]


infin

sum

119899=0

[119892 (0 2119899 1 0 119905) + 119892 (0 2119899 + 2 1 0 119905)]

+ 119863lowastSh0

Nu1= minus

120597119879

120597119910

10038161003816100381610038161003816100381610038161003816119910=0


119888119905)]

2119870119888119905radic120587119905

times [2

infin

sum

119899=0

exp(minus(2119899 + 1)

2

4119905)

minus1

119905

infin

sum

119899=0

(2119899 + 1)2 exp(minus

(2119899 + 1)2

4119905)]

+ 2 (119863lowast+ 1)

infin

sum

119899=0

[119892 (0 2119899 + 1 1 0 119905)] minus 119863lowastSh1

(18)


1205910= minus

120597119906

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

= 1198608Sh0+

119863lowast

2119905radic120587119905


119888119905)

119870119888(119870119888minus 119872 minus (1119870

119901))

+1

119870119888(119872 + (1119870

119901))


(119872 + (1119870119901)) (119870119888minus 119872 minus (1119870

119901))

]

times [

infin

sum

119899=0

exp(minus1198992

119905) + exp(minus

(119899 + 1)2

119905)

minus2

119905

infin

sum

119899=0

1198992 exp(minus

1198992

119905)

+(119899 + 1)2 exp(minus

(119899 + 1)2

119905)]

+ 1198606

infin

sum

119899=0

[119892 (0 2119899 1 0 119905) + 119892 (0 2119899 + 2 1 0 119905)]

minus 1198607

infin

sum

119899=0

[119892(119872 +1

119870119901

2119899 1 0 119905)

+119892(119872 +1

119870119901

2119899 + 2 1 0 119905)]

(19)1205911= minus

120597119906

120597119910

10038161003816100381610038161003816100381610038161003816119910=1

= 1198608Sh1+

119863lowast

119905radic120587119905

times [

exp (minus119872 minus (1119870119901)) 119905

(119872 + (1119870119901)) (119870119888minus 119872 minus (1119870

119901))

minus1

119870119888(119872 + (1119870

119901))


119888119905)

119870119888(119870119888minus 119872 minus (1119870

119901))

]

times

infin

sum

119899=0

[exp(minus(2119899 + 1)

2

119905)

minus2(2119899 + 1)

2

119905exp(minus

(2119899 + 1)2

119905)]

minus 21198606

infin

sum

119899=0

[119892 (0 2119899 + 1 1 0 119905)]

times 21198607

infin

sum

119899=0

[119892(119872 +1

119870119901

2119899 + 1 1 0 119905)]

(20)


1198892119906

1198891199102+ 119873119862 + 119879 minus (119872 +

1

119870119901

)119906 = 0

1198892119862

1198891199102minus 119878119888119870119888119862 = 0

1198892119879

1198891199102+ 119863lowast 1198892119862

1198891199102= 0

(21)


119862 =sinh (radic119878

119888119870119888(1 minus 119910))

sinh (radic119878119888119870119888)


lowast119862

Journal of Fluids 7

Curve

I 04 05 022 2 2 18 2 100

II 02 05 022 2 2 18 2 100

III 04 2 022 2 2 18 2 100

IV 04 05 03 2 2 18 2 100

V 04 05 022 1 2 18 2 100

VI 04 05 022 2 4 18 2 100

VII 04 05 022 2 2 2 2 100

VIII 04 05 022 2 2 18 4 100

IX 04 05 022 2 2 18 2 05


04

02

0

minus02

minus04

minus06

minus08

minus1

u

0 02 04 06 08 1

II

IVI

V III

IXVII

VIII

IV

X

y


mdash


119906 = 1198609

sinhradic(119872 + (1119870119901)) (1 minus 119910)

sinhradic119872 + (1119870119901)

+ 1198605119862 + 119860

6(1 minus 119910)

(22)





119888119870119888


119888119870119888

(23)


Nu0= 119863lowast+ 1 + 119863

lowastSh0

Nu1= minus (119863


lowastSh1

(24)


1205910= minus1198609radic119872 +

1

119870119901

cothradic119872 +1

119870119901

+ 1198605Sh0minus 1198606

1205911=

1198609radic119872 + (1119870

119901)

sinhradic119872 + (1119870119901)

minus 1198605Sh1+ 1198606

(25)

Mass Flux Consider

119876 = 1198609radic119872 +

1

119870119901

coshradic119872 + (1119870119901) minus 1

sinhradic119872 + (1119870119901)

+1198605

radic119878119888119870119888

coshradic119878119888119870119888minus 1

sinhradic119878119888119870119888

+1198606

2

(26)



119888)











119888)


8 Journal of Fluids

1000

500

0

minus500

minus1000

minus1500

minus2000

u

0 02 04 06 08 1 12y

Sc = 5 Kc = 0 Kp = 100

Sc = 150 Kc = 0 Kp = 100

Sc = 200 Kc = 0 Kp = 100

Sc = 200 Kc = 0 Kp = 05


lowast= 2

0 02 04 06 08 1 12y

035

03

025

02

015

01

005

0

u

VIII

VI

I II III VII

V

IV

Curve

I 05 100 1 2 2 100

II 05 022 2 2 2 100

III 05 152 1 2 2 100

IV 05 022 2 2 2 05

V 2 022 2 2 2 100

VI 05 022 2 2 30 100

VII 05 03 1 2 2 100VIII 05 022 2 4 2 100

M Sc Kc N KpDlowast












119888 that is 119878

119888= 150 and

119878119888


119888lt 10 and 119870




Journal of Fluids 9

CurveI 04 022 20 700II 04 022 00 700III

III

04 00 700IV 04 20 700V 02 20 700VI 04 022 00 071VII 04 022 02 071VII 04 00 071IX 04 02 071X 02 02 071

t Sc Kc Pr

6

5

4

3

2

1

00 02 04 06 08 1

y

T

V I II

IV

VI-X

WaterAir

030030030

030030030


119903= 70 (water) and 119875

119903= 071 (air)

0

02

04

06

08

1

12

0 02 04 06 08 1

T

y

IIII

II

Curve t Kc

I 04 20II 04 10III 02 20


119888= 1)















119903 119905 and


119903= 70 (water)





002040608

1121416

0 02 04 06 08 1

T

y

IV

II

I

III

Curve Sc Kc Dlowast

I 022 2 2II 030 2 2III 022 1 2IV 022 2 30


CurveI 04 100 0II 04 617 0III 04 100 2IV 04 617 2V 02 100 2

t Kc

minus02

0

02

04

06

08

1

0 02 04 06 08 1

C

I

III

II IVV

y

Sc


0010203040506070809

1

0 02 04 06 08 1

C

y

III

III

Curve t Kc

I 04 20II 04 10III 02 20




= 1 and 119878119888


minus02

0

02

04

06

08

1

0 02 04 06 08 1

C

y

Curve Sc Kc

I 100 2II 617 1III 617 2





119888(ie 119878










Sl no 119905 119872 119878119888

119870119888

119873 119875119903

119863lowast

119870119901

1205910

1205911



Sl no 119872 119878119888

119870119888

119873 119863lowast

119870119901

1205910

1205911

1 05 022 2 2 2 100 1732053 09325152 2 022 2 2 2 100 166425 07954363 05 03 2 2 2 100 1732052 09325154 05 022 1 2 2 100 173205 09325155 05 022 2 4 2 100 1664279 08692876 05 022 2 2 4 100 2144507 13763827 05 022 2 2 2 05 142312 0524363


Sl no 119905 119878119888

119870119888

119875119903

119863lowast Nu

0Nu1





119888 119870119888 and 119863





119888





Sl no 119905 119878119888

119870119888

Sh0

Sh1

1 04 100 2 59757644 minus06128007882 02 100 2 6313751515 0158384443 04 617 2 7026366229 04705642814 02 100 0 567128182 minus0466307658


119888 and 119870




119888increases





Sl no 119905 119872 119878119888

119870c 119873 119875119903

119863lowast

119870119901

119876

1 04 05 022 2 2 18 2 100 01978362 02 05 022 2 2 18 2 100 01765723 04 2 022 2 2 18 2 100 00576554 04 05 03 2 2 18 2 100 05684755 04 05 022 1 2 18 2 100 00911936 04 05 022 2 4 18 2 100 01666677 04 05 022 2 2 2 2 100 02649398 04 05 022 2 2 18 4 100 03645839 04 05 022 2 2 18 2 05 0034241


Sl no 119872 119878119888

119870119888

119873 119863lowast

119870119901

119876

1 05 022 2 2 2 100 01190522 2 022 2 2 2 100 01042073 05 03 2 2 2 100 01190524 05 022 1 2 2 100 01190525 05 022 2 4 2 100 01950776 05 022 2 2 4 100 01223927 05 022 2 2 2 05 0094632







119901 In case of 119878

119888



119888and 119863


5 Conclusion











Appendices

A


erfc (x) =2

radic120587int

infin

119909

119890minus1205822

119889120582

119891 (1199091 1199092 1199093 1199094 1199095)

=1

2exp(

11990941199095

1199093

)


1199094

1199093

+ 1199091)


times erfc( 1199092

2radic1199095

minus radic(1199094

1199093

+ 1199091)1199095)

+ exp(1199092radic

1199094

1199093

+ 1199091)

times erfc( 1199092

2radic1199095

+ radic(1199094

1199093

+ 1199091)1199095)]

119892 (1199091 1199092 1199093 1199094 1199095)

=1

2radic

1199094

1199093

+ 1199091exp(

11990941199095

1199093

)


1199094

1199093

+ 1199091)

times erfc( 1199092

2radic1199095

+ radic(1199094

1199093

+ 1199091)1199095)


1199094

1199093

+ 1199091)

times erfc( 1199092

2radic1199095

minus radic(1199094

1199093

+ 1199091)1199095)]

minus2


1199092

3

41199095

+ 11990911199095))

(A1)

B


119878119888

119875119903minus 119878119888

+ 119875119903minus 119878119888]

1198602= minus

1 minus 119878119888

119878119888119870119888minus 119872 minus (1119870

119901)

times [119873 +


119901))

119878119888119870119888(1 minus 119875

119903) + (119872 + (1119870

119901)) (119875119903minus 119878119888)

]

1198603=

1 minus 119875119903

119872 + (1119870119901)

times [1 +


119901))

119878119888119870119888(1 minus 119875

119903) + (119872 + (1119870

119901)) (119875119903minus 119878119888)

]

1198604=

119863lowast119875119903(119875119903minus 119878119888)

119878119888119870119888(1 minus 119875

119903) + (119872 + (1119870

119901)) (119875119903minus 119878119888)

1198605=


119878119888119870119888minus 119872 minus (1119870

119901)

1198606=

119863lowast+ 1

119872 + (1119870119901)

1198607=

119870119888(119863lowast+ 1) minus (119872 + (1119870

119901)) (119873 + 1)

(119872 + (1119870119901)) (119870119888minus 119872 minus (1119870

119901))

1198608=


119870119888minus 119872 minus (1119870

119901)

1198609=

minus119878119888119870119888(119863lowast+ 1) + (119872 + (1119870

119901)) (119873 + 1)

(119872 + (1119870119901)) (119878119888119870119888minus 119872 minus (1119870

119901))

References






























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



4 Journal of Fluids

minus 1198604

infin

sum

119899=0

[119891(0 119886119899radic119875119903 119875119903minus 119878119888 119878119888119870119888 119905)


+ (1198605minus 1198606)

infin

sum

119899=0

[119891(119872 +1

119870119901

119886119899 1 0 119905)

minus119891(119872 +1

119870119901

119887119899 1 0 119905)]

+ 1198606

infin

sum

119899=0

[119891(0 119886119899radic119875119903 1 0 119905) minus 119891(0 119887

119899radic119875119903 1 0 119905)]

(11)where 119886

119899= 2119899 + 119910 and 119887

119899= 2119899 + 2 minus 119910


1 1198602 1198603 1198604 1198605 and 119860

6are given in

Appendix B



Sh0=

120597119862

120597119910

10038161003816100381610038161003816100381610038161003816119910=0


infin

sum

119899=0

[119892 (119870119888 2119899radic119878

119888 1 0 119905)

+119892 (119870119888 (2119899 + 2)radic119878

119888 1 0 119905)]

Sh1=

120597119862

120597119910

10038161003816100381610038161003816100381610038161003816119910=1

= 2radic119878119888

infin

sum

119899=0

[119892 (119870119888 (2119899 + 1)radic119878

119888 1 0 119905)]


Nu0= minus

120597119879

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

= 1198601radic119878119888

infin

sum

119899=0

[119892 (119870119888 2119899radic119878

119888 119875119903minus 119878119888 119878119888119870119888 119905)

+119892 (119870119888 (2119899 + 2)radic119878

119888 119875119903minus 119878119888 119878119888119870119888 119905)]

minus 1198601radic119875119903

infin

sum

119899=0

[119892 (0 2119899radic119875119903 119875119903minus 119878119888 119878119888119870119888 119905)

+119892 (0 (2119899 + 2)radic119875119903 119875119903minus 119878119888 119878119888119870119888 119905)]


119903

infin

sum

119899=0

[119892 (0 2119899radic119875119903 1 0 119905)

+119892 (0 (2119899 + 2)radic119875119903 1 0 119905)]


Nu1=

120597119879

120597119910

10038161003816100381610038161003816100381610038161003816119910=1

= 21198601radic119875119903

infin

sum

119899=0

[119892 (0 (2119899 + 1)radic119875119903 119875119903minus 119878119888 119878119888119870119888 119905)]

minus 21198601radic119878119888

infin

sum

119899=0

[119892 (119870119888 (2119899 + 1)radic119878

119888 119875119903minus 119878119888 119878119888119870119888 119905)]

+ 2 (119863lowast+ 1)radic119875

119903

infin

sum

119899=0

[119892 (0 (2119899 + 1)radic119875119903 1 0 119905)]


(13)


1205910= minus

120597119906

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

= 1198602

infin

sum

119899=0

[119892(119872 +1

119870119901


1

119870119901

119905)

+ 119892(119872 +1

119870119901

2119899 + 2 1 minus 119878119888 119878119888119870119888

minus119872 minus1

119870119901

119905)]

minus 1198602radic119878119888

infin

sum

119899=0

[119892(119870119888 2119899radic119878

119888 1 minus 119878

119888 119878119888119870119888minus 119872 minus

1

119870119901

119905)

+ 119892(119870119888 (2119899 + 2)radic119878

119888 1 minus 119878

119888 119878119888119870119888

minus119872 minus1

119870119901

119905)]

+ 1198603

infin

sum

119899=0

[119892(119872 +1

119870119901


1

119870119901

119905)

+ 119892(119872 +1

119870119901

2119899 + 2 1 minus 119875119903

minus119872 minus1

119870119901

119905)]

minus 1198603radic119875119903

infin

sum

119899=0

[119892(119870119888 2119899radic119875

119903 1 minus 119875


1

119870119901

119905)

+ 119892(119870119888 (2119899 + 2)radic119875

119903 1 minus 119875

119903

minus119872 minus1

119870119901

119905)]

Journal of Fluids 5

+ 21198604

infin

sum

119899=0

[119892(119872 +1

119870119901

2119899 119875119903minus 119878119888 119878119888119870119888 119905)

+ 119892(119872 +1

119870119901

2119899 + 2 119875119903minus 119878119888 119878119888119870119888 119905)]

minus 1198604radic119878119862

infin

sum

119899=0

[119892 (119870119888 2119899radic119878

119888 119875119903minus 119878119888 119878119888119870119888 119905)

+ 119892 (119870119888 (2119899 + 2)radic119878

119888 119875119903minus 119878119888 119878119888119870119888 119905)]

minus 1198604radic119875119903

infin

sum

119899=0

[119892 (0 2119899radic119875119903 119875119903minus 119878119888 119878119888119870119888 119905)

+119892 (0 (2119899 + 2)radic119875119903 119875119903minus 119878119888 119878119888119870119888 119905)]

+ (1198605minus 1198606)

infin

sum

119899=0

[119892(119872 +1

119870119901

2119899 1 0 119905)

+ 119892(119872 +1

119870119901

2119899 + 2 1 0 119905)]

+ 1198605Sh0+ 1198606radic119875119903

infin

sum

119899=0

[119892 (0 2119899radic119875119903 1 0 119905)

+ 119892 (0 (2119899 + 2)radic119875119903 1 0 119905)]

1205911= minus

120597119906

120597119910

10038161003816100381610038161003816100381610038161003816119910=1

= minus21198602

infin

sum

119899=0

[119892(119872 +1

119870119901

2119899 + 1 1 minus 119878119888 119878119888119870119888

minus119872 minus1

119870119901

119905)]

+ 21198602radic119878119888

infin

sum

119899=0

[119892 (119870119888 (2119899 + 1)radic119878

119888 1 minus 119878

119888 119878119888119870119888

minus119872 119905)]

minus 21198603

infin

sum

119899=0

[119892(119872 +1

119870119901


1

119870119901

119905)]

+ 21198603radic119875119903

infin

sum

119899=0

[119892(119870119888 (2119899 + 1)radic119875

119903 1 minus 119875

119903 minus119872

minus1

119870119901

119905)]

minus 41198604

infin

sum

119899=0

[119892(119872 +1

119870119901

2119899 + 1 119875119903minus 119878119888 119878119888119870119888 119905)]

+ 21198604radic119878119888

infin

sum

119899=0

[119892 (119870119888 (2119899 + 1)radic119878

119888 119875119903minus 119878119888 119878119888119870119888 119905)]

+ 21198604radic119875119903

infin

sum

119899=0

[119892 (0 (2119899 + 1)radic119875119903 119875119903minus 119878119888 119878119888119870119888 119905)]

minus 2 (1198605minus 1198606)

infin

sum

119899=0

[119892(119872 +1

119870119901

2119899 + 1 1 0 119905)]

minus 21198606radic119875119903

infin

sum

119899=0

[119892 (0 (2119899 + 1)radic119875119903 1 0 119905)] + 119860

5Sh1



119876 = int

1

0

119906 119889119910 (15)




119862 =

infin

sum

119899=0

[119891 (0 119886119899 1 0 119905) minus 119891 (0 119887

119899 1 0 119905)]

119879 =


minus119870119888119905)

2119870119888119905radic120587119905

infin

sum

119899=0

[119886119899exp(minus

1198862

119899

4119905) minus 119887119899exp(minus

1198872

119899

4119905)]

+ (119863lowast+ 1)

infin

sum

119899=0

[119891 (0 119886119899 1 0 119905) minus 119891 (0 119887

119899 1 0 119905)] minus 119863

lowast119862

119906 = 1198608119862 +

119863lowast

2119905radic120587119905[

exp (minus119870119888119905)

119870119888(119870119888minus 119872 minus (1119870

119901))

+1

(119872 + (1119870119901))119870119888


(119872 + (1119870119901)) (119870119888minus 119872 minus (1119870

119901))

]

times

infin

sum

119899=0

[119886119899exp(minus

1198862

119899

4119905) minus 119887119899exp(minus

1198872

119899

4119905)]

+ 1198606

infin

sum

119899=0

[119891 (0 119886119899 1 0 119905) minus 119891 (0 119887

119899 1 0 119905)]

(16)where 119860

7and 119860




Sh0= minus

120597119862

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

= minus

infin

sum

119899=0

[119892 (119870119888 2119899 1 0 119905) minus 119892 (119870

119888 2119899 + 2 1 0 119905)]

Sh1= minus

120597119862

120597119910

10038161003816100381610038161003816100381610038161003816119910=1

= 2

infin

sum

119899=0

[119892 (119870119888 2119899 + 1 1 0 119905)]

(17)

6 Journal of Fluids


Nu0= minus

120597119879

120597119910

10038161003816100381610038161003816100381610038161003816119910=0


119888119905)]

2119870119888119905radic120587119905

times [

infin

sum

119899=0

exp(minus1198992

119905) + exp(minus

(119899 + 1)2

119905)

minus2

119905

infin

sum

119899=0

1198992 exp(minus

1198992

119905)

+(119899 + 1)2 exp(minus

(119899 + 1)2

119905)]


infin

sum

119899=0

[119892 (0 2119899 1 0 119905) + 119892 (0 2119899 + 2 1 0 119905)]

+ 119863lowastSh0

Nu1= minus

120597119879

120597119910

10038161003816100381610038161003816100381610038161003816119910=0


119888119905)]

2119870119888119905radic120587119905

times [2

infin

sum

119899=0

exp(minus(2119899 + 1)

2

4119905)

minus1

119905

infin

sum

119899=0

(2119899 + 1)2 exp(minus

(2119899 + 1)2

4119905)]

+ 2 (119863lowast+ 1)

infin

sum

119899=0

[119892 (0 2119899 + 1 1 0 119905)] minus 119863lowastSh1

(18)


1205910= minus

120597119906

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

= 1198608Sh0+

119863lowast

2119905radic120587119905


119888119905)

119870119888(119870119888minus 119872 minus (1119870

119901))

+1

119870119888(119872 + (1119870

119901))


(119872 + (1119870119901)) (119870119888minus 119872 minus (1119870

119901))

]

times [

infin

sum

119899=0

exp(minus1198992

119905) + exp(minus

(119899 + 1)2

119905)

minus2

119905

infin

sum

119899=0

1198992 exp(minus

1198992

119905)

+(119899 + 1)2 exp(minus

(119899 + 1)2

119905)]

+ 1198606

infin

sum

119899=0

[119892 (0 2119899 1 0 119905) + 119892 (0 2119899 + 2 1 0 119905)]

minus 1198607

infin

sum

119899=0

[119892(119872 +1

119870119901

2119899 1 0 119905)

+119892(119872 +1

119870119901

2119899 + 2 1 0 119905)]

(19)1205911= minus

120597119906

120597119910

10038161003816100381610038161003816100381610038161003816119910=1

= 1198608Sh1+

119863lowast

119905radic120587119905

times [

exp (minus119872 minus (1119870119901)) 119905

(119872 + (1119870119901)) (119870119888minus 119872 minus (1119870

119901))

minus1

119870119888(119872 + (1119870

119901))


119888119905)

119870119888(119870119888minus 119872 minus (1119870

119901))

]

times

infin

sum

119899=0

[exp(minus(2119899 + 1)

2

119905)

minus2(2119899 + 1)

2

119905exp(minus

(2119899 + 1)2

119905)]

minus 21198606

infin

sum

119899=0

[119892 (0 2119899 + 1 1 0 119905)]

times 21198607

infin

sum

119899=0

[119892(119872 +1

119870119901

2119899 + 1 1 0 119905)]

(20)


1198892119906

1198891199102+ 119873119862 + 119879 minus (119872 +

1

119870119901

)119906 = 0

1198892119862

1198891199102minus 119878119888119870119888119862 = 0

1198892119879

1198891199102+ 119863lowast 1198892119862

1198891199102= 0

(21)


119862 =sinh (radic119878

119888119870119888(1 minus 119910))

sinh (radic119878119888119870119888)


lowast119862

Journal of Fluids 7

Curve

I 04 05 022 2 2 18 2 100

II 02 05 022 2 2 18 2 100

III 04 2 022 2 2 18 2 100

IV 04 05 03 2 2 18 2 100

V 04 05 022 1 2 18 2 100

VI 04 05 022 2 4 18 2 100

VII 04 05 022 2 2 2 2 100

VIII 04 05 022 2 2 18 4 100

IX 04 05 022 2 2 18 2 05


04

02

0

minus02

minus04

minus06

minus08

minus1

u

0 02 04 06 08 1

II

IVI

V III

IXVII

VIII

IV

X

y


mdash


119906 = 1198609

sinhradic(119872 + (1119870119901)) (1 minus 119910)

sinhradic119872 + (1119870119901)

+ 1198605119862 + 119860

6(1 minus 119910)

(22)





119888119870119888


119888119870119888

(23)


Nu0= 119863lowast+ 1 + 119863

lowastSh0

Nu1= minus (119863


lowastSh1

(24)


1205910= minus1198609radic119872 +

1

119870119901

cothradic119872 +1

119870119901

+ 1198605Sh0minus 1198606

1205911=

1198609radic119872 + (1119870

119901)

sinhradic119872 + (1119870119901)

minus 1198605Sh1+ 1198606

(25)

Mass Flux Consider

119876 = 1198609radic119872 +

1

119870119901

coshradic119872 + (1119870119901) minus 1

sinhradic119872 + (1119870119901)

+1198605

radic119878119888119870119888

coshradic119878119888119870119888minus 1

sinhradic119878119888119870119888

+1198606

2

(26)



119888)











119888)


8 Journal of Fluids

1000

500

0

minus500

minus1000

minus1500

minus2000

u

0 02 04 06 08 1 12y

Sc = 5 Kc = 0 Kp = 100

Sc = 150 Kc = 0 Kp = 100

Sc = 200 Kc = 0 Kp = 100

Sc = 200 Kc = 0 Kp = 05


lowast= 2

0 02 04 06 08 1 12y

035

03

025

02

015

01

005

0

u

VIII

VI

I II III VII

V

IV

Curve

I 05 100 1 2 2 100

II 05 022 2 2 2 100

III 05 152 1 2 2 100

IV 05 022 2 2 2 05

V 2 022 2 2 2 100

VI 05 022 2 2 30 100

VII 05 03 1 2 2 100VIII 05 022 2 4 2 100

M Sc Kc N KpDlowast












119888 that is 119878

119888= 150 and

119878119888


119888lt 10 and 119870




Journal of Fluids 9

CurveI 04 022 20 700II 04 022 00 700III

III

04 00 700IV 04 20 700V 02 20 700VI 04 022 00 071VII 04 022 02 071VII 04 00 071IX 04 02 071X 02 02 071

t Sc Kc Pr

6

5

4

3

2

1

00 02 04 06 08 1

y

T

V I II

IV

VI-X

WaterAir

030030030

030030030


119903= 70 (water) and 119875

119903= 071 (air)

0

02

04

06

08

1

12

0 02 04 06 08 1

T

y

IIII

II

Curve t Kc

I 04 20II 04 10III 02 20


119888= 1)















119903 119905 and


119903= 70 (water)





002040608

1121416

0 02 04 06 08 1

T

y

IV

II

I

III

Curve Sc Kc Dlowast

I 022 2 2II 030 2 2III 022 1 2IV 022 2 30


CurveI 04 100 0II 04 617 0III 04 100 2IV 04 617 2V 02 100 2

t Kc

minus02

0

02

04

06

08

1

0 02 04 06 08 1

C

I

III

II IVV

y

Sc


0010203040506070809

1

0 02 04 06 08 1

C

y

III

III

Curve t Kc

I 04 20II 04 10III 02 20




= 1 and 119878119888


minus02

0

02

04

06

08

1

0 02 04 06 08 1

C

y

Curve Sc Kc

I 100 2II 617 1III 617 2





119888(ie 119878










Sl no 119905 119872 119878119888

119870119888

119873 119875119903

119863lowast

119870119901

1205910

1205911



Sl no 119872 119878119888

119870119888

119873 119863lowast

119870119901

1205910

1205911

1 05 022 2 2 2 100 1732053 09325152 2 022 2 2 2 100 166425 07954363 05 03 2 2 2 100 1732052 09325154 05 022 1 2 2 100 173205 09325155 05 022 2 4 2 100 1664279 08692876 05 022 2 2 4 100 2144507 13763827 05 022 2 2 2 05 142312 0524363


Sl no 119905 119878119888

119870119888

119875119903

119863lowast Nu

0Nu1





119888 119870119888 and 119863





119888





Sl no 119905 119878119888

119870119888

Sh0

Sh1

1 04 100 2 59757644 minus06128007882 02 100 2 6313751515 0158384443 04 617 2 7026366229 04705642814 02 100 0 567128182 minus0466307658


119888 and 119870




119888increases





Sl no 119905 119872 119878119888

119870c 119873 119875119903

119863lowast

119870119901

119876

1 04 05 022 2 2 18 2 100 01978362 02 05 022 2 2 18 2 100 01765723 04 2 022 2 2 18 2 100 00576554 04 05 03 2 2 18 2 100 05684755 04 05 022 1 2 18 2 100 00911936 04 05 022 2 4 18 2 100 01666677 04 05 022 2 2 2 2 100 02649398 04 05 022 2 2 18 4 100 03645839 04 05 022 2 2 18 2 05 0034241


Sl no 119872 119878119888

119870119888

119873 119863lowast

119870119901

119876

1 05 022 2 2 2 100 01190522 2 022 2 2 2 100 01042073 05 03 2 2 2 100 01190524 05 022 1 2 2 100 01190525 05 022 2 4 2 100 01950776 05 022 2 2 4 100 01223927 05 022 2 2 2 05 0094632







119901 In case of 119878

119888



119888and 119863


5 Conclusion











Appendices

A


erfc (x) =2

radic120587int

infin

119909

119890minus1205822

119889120582

119891 (1199091 1199092 1199093 1199094 1199095)

=1

2exp(

11990941199095

1199093

)


1199094

1199093

+ 1199091)


times erfc( 1199092

2radic1199095

minus radic(1199094

1199093

+ 1199091)1199095)

+ exp(1199092radic

1199094

1199093

+ 1199091)

times erfc( 1199092

2radic1199095

+ radic(1199094

1199093

+ 1199091)1199095)]

119892 (1199091 1199092 1199093 1199094 1199095)

=1

2radic

1199094

1199093

+ 1199091exp(

11990941199095

1199093

)


1199094

1199093

+ 1199091)

times erfc( 1199092

2radic1199095

+ radic(1199094

1199093

+ 1199091)1199095)


1199094

1199093

+ 1199091)

times erfc( 1199092

2radic1199095

minus radic(1199094

1199093

+ 1199091)1199095)]

minus2


1199092

3

41199095

+ 11990911199095))

(A1)

B


119878119888

119875119903minus 119878119888

+ 119875119903minus 119878119888]

1198602= minus

1 minus 119878119888

119878119888119870119888minus 119872 minus (1119870

119901)

times [119873 +


119901))

119878119888119870119888(1 minus 119875

119903) + (119872 + (1119870

119901)) (119875119903minus 119878119888)

]

1198603=

1 minus 119875119903

119872 + (1119870119901)

times [1 +


119901))

119878119888119870119888(1 minus 119875

119903) + (119872 + (1119870

119901)) (119875119903minus 119878119888)

]

1198604=

119863lowast119875119903(119875119903minus 119878119888)

119878119888119870119888(1 minus 119875

119903) + (119872 + (1119870

119901)) (119875119903minus 119878119888)

1198605=


119878119888119870119888minus 119872 minus (1119870

119901)

1198606=

119863lowast+ 1

119872 + (1119870119901)

1198607=

119870119888(119863lowast+ 1) minus (119872 + (1119870

119901)) (119873 + 1)

(119872 + (1119870119901)) (119870119888minus 119872 minus (1119870

119901))

1198608=


119870119888minus 119872 minus (1119870

119901)

1198609=

minus119878119888119870119888(119863lowast+ 1) + (119872 + (1119870

119901)) (119873 + 1)

(119872 + (1119870119901)) (119878119888119870119888minus 119872 minus (1119870

119901))

References






























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Journal of Fluids 5

+ 21198604

infin

sum

119899=0

[119892(119872 +1

119870119901

2119899 119875119903minus 119878119888 119878119888119870119888 119905)

+ 119892(119872 +1

119870119901

2119899 + 2 119875119903minus 119878119888 119878119888119870119888 119905)]

minus 1198604radic119878119862

infin

sum

119899=0

[119892 (119870119888 2119899radic119878

119888 119875119903minus 119878119888 119878119888119870119888 119905)

+ 119892 (119870119888 (2119899 + 2)radic119878

119888 119875119903minus 119878119888 119878119888119870119888 119905)]

minus 1198604radic119875119903

infin

sum

119899=0

[119892 (0 2119899radic119875119903 119875119903minus 119878119888 119878119888119870119888 119905)

+119892 (0 (2119899 + 2)radic119875119903 119875119903minus 119878119888 119878119888119870119888 119905)]

+ (1198605minus 1198606)

infin

sum

119899=0

[119892(119872 +1

119870119901

2119899 1 0 119905)

+ 119892(119872 +1

119870119901

2119899 + 2 1 0 119905)]

+ 1198605Sh0+ 1198606radic119875119903

infin

sum

119899=0

[119892 (0 2119899radic119875119903 1 0 119905)

+ 119892 (0 (2119899 + 2)radic119875119903 1 0 119905)]

1205911= minus

120597119906

120597119910

10038161003816100381610038161003816100381610038161003816119910=1

= minus21198602

infin

sum

119899=0

[119892(119872 +1

119870119901

2119899 + 1 1 minus 119878119888 119878119888119870119888

minus119872 minus1

119870119901

119905)]

+ 21198602radic119878119888

infin

sum

119899=0

[119892 (119870119888 (2119899 + 1)radic119878

119888 1 minus 119878

119888 119878119888119870119888

minus119872 119905)]

minus 21198603

infin

sum

119899=0

[119892(119872 +1

119870119901


1

119870119901

119905)]

+ 21198603radic119875119903

infin

sum

119899=0

[119892(119870119888 (2119899 + 1)radic119875

119903 1 minus 119875

119903 minus119872

minus1

119870119901

119905)]

minus 41198604

infin

sum

119899=0

[119892(119872 +1

119870119901

2119899 + 1 119875119903minus 119878119888 119878119888119870119888 119905)]

+ 21198604radic119878119888

infin

sum

119899=0

[119892 (119870119888 (2119899 + 1)radic119878

119888 119875119903minus 119878119888 119878119888119870119888 119905)]

+ 21198604radic119875119903

infin

sum

119899=0

[119892 (0 (2119899 + 1)radic119875119903 119875119903minus 119878119888 119878119888119870119888 119905)]

minus 2 (1198605minus 1198606)

infin

sum

119899=0

[119892(119872 +1

119870119901

2119899 + 1 1 0 119905)]

minus 21198606radic119875119903

infin

sum

119899=0

[119892 (0 (2119899 + 1)radic119875119903 1 0 119905)] + 119860

5Sh1



119876 = int

1

0

119906 119889119910 (15)




119862 =

infin

sum

119899=0

[119891 (0 119886119899 1 0 119905) minus 119891 (0 119887

119899 1 0 119905)]

119879 =


minus119870119888119905)

2119870119888119905radic120587119905

infin

sum

119899=0

[119886119899exp(minus

1198862

119899

4119905) minus 119887119899exp(minus

1198872

119899

4119905)]

+ (119863lowast+ 1)

infin

sum

119899=0

[119891 (0 119886119899 1 0 119905) minus 119891 (0 119887

119899 1 0 119905)] minus 119863

lowast119862

119906 = 1198608119862 +

119863lowast

2119905radic120587119905[

exp (minus119870119888119905)

119870119888(119870119888minus 119872 minus (1119870

119901))

+1

(119872 + (1119870119901))119870119888


(119872 + (1119870119901)) (119870119888minus 119872 minus (1119870

119901))

]

times

infin

sum

119899=0

[119886119899exp(minus

1198862

119899

4119905) minus 119887119899exp(minus

1198872

119899

4119905)]

+ 1198606

infin

sum

119899=0

[119891 (0 119886119899 1 0 119905) minus 119891 (0 119887

119899 1 0 119905)]

(16)where 119860

7and 119860




Sh0= minus

120597119862

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

= minus

infin

sum

119899=0

[119892 (119870119888 2119899 1 0 119905) minus 119892 (119870

119888 2119899 + 2 1 0 119905)]

Sh1= minus

120597119862

120597119910

10038161003816100381610038161003816100381610038161003816119910=1

= 2

infin

sum

119899=0

[119892 (119870119888 2119899 + 1 1 0 119905)]

(17)

6 Journal of Fluids


Nu0= minus

120597119879

120597119910

10038161003816100381610038161003816100381610038161003816119910=0


119888119905)]

2119870119888119905radic120587119905

times [

infin

sum

119899=0

exp(minus1198992

119905) + exp(minus

(119899 + 1)2

119905)

minus2

119905

infin

sum

119899=0

1198992 exp(minus

1198992

119905)

+(119899 + 1)2 exp(minus

(119899 + 1)2

119905)]


infin

sum

119899=0

[119892 (0 2119899 1 0 119905) + 119892 (0 2119899 + 2 1 0 119905)]

+ 119863lowastSh0

Nu1= minus

120597119879

120597119910

10038161003816100381610038161003816100381610038161003816119910=0


119888119905)]

2119870119888119905radic120587119905

times [2

infin

sum

119899=0

exp(minus(2119899 + 1)

2

4119905)

minus1

119905

infin

sum

119899=0

(2119899 + 1)2 exp(minus

(2119899 + 1)2

4119905)]

+ 2 (119863lowast+ 1)

infin

sum

119899=0

[119892 (0 2119899 + 1 1 0 119905)] minus 119863lowastSh1

(18)


1205910= minus

120597119906

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

= 1198608Sh0+

119863lowast

2119905radic120587119905


119888119905)

119870119888(119870119888minus 119872 minus (1119870

119901))

+1

119870119888(119872 + (1119870

119901))


(119872 + (1119870119901)) (119870119888minus 119872 minus (1119870

119901))

]

times [

infin

sum

119899=0

exp(minus1198992

119905) + exp(minus

(119899 + 1)2

119905)

minus2

119905

infin

sum

119899=0

1198992 exp(minus

1198992

119905)

+(119899 + 1)2 exp(minus

(119899 + 1)2

119905)]

+ 1198606

infin

sum

119899=0

[119892 (0 2119899 1 0 119905) + 119892 (0 2119899 + 2 1 0 119905)]

minus 1198607

infin

sum

119899=0

[119892(119872 +1

119870119901

2119899 1 0 119905)

+119892(119872 +1

119870119901

2119899 + 2 1 0 119905)]

(19)1205911= minus

120597119906

120597119910

10038161003816100381610038161003816100381610038161003816119910=1

= 1198608Sh1+

119863lowast

119905radic120587119905

times [

exp (minus119872 minus (1119870119901)) 119905

(119872 + (1119870119901)) (119870119888minus 119872 minus (1119870

119901))

minus1

119870119888(119872 + (1119870

119901))


119888119905)

119870119888(119870119888minus 119872 minus (1119870

119901))

]

times

infin

sum

119899=0

[exp(minus(2119899 + 1)

2

119905)

minus2(2119899 + 1)

2

119905exp(minus

(2119899 + 1)2

119905)]

minus 21198606

infin

sum

119899=0

[119892 (0 2119899 + 1 1 0 119905)]

times 21198607

infin

sum

119899=0

[119892(119872 +1

119870119901

2119899 + 1 1 0 119905)]

(20)


1198892119906

1198891199102+ 119873119862 + 119879 minus (119872 +

1

119870119901

)119906 = 0

1198892119862

1198891199102minus 119878119888119870119888119862 = 0

1198892119879

1198891199102+ 119863lowast 1198892119862

1198891199102= 0

(21)


119862 =sinh (radic119878

119888119870119888(1 minus 119910))

sinh (radic119878119888119870119888)


lowast119862

Journal of Fluids 7

Curve

I 04 05 022 2 2 18 2 100

II 02 05 022 2 2 18 2 100

III 04 2 022 2 2 18 2 100

IV 04 05 03 2 2 18 2 100

V 04 05 022 1 2 18 2 100

VI 04 05 022 2 4 18 2 100

VII 04 05 022 2 2 2 2 100

VIII 04 05 022 2 2 18 4 100

IX 04 05 022 2 2 18 2 05


04

02

0

minus02

minus04

minus06

minus08

minus1

u

0 02 04 06 08 1

II

IVI

V III

IXVII

VIII

IV

X

y


mdash


119906 = 1198609

sinhradic(119872 + (1119870119901)) (1 minus 119910)

sinhradic119872 + (1119870119901)

+ 1198605119862 + 119860

6(1 minus 119910)

(22)





119888119870119888


119888119870119888

(23)


Nu0= 119863lowast+ 1 + 119863

lowastSh0

Nu1= minus (119863


lowastSh1

(24)


1205910= minus1198609radic119872 +

1

119870119901

cothradic119872 +1

119870119901

+ 1198605Sh0minus 1198606

1205911=

1198609radic119872 + (1119870

119901)

sinhradic119872 + (1119870119901)

minus 1198605Sh1+ 1198606

(25)

Mass Flux Consider

119876 = 1198609radic119872 +

1

119870119901

coshradic119872 + (1119870119901) minus 1

sinhradic119872 + (1119870119901)

+1198605

radic119878119888119870119888

coshradic119878119888119870119888minus 1

sinhradic119878119888119870119888

+1198606

2

(26)



119888)











119888)


8 Journal of Fluids

1000

500

0

minus500

minus1000

minus1500

minus2000

u

0 02 04 06 08 1 12y

Sc = 5 Kc = 0 Kp = 100

Sc = 150 Kc = 0 Kp = 100

Sc = 200 Kc = 0 Kp = 100

Sc = 200 Kc = 0 Kp = 05


lowast= 2

0 02 04 06 08 1 12y

035

03

025

02

015

01

005

0

u

VIII

VI

I II III VII

V

IV

Curve

I 05 100 1 2 2 100

II 05 022 2 2 2 100

III 05 152 1 2 2 100

IV 05 022 2 2 2 05

V 2 022 2 2 2 100

VI 05 022 2 2 30 100

VII 05 03 1 2 2 100VIII 05 022 2 4 2 100

M Sc Kc N KpDlowast












119888 that is 119878

119888= 150 and

119878119888


119888lt 10 and 119870




Journal of Fluids 9

CurveI 04 022 20 700II 04 022 00 700III

III

04 00 700IV 04 20 700V 02 20 700VI 04 022 00 071VII 04 022 02 071VII 04 00 071IX 04 02 071X 02 02 071

t Sc Kc Pr

6

5

4

3

2

1

00 02 04 06 08 1

y

T

V I II

IV

VI-X

WaterAir

030030030

030030030


119903= 70 (water) and 119875

119903= 071 (air)

0

02

04

06

08

1

12

0 02 04 06 08 1

T

y

IIII

II

Curve t Kc

I 04 20II 04 10III 02 20


119888= 1)















119903 119905 and


119903= 70 (water)





002040608

1121416

0 02 04 06 08 1

T

y

IV

II

I

III

Curve Sc Kc Dlowast

I 022 2 2II 030 2 2III 022 1 2IV 022 2 30


CurveI 04 100 0II 04 617 0III 04 100 2IV 04 617 2V 02 100 2

t Kc

minus02

0

02

04

06

08

1

0 02 04 06 08 1

C

I

III

II IVV

y

Sc


0010203040506070809

1

0 02 04 06 08 1

C

y

III

III

Curve t Kc

I 04 20II 04 10III 02 20




= 1 and 119878119888


minus02

0

02

04

06

08

1

0 02 04 06 08 1

C

y

Curve Sc Kc

I 100 2II 617 1III 617 2





119888(ie 119878










Sl no 119905 119872 119878119888

119870119888

119873 119875119903

119863lowast

119870119901

1205910

1205911



Sl no 119872 119878119888

119870119888

119873 119863lowast

119870119901

1205910

1205911

1 05 022 2 2 2 100 1732053 09325152 2 022 2 2 2 100 166425 07954363 05 03 2 2 2 100 1732052 09325154 05 022 1 2 2 100 173205 09325155 05 022 2 4 2 100 1664279 08692876 05 022 2 2 4 100 2144507 13763827 05 022 2 2 2 05 142312 0524363


Sl no 119905 119878119888

119870119888

119875119903

119863lowast Nu

0Nu1





119888 119870119888 and 119863





119888





Sl no 119905 119878119888

119870119888

Sh0

Sh1

1 04 100 2 59757644 minus06128007882 02 100 2 6313751515 0158384443 04 617 2 7026366229 04705642814 02 100 0 567128182 minus0466307658


119888 and 119870




119888increases





Sl no 119905 119872 119878119888

119870c 119873 119875119903

119863lowast

119870119901

119876

1 04 05 022 2 2 18 2 100 01978362 02 05 022 2 2 18 2 100 01765723 04 2 022 2 2 18 2 100 00576554 04 05 03 2 2 18 2 100 05684755 04 05 022 1 2 18 2 100 00911936 04 05 022 2 4 18 2 100 01666677 04 05 022 2 2 2 2 100 02649398 04 05 022 2 2 18 4 100 03645839 04 05 022 2 2 18 2 05 0034241


Sl no 119872 119878119888

119870119888

119873 119863lowast

119870119901

119876

1 05 022 2 2 2 100 01190522 2 022 2 2 2 100 01042073 05 03 2 2 2 100 01190524 05 022 1 2 2 100 01190525 05 022 2 4 2 100 01950776 05 022 2 2 4 100 01223927 05 022 2 2 2 05 0094632







119901 In case of 119878

119888



119888and 119863


5 Conclusion











Appendices

A


erfc (x) =2

radic120587int

infin

119909

119890minus1205822

119889120582

119891 (1199091 1199092 1199093 1199094 1199095)

=1

2exp(

11990941199095

1199093

)


1199094

1199093

+ 1199091)


times erfc( 1199092

2radic1199095

minus radic(1199094

1199093

+ 1199091)1199095)

+ exp(1199092radic

1199094

1199093

+ 1199091)

times erfc( 1199092

2radic1199095

+ radic(1199094

1199093

+ 1199091)1199095)]

119892 (1199091 1199092 1199093 1199094 1199095)

=1

2radic

1199094

1199093

+ 1199091exp(

11990941199095

1199093

)


1199094

1199093

+ 1199091)

times erfc( 1199092

2radic1199095

+ radic(1199094

1199093

+ 1199091)1199095)


1199094

1199093

+ 1199091)

times erfc( 1199092

2radic1199095

minus radic(1199094

1199093

+ 1199091)1199095)]

minus2


1199092

3

41199095

+ 11990911199095))

(A1)

B


119878119888

119875119903minus 119878119888

+ 119875119903minus 119878119888]

1198602= minus

1 minus 119878119888

119878119888119870119888minus 119872 minus (1119870

119901)

times [119873 +


119901))

119878119888119870119888(1 minus 119875

119903) + (119872 + (1119870

119901)) (119875119903minus 119878119888)

]

1198603=

1 minus 119875119903

119872 + (1119870119901)

times [1 +


119901))

119878119888119870119888(1 minus 119875

119903) + (119872 + (1119870

119901)) (119875119903minus 119878119888)

]

1198604=

119863lowast119875119903(119875119903minus 119878119888)

119878119888119870119888(1 minus 119875

119903) + (119872 + (1119870

119901)) (119875119903minus 119878119888)

1198605=


119878119888119870119888minus 119872 minus (1119870

119901)

1198606=

119863lowast+ 1

119872 + (1119870119901)

1198607=

119870119888(119863lowast+ 1) minus (119872 + (1119870

119901)) (119873 + 1)

(119872 + (1119870119901)) (119870119888minus 119872 minus (1119870

119901))

1198608=


119870119888minus 119872 minus (1119870

119901)

1198609=

minus119878119888119870119888(119863lowast+ 1) + (119872 + (1119870

119901)) (119873 + 1)

(119872 + (1119870119901)) (119878119888119870119888minus 119872 minus (1119870

119901))

References






























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



6 Journal of Fluids


Nu0= minus

120597119879

120597119910

10038161003816100381610038161003816100381610038161003816119910=0


119888119905)]

2119870119888119905radic120587119905

times [

infin

sum

119899=0

exp(minus1198992

119905) + exp(minus

(119899 + 1)2

119905)

minus2

119905

infin

sum

119899=0

1198992 exp(minus

1198992

119905)

+(119899 + 1)2 exp(minus

(119899 + 1)2

119905)]


infin

sum

119899=0

[119892 (0 2119899 1 0 119905) + 119892 (0 2119899 + 2 1 0 119905)]

+ 119863lowastSh0

Nu1= minus

120597119879

120597119910

10038161003816100381610038161003816100381610038161003816119910=0


119888119905)]

2119870119888119905radic120587119905

times [2

infin

sum

119899=0

exp(minus(2119899 + 1)

2

4119905)

minus1

119905

infin

sum

119899=0

(2119899 + 1)2 exp(minus

(2119899 + 1)2

4119905)]

+ 2 (119863lowast+ 1)

infin

sum

119899=0

[119892 (0 2119899 + 1 1 0 119905)] minus 119863lowastSh1

(18)


1205910= minus

120597119906

120597119910

10038161003816100381610038161003816100381610038161003816119910=0

= 1198608Sh0+

119863lowast

2119905radic120587119905


119888119905)

119870119888(119870119888minus 119872 minus (1119870

119901))

+1

119870119888(119872 + (1119870

119901))


(119872 + (1119870119901)) (119870119888minus 119872 minus (1119870

119901))

]

times [

infin

sum

119899=0

exp(minus1198992

119905) + exp(minus

(119899 + 1)2

119905)

minus2

119905

infin

sum

119899=0

1198992 exp(minus

1198992

119905)

+(119899 + 1)2 exp(minus

(119899 + 1)2

119905)]

+ 1198606

infin

sum

119899=0

[119892 (0 2119899 1 0 119905) + 119892 (0 2119899 + 2 1 0 119905)]

minus 1198607

infin

sum

119899=0

[119892(119872 +1

119870119901

2119899 1 0 119905)

+119892(119872 +1

119870119901

2119899 + 2 1 0 119905)]

(19)1205911= minus

120597119906

120597119910

10038161003816100381610038161003816100381610038161003816119910=1

= 1198608Sh1+

119863lowast

119905radic120587119905

times [

exp (minus119872 minus (1119870119901)) 119905

(119872 + (1119870119901)) (119870119888minus 119872 minus (1119870

119901))

minus1

119870119888(119872 + (1119870

119901))


119888119905)

119870119888(119870119888minus 119872 minus (1119870

119901))

]

times

infin

sum

119899=0

[exp(minus(2119899 + 1)

2

119905)

minus2(2119899 + 1)

2

119905exp(minus

(2119899 + 1)2

119905)]

minus 21198606

infin

sum

119899=0

[119892 (0 2119899 + 1 1 0 119905)]

times 21198607

infin

sum

119899=0

[119892(119872 +1

119870119901

2119899 + 1 1 0 119905)]

(20)


1198892119906

1198891199102+ 119873119862 + 119879 minus (119872 +

1

119870119901

)119906 = 0

1198892119862

1198891199102minus 119878119888119870119888119862 = 0

1198892119879

1198891199102+ 119863lowast 1198892119862

1198891199102= 0

(21)


119862 =sinh (radic119878

119888119870119888(1 minus 119910))

sinh (radic119878119888119870119888)


lowast119862

Journal of Fluids 7

Curve

I 04 05 022 2 2 18 2 100

II 02 05 022 2 2 18 2 100

III 04 2 022 2 2 18 2 100

IV 04 05 03 2 2 18 2 100

V 04 05 022 1 2 18 2 100

VI 04 05 022 2 4 18 2 100

VII 04 05 022 2 2 2 2 100

VIII 04 05 022 2 2 18 4 100

IX 04 05 022 2 2 18 2 05


04

02

0

minus02

minus04

minus06

minus08

minus1

u

0 02 04 06 08 1

II

IVI

V III

IXVII

VIII

IV

X

y


mdash


119906 = 1198609

sinhradic(119872 + (1119870119901)) (1 minus 119910)

sinhradic119872 + (1119870119901)

+ 1198605119862 + 119860

6(1 minus 119910)

(22)





119888119870119888


119888119870119888

(23)


Nu0= 119863lowast+ 1 + 119863

lowastSh0

Nu1= minus (119863


lowastSh1

(24)


1205910= minus1198609radic119872 +

1

119870119901

cothradic119872 +1

119870119901

+ 1198605Sh0minus 1198606

1205911=

1198609radic119872 + (1119870

119901)

sinhradic119872 + (1119870119901)

minus 1198605Sh1+ 1198606

(25)

Mass Flux Consider

119876 = 1198609radic119872 +

1

119870119901

coshradic119872 + (1119870119901) minus 1

sinhradic119872 + (1119870119901)

+1198605

radic119878119888119870119888

coshradic119878119888119870119888minus 1

sinhradic119878119888119870119888

+1198606

2

(26)



119888)











119888)


8 Journal of Fluids

1000

500

0

minus500

minus1000

minus1500

minus2000

u

0 02 04 06 08 1 12y

Sc = 5 Kc = 0 Kp = 100

Sc = 150 Kc = 0 Kp = 100

Sc = 200 Kc = 0 Kp = 100

Sc = 200 Kc = 0 Kp = 05


lowast= 2

0 02 04 06 08 1 12y

035

03

025

02

015

01

005

0

u

VIII

VI

I II III VII

V

IV

Curve

I 05 100 1 2 2 100

II 05 022 2 2 2 100

III 05 152 1 2 2 100

IV 05 022 2 2 2 05

V 2 022 2 2 2 100

VI 05 022 2 2 30 100

VII 05 03 1 2 2 100VIII 05 022 2 4 2 100

M Sc Kc N KpDlowast












119888 that is 119878

119888= 150 and

119878119888


119888lt 10 and 119870




Journal of Fluids 9

CurveI 04 022 20 700II 04 022 00 700III

III

04 00 700IV 04 20 700V 02 20 700VI 04 022 00 071VII 04 022 02 071VII 04 00 071IX 04 02 071X 02 02 071

t Sc Kc Pr

6

5

4

3

2

1

00 02 04 06 08 1

y

T

V I II

IV

VI-X

WaterAir

030030030

030030030


119903= 70 (water) and 119875

119903= 071 (air)

0

02

04

06

08

1

12

0 02 04 06 08 1

T

y

IIII

II

Curve t Kc

I 04 20II 04 10III 02 20


119888= 1)















119903 119905 and


119903= 70 (water)





002040608

1121416

0 02 04 06 08 1

T

y

IV

II

I

III

Curve Sc Kc Dlowast

I 022 2 2II 030 2 2III 022 1 2IV 022 2 30


CurveI 04 100 0II 04 617 0III 04 100 2IV 04 617 2V 02 100 2

t Kc

minus02

0

02

04

06

08

1

0 02 04 06 08 1

C

I

III

II IVV

y

Sc


0010203040506070809

1

0 02 04 06 08 1

C

y

III

III

Curve t Kc

I 04 20II 04 10III 02 20




= 1 and 119878119888


minus02

0

02

04

06

08

1

0 02 04 06 08 1

C

y

Curve Sc Kc

I 100 2II 617 1III 617 2





119888(ie 119878










Sl no 119905 119872 119878119888

119870119888

119873 119875119903

119863lowast

119870119901

1205910

1205911



Sl no 119872 119878119888

119870119888

119873 119863lowast

119870119901

1205910

1205911

1 05 022 2 2 2 100 1732053 09325152 2 022 2 2 2 100 166425 07954363 05 03 2 2 2 100 1732052 09325154 05 022 1 2 2 100 173205 09325155 05 022 2 4 2 100 1664279 08692876 05 022 2 2 4 100 2144507 13763827 05 022 2 2 2 05 142312 0524363


Sl no 119905 119878119888

119870119888

119875119903

119863lowast Nu

0Nu1





119888 119870119888 and 119863





119888





Sl no 119905 119878119888

119870119888

Sh0

Sh1

1 04 100 2 59757644 minus06128007882 02 100 2 6313751515 0158384443 04 617 2 7026366229 04705642814 02 100 0 567128182 minus0466307658


119888 and 119870




119888increases





Sl no 119905 119872 119878119888

119870c 119873 119875119903

119863lowast

119870119901

119876

1 04 05 022 2 2 18 2 100 01978362 02 05 022 2 2 18 2 100 01765723 04 2 022 2 2 18 2 100 00576554 04 05 03 2 2 18 2 100 05684755 04 05 022 1 2 18 2 100 00911936 04 05 022 2 4 18 2 100 01666677 04 05 022 2 2 2 2 100 02649398 04 05 022 2 2 18 4 100 03645839 04 05 022 2 2 18 2 05 0034241


Sl no 119872 119878119888

119870119888

119873 119863lowast

119870119901

119876

1 05 022 2 2 2 100 01190522 2 022 2 2 2 100 01042073 05 03 2 2 2 100 01190524 05 022 1 2 2 100 01190525 05 022 2 4 2 100 01950776 05 022 2 2 4 100 01223927 05 022 2 2 2 05 0094632







119901 In case of 119878

119888



119888and 119863


5 Conclusion











Appendices

A


erfc (x) =2

radic120587int

infin

119909

119890minus1205822

119889120582

119891 (1199091 1199092 1199093 1199094 1199095)

=1

2exp(

11990941199095

1199093

)


1199094

1199093

+ 1199091)


times erfc( 1199092

2radic1199095

minus radic(1199094

1199093

+ 1199091)1199095)

+ exp(1199092radic

1199094

1199093

+ 1199091)

times erfc( 1199092

2radic1199095

+ radic(1199094

1199093

+ 1199091)1199095)]

119892 (1199091 1199092 1199093 1199094 1199095)

=1

2radic

1199094

1199093

+ 1199091exp(

11990941199095

1199093

)


1199094

1199093

+ 1199091)

times erfc( 1199092

2radic1199095

+ radic(1199094

1199093

+ 1199091)1199095)


1199094

1199093

+ 1199091)

times erfc( 1199092

2radic1199095

minus radic(1199094

1199093

+ 1199091)1199095)]

minus2


1199092

3

41199095

+ 11990911199095))

(A1)

B


119878119888

119875119903minus 119878119888

+ 119875119903minus 119878119888]

1198602= minus

1 minus 119878119888

119878119888119870119888minus 119872 minus (1119870

119901)

times [119873 +


119901))

119878119888119870119888(1 minus 119875

119903) + (119872 + (1119870

119901)) (119875119903minus 119878119888)

]

1198603=

1 minus 119875119903

119872 + (1119870119901)

times [1 +


119901))

119878119888119870119888(1 minus 119875

119903) + (119872 + (1119870

119901)) (119875119903minus 119878119888)

]

1198604=

119863lowast119875119903(119875119903minus 119878119888)

119878119888119870119888(1 minus 119875

119903) + (119872 + (1119870

119901)) (119875119903minus 119878119888)

1198605=


119878119888119870119888minus 119872 minus (1119870

119901)

1198606=

119863lowast+ 1

119872 + (1119870119901)

1198607=

119870119888(119863lowast+ 1) minus (119872 + (1119870

119901)) (119873 + 1)

(119872 + (1119870119901)) (119870119888minus 119872 minus (1119870

119901))

1198608=


119870119888minus 119872 minus (1119870

119901)

1198609=

minus119878119888119870119888(119863lowast+ 1) + (119872 + (1119870

119901)) (119873 + 1)

(119872 + (1119870119901)) (119878119888119870119888minus 119872 minus (1119870

119901))

References






























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Journal of Fluids 7

Curve

I 04 05 022 2 2 18 2 100

II 02 05 022 2 2 18 2 100

III 04 2 022 2 2 18 2 100

IV 04 05 03 2 2 18 2 100

V 04 05 022 1 2 18 2 100

VI 04 05 022 2 4 18 2 100

VII 04 05 022 2 2 2 2 100

VIII 04 05 022 2 2 18 4 100

IX 04 05 022 2 2 18 2 05


04

02

0

minus02

minus04

minus06

minus08

minus1

u

0 02 04 06 08 1

II

IVI

V III

IXVII

VIII

IV

X

y


mdash


119906 = 1198609

sinhradic(119872 + (1119870119901)) (1 minus 119910)

sinhradic119872 + (1119870119901)

+ 1198605119862 + 119860

6(1 minus 119910)

(22)





119888119870119888


119888119870119888

(23)


Nu0= 119863lowast+ 1 + 119863

lowastSh0

Nu1= minus (119863


lowastSh1

(24)


1205910= minus1198609radic119872 +

1

119870119901

cothradic119872 +1

119870119901

+ 1198605Sh0minus 1198606

1205911=

1198609radic119872 + (1119870

119901)

sinhradic119872 + (1119870119901)

minus 1198605Sh1+ 1198606

(25)

Mass Flux Consider

119876 = 1198609radic119872 +

1

119870119901

coshradic119872 + (1119870119901) minus 1

sinhradic119872 + (1119870119901)

+1198605

radic119878119888119870119888

coshradic119878119888119870119888minus 1

sinhradic119878119888119870119888

+1198606

2

(26)



119888)











119888)


8 Journal of Fluids

1000

500

0

minus500

minus1000

minus1500

minus2000

u

0 02 04 06 08 1 12y

Sc = 5 Kc = 0 Kp = 100

Sc = 150 Kc = 0 Kp = 100

Sc = 200 Kc = 0 Kp = 100

Sc = 200 Kc = 0 Kp = 05


lowast= 2

0 02 04 06 08 1 12y

035

03

025

02

015

01

005

0

u

VIII

VI

I II III VII

V

IV

Curve

I 05 100 1 2 2 100

II 05 022 2 2 2 100

III 05 152 1 2 2 100

IV 05 022 2 2 2 05

V 2 022 2 2 2 100

VI 05 022 2 2 30 100

VII 05 03 1 2 2 100VIII 05 022 2 4 2 100

M Sc Kc N KpDlowast












119888 that is 119878

119888= 150 and

119878119888


119888lt 10 and 119870




Journal of Fluids 9

CurveI 04 022 20 700II 04 022 00 700III

III

04 00 700IV 04 20 700V 02 20 700VI 04 022 00 071VII 04 022 02 071VII 04 00 071IX 04 02 071X 02 02 071

t Sc Kc Pr

6

5

4

3

2

1

00 02 04 06 08 1

y

T

V I II

IV

VI-X

WaterAir

030030030

030030030


119903= 70 (water) and 119875

119903= 071 (air)

0

02

04

06

08

1

12

0 02 04 06 08 1

T

y

IIII

II

Curve t Kc

I 04 20II 04 10III 02 20


119888= 1)















119903 119905 and


119903= 70 (water)





002040608

1121416

0 02 04 06 08 1

T

y

IV

II

I

III

Curve Sc Kc Dlowast

I 022 2 2II 030 2 2III 022 1 2IV 022 2 30


CurveI 04 100 0II 04 617 0III 04 100 2IV 04 617 2V 02 100 2

t Kc

minus02

0

02

04

06

08

1

0 02 04 06 08 1

C

I

III

II IVV

y

Sc


0010203040506070809

1

0 02 04 06 08 1

C

y

III

III

Curve t Kc

I 04 20II 04 10III 02 20




= 1 and 119878119888


minus02

0

02

04

06

08

1

0 02 04 06 08 1

C

y

Curve Sc Kc

I 100 2II 617 1III 617 2





119888(ie 119878










Sl no 119905 119872 119878119888

119870119888

119873 119875119903

119863lowast

119870119901

1205910

1205911



Sl no 119872 119878119888

119870119888

119873 119863lowast

119870119901

1205910

1205911

1 05 022 2 2 2 100 1732053 09325152 2 022 2 2 2 100 166425 07954363 05 03 2 2 2 100 1732052 09325154 05 022 1 2 2 100 173205 09325155 05 022 2 4 2 100 1664279 08692876 05 022 2 2 4 100 2144507 13763827 05 022 2 2 2 05 142312 0524363


Sl no 119905 119878119888

119870119888

119875119903

119863lowast Nu

0Nu1





119888 119870119888 and 119863





119888





Sl no 119905 119878119888

119870119888

Sh0

Sh1

1 04 100 2 59757644 minus06128007882 02 100 2 6313751515 0158384443 04 617 2 7026366229 04705642814 02 100 0 567128182 minus0466307658


119888 and 119870




119888increases





Sl no 119905 119872 119878119888

119870c 119873 119875119903

119863lowast

119870119901

119876

1 04 05 022 2 2 18 2 100 01978362 02 05 022 2 2 18 2 100 01765723 04 2 022 2 2 18 2 100 00576554 04 05 03 2 2 18 2 100 05684755 04 05 022 1 2 18 2 100 00911936 04 05 022 2 4 18 2 100 01666677 04 05 022 2 2 2 2 100 02649398 04 05 022 2 2 18 4 100 03645839 04 05 022 2 2 18 2 05 0034241


Sl no 119872 119878119888

119870119888

119873 119863lowast

119870119901

119876

1 05 022 2 2 2 100 01190522 2 022 2 2 2 100 01042073 05 03 2 2 2 100 01190524 05 022 1 2 2 100 01190525 05 022 2 4 2 100 01950776 05 022 2 2 4 100 01223927 05 022 2 2 2 05 0094632







119901 In case of 119878

119888



119888and 119863


5 Conclusion











Appendices

A


erfc (x) =2

radic120587int

infin

119909

119890minus1205822

119889120582

119891 (1199091 1199092 1199093 1199094 1199095)

=1

2exp(

11990941199095

1199093

)


1199094

1199093

+ 1199091)


times erfc( 1199092

2radic1199095

minus radic(1199094

1199093

+ 1199091)1199095)

+ exp(1199092radic

1199094

1199093

+ 1199091)

times erfc( 1199092

2radic1199095

+ radic(1199094

1199093

+ 1199091)1199095)]

119892 (1199091 1199092 1199093 1199094 1199095)

=1

2radic

1199094

1199093

+ 1199091exp(

11990941199095

1199093

)


1199094

1199093

+ 1199091)

times erfc( 1199092

2radic1199095

+ radic(1199094

1199093

+ 1199091)1199095)


1199094

1199093

+ 1199091)

times erfc( 1199092

2radic1199095

minus radic(1199094

1199093

+ 1199091)1199095)]

minus2


1199092

3

41199095

+ 11990911199095))

(A1)

B


119878119888

119875119903minus 119878119888

+ 119875119903minus 119878119888]

1198602= minus

1 minus 119878119888

119878119888119870119888minus 119872 minus (1119870

119901)

times [119873 +


119901))

119878119888119870119888(1 minus 119875

119903) + (119872 + (1119870

119901)) (119875119903minus 119878119888)

]

1198603=

1 minus 119875119903

119872 + (1119870119901)

times [1 +


119901))

119878119888119870119888(1 minus 119875

119903) + (119872 + (1119870

119901)) (119875119903minus 119878119888)

]

1198604=

119863lowast119875119903(119875119903minus 119878119888)

119878119888119870119888(1 minus 119875

119903) + (119872 + (1119870

119901)) (119875119903minus 119878119888)

1198605=


119878119888119870119888minus 119872 minus (1119870

119901)

1198606=

119863lowast+ 1

119872 + (1119870119901)

1198607=

119870119888(119863lowast+ 1) minus (119872 + (1119870

119901)) (119873 + 1)

(119872 + (1119870119901)) (119870119888minus 119872 minus (1119870

119901))

1198608=


119870119888minus 119872 minus (1119870

119901)

1198609=

minus119878119888119870119888(119863lowast+ 1) + (119872 + (1119870

119901)) (119873 + 1)

(119872 + (1119870119901)) (119878119888119870119888minus 119872 minus (1119870

119901))

References






























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



8 Journal of Fluids

1000

500

0

minus500

minus1000

minus1500

minus2000

u

0 02 04 06 08 1 12y

Sc = 5 Kc = 0 Kp = 100

Sc = 150 Kc = 0 Kp = 100

Sc = 200 Kc = 0 Kp = 100

Sc = 200 Kc = 0 Kp = 05


lowast= 2

0 02 04 06 08 1 12y

035

03

025

02

015

01

005

0

u

VIII

VI

I II III VII

V

IV

Curve

I 05 100 1 2 2 100

II 05 022 2 2 2 100

III 05 152 1 2 2 100

IV 05 022 2 2 2 05

V 2 022 2 2 2 100

VI 05 022 2 2 30 100

VII 05 03 1 2 2 100VIII 05 022 2 4 2 100

M Sc Kc N KpDlowast












119888 that is 119878

119888= 150 and

119878119888


119888lt 10 and 119870




Journal of Fluids 9

CurveI 04 022 20 700II 04 022 00 700III

III

04 00 700IV 04 20 700V 02 20 700VI 04 022 00 071VII 04 022 02 071VII 04 00 071IX 04 02 071X 02 02 071

t Sc Kc Pr

6

5

4

3

2

1

00 02 04 06 08 1

y

T

V I II

IV

VI-X

WaterAir

030030030

030030030


119903= 70 (water) and 119875

119903= 071 (air)

0

02

04

06

08

1

12

0 02 04 06 08 1

T

y

IIII

II

Curve t Kc

I 04 20II 04 10III 02 20


119888= 1)















119903 119905 and


119903= 70 (water)





002040608

1121416

0 02 04 06 08 1

T

y

IV

II

I

III

Curve Sc Kc Dlowast

I 022 2 2II 030 2 2III 022 1 2IV 022 2 30


CurveI 04 100 0II 04 617 0III 04 100 2IV 04 617 2V 02 100 2

t Kc

minus02

0

02

04

06

08

1

0 02 04 06 08 1

C

I

III

II IVV

y

Sc


0010203040506070809

1

0 02 04 06 08 1

C

y

III

III

Curve t Kc

I 04 20II 04 10III 02 20




= 1 and 119878119888


minus02

0

02

04

06

08

1

0 02 04 06 08 1

C

y

Curve Sc Kc

I 100 2II 617 1III 617 2





119888(ie 119878










Sl no 119905 119872 119878119888

119870119888

119873 119875119903

119863lowast

119870119901

1205910

1205911



Sl no 119872 119878119888

119870119888

119873 119863lowast

119870119901

1205910

1205911

1 05 022 2 2 2 100 1732053 09325152 2 022 2 2 2 100 166425 07954363 05 03 2 2 2 100 1732052 09325154 05 022 1 2 2 100 173205 09325155 05 022 2 4 2 100 1664279 08692876 05 022 2 2 4 100 2144507 13763827 05 022 2 2 2 05 142312 0524363


Sl no 119905 119878119888

119870119888

119875119903

119863lowast Nu

0Nu1





119888 119870119888 and 119863





119888





Sl no 119905 119878119888

119870119888

Sh0

Sh1

1 04 100 2 59757644 minus06128007882 02 100 2 6313751515 0158384443 04 617 2 7026366229 04705642814 02 100 0 567128182 minus0466307658


119888 and 119870




119888increases





Sl no 119905 119872 119878119888

119870c 119873 119875119903

119863lowast

119870119901

119876

1 04 05 022 2 2 18 2 100 01978362 02 05 022 2 2 18 2 100 01765723 04 2 022 2 2 18 2 100 00576554 04 05 03 2 2 18 2 100 05684755 04 05 022 1 2 18 2 100 00911936 04 05 022 2 4 18 2 100 01666677 04 05 022 2 2 2 2 100 02649398 04 05 022 2 2 18 4 100 03645839 04 05 022 2 2 18 2 05 0034241


Sl no 119872 119878119888

119870119888

119873 119863lowast

119870119901

119876

1 05 022 2 2 2 100 01190522 2 022 2 2 2 100 01042073 05 03 2 2 2 100 01190524 05 022 1 2 2 100 01190525 05 022 2 4 2 100 01950776 05 022 2 2 4 100 01223927 05 022 2 2 2 05 0094632







119901 In case of 119878

119888



119888and 119863


5 Conclusion











Appendices

A


erfc (x) =2

radic120587int

infin

119909

119890minus1205822

119889120582

119891 (1199091 1199092 1199093 1199094 1199095)

=1

2exp(

11990941199095

1199093

)


1199094

1199093

+ 1199091)


times erfc( 1199092

2radic1199095

minus radic(1199094

1199093

+ 1199091)1199095)

+ exp(1199092radic

1199094

1199093

+ 1199091)

times erfc( 1199092

2radic1199095

+ radic(1199094

1199093

+ 1199091)1199095)]

119892 (1199091 1199092 1199093 1199094 1199095)

=1

2radic

1199094

1199093

+ 1199091exp(

11990941199095

1199093

)


1199094

1199093

+ 1199091)

times erfc( 1199092

2radic1199095

+ radic(1199094

1199093

+ 1199091)1199095)


1199094

1199093

+ 1199091)

times erfc( 1199092

2radic1199095

minus radic(1199094

1199093

+ 1199091)1199095)]

minus2


1199092

3

41199095

+ 11990911199095))

(A1)

B


119878119888

119875119903minus 119878119888

+ 119875119903minus 119878119888]

1198602= minus

1 minus 119878119888

119878119888119870119888minus 119872 minus (1119870

119901)

times [119873 +


119901))

119878119888119870119888(1 minus 119875

119903) + (119872 + (1119870

119901)) (119875119903minus 119878119888)

]

1198603=

1 minus 119875119903

119872 + (1119870119901)

times [1 +


119901))

119878119888119870119888(1 minus 119875

119903) + (119872 + (1119870

119901)) (119875119903minus 119878119888)

]

1198604=

119863lowast119875119903(119875119903minus 119878119888)

119878119888119870119888(1 minus 119875

119903) + (119872 + (1119870

119901)) (119875119903minus 119878119888)

1198605=


119878119888119870119888minus 119872 minus (1119870

119901)

1198606=

119863lowast+ 1

119872 + (1119870119901)

1198607=

119870119888(119863lowast+ 1) minus (119872 + (1119870

119901)) (119873 + 1)

(119872 + (1119870119901)) (119870119888minus 119872 minus (1119870

119901))

1198608=


119870119888minus 119872 minus (1119870

119901)

1198609=

minus119878119888119870119888(119863lowast+ 1) + (119872 + (1119870

119901)) (119873 + 1)

(119872 + (1119870119901)) (119878119888119870119888minus 119872 minus (1119870

119901))

References






























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Journal of Fluids 9

CurveI 04 022 20 700II 04 022 00 700III

III

04 00 700IV 04 20 700V 02 20 700VI 04 022 00 071VII 04 022 02 071VII 04 00 071IX 04 02 071X 02 02 071

t Sc Kc Pr

6

5

4

3

2

1

00 02 04 06 08 1

y

T

V I II

IV

VI-X

WaterAir

030030030

030030030


119903= 70 (water) and 119875

119903= 071 (air)

0

02

04

06

08

1

12

0 02 04 06 08 1

T

y

IIII

II

Curve t Kc

I 04 20II 04 10III 02 20


119888= 1)















119903 119905 and


119903= 70 (water)





002040608

1121416

0 02 04 06 08 1

T

y

IV

II

I

III

Curve Sc Kc Dlowast

I 022 2 2II 030 2 2III 022 1 2IV 022 2 30


CurveI 04 100 0II 04 617 0III 04 100 2IV 04 617 2V 02 100 2

t Kc

minus02

0

02

04

06

08

1

0 02 04 06 08 1

C

I

III

II IVV

y

Sc


0010203040506070809

1

0 02 04 06 08 1

C

y

III

III

Curve t Kc

I 04 20II 04 10III 02 20




= 1 and 119878119888


minus02

0

02

04

06

08

1

0 02 04 06 08 1

C

y

Curve Sc Kc

I 100 2II 617 1III 617 2





119888(ie 119878










Sl no 119905 119872 119878119888

119870119888

119873 119875119903

119863lowast

119870119901

1205910

1205911



Sl no 119872 119878119888

119870119888

119873 119863lowast

119870119901

1205910

1205911

1 05 022 2 2 2 100 1732053 09325152 2 022 2 2 2 100 166425 07954363 05 03 2 2 2 100 1732052 09325154 05 022 1 2 2 100 173205 09325155 05 022 2 4 2 100 1664279 08692876 05 022 2 2 4 100 2144507 13763827 05 022 2 2 2 05 142312 0524363


Sl no 119905 119878119888

119870119888

119875119903

119863lowast Nu

0Nu1





119888 119870119888 and 119863





119888





Sl no 119905 119878119888

119870119888

Sh0

Sh1

1 04 100 2 59757644 minus06128007882 02 100 2 6313751515 0158384443 04 617 2 7026366229 04705642814 02 100 0 567128182 minus0466307658


119888 and 119870




119888increases





Sl no 119905 119872 119878119888

119870c 119873 119875119903

119863lowast

119870119901

119876

1 04 05 022 2 2 18 2 100 01978362 02 05 022 2 2 18 2 100 01765723 04 2 022 2 2 18 2 100 00576554 04 05 03 2 2 18 2 100 05684755 04 05 022 1 2 18 2 100 00911936 04 05 022 2 4 18 2 100 01666677 04 05 022 2 2 2 2 100 02649398 04 05 022 2 2 18 4 100 03645839 04 05 022 2 2 18 2 05 0034241


Sl no 119872 119878119888

119870119888

119873 119863lowast

119870119901

119876

1 05 022 2 2 2 100 01190522 2 022 2 2 2 100 01042073 05 03 2 2 2 100 01190524 05 022 1 2 2 100 01190525 05 022 2 4 2 100 01950776 05 022 2 2 4 100 01223927 05 022 2 2 2 05 0094632







119901 In case of 119878

119888



119888and 119863


5 Conclusion











Appendices

A


erfc (x) =2

radic120587int

infin

119909

119890minus1205822

119889120582

119891 (1199091 1199092 1199093 1199094 1199095)

=1

2exp(

11990941199095

1199093

)


1199094

1199093

+ 1199091)


times erfc( 1199092

2radic1199095

minus radic(1199094

1199093

+ 1199091)1199095)

+ exp(1199092radic

1199094

1199093

+ 1199091)

times erfc( 1199092

2radic1199095

+ radic(1199094

1199093

+ 1199091)1199095)]

119892 (1199091 1199092 1199093 1199094 1199095)

=1

2radic

1199094

1199093

+ 1199091exp(

11990941199095

1199093

)


1199094

1199093

+ 1199091)

times erfc( 1199092

2radic1199095

+ radic(1199094

1199093

+ 1199091)1199095)


1199094

1199093

+ 1199091)

times erfc( 1199092

2radic1199095

minus radic(1199094

1199093

+ 1199091)1199095)]

minus2


1199092

3

41199095

+ 11990911199095))

(A1)

B


119878119888

119875119903minus 119878119888

+ 119875119903minus 119878119888]

1198602= minus

1 minus 119878119888

119878119888119870119888minus 119872 minus (1119870

119901)

times [119873 +


119901))

119878119888119870119888(1 minus 119875

119903) + (119872 + (1119870

119901)) (119875119903minus 119878119888)

]

1198603=

1 minus 119875119903

119872 + (1119870119901)

times [1 +


119901))

119878119888119870119888(1 minus 119875

119903) + (119872 + (1119870

119901)) (119875119903minus 119878119888)

]

1198604=

119863lowast119875119903(119875119903minus 119878119888)

119878119888119870119888(1 minus 119875

119903) + (119872 + (1119870

119901)) (119875119903minus 119878119888)

1198605=


119878119888119870119888minus 119872 minus (1119870

119901)

1198606=

119863lowast+ 1

119872 + (1119870119901)

1198607=

119870119888(119863lowast+ 1) minus (119872 + (1119870

119901)) (119873 + 1)

(119872 + (1119870119901)) (119870119888minus 119872 minus (1119870

119901))

1198608=


119870119888minus 119872 minus (1119870

119901)

1198609=

minus119878119888119870119888(119863lowast+ 1) + (119872 + (1119870

119901)) (119873 + 1)

(119872 + (1119870119901)) (119878119888119870119888minus 119872 minus (1119870

119901))

References






























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




002040608

1121416

0 02 04 06 08 1

T

y

IV

II

I

III

Curve Sc Kc Dlowast

I 022 2 2II 030 2 2III 022 1 2IV 022 2 30


CurveI 04 100 0II 04 617 0III 04 100 2IV 04 617 2V 02 100 2

t Kc

minus02

0

02

04

06

08

1

0 02 04 06 08 1

C

I

III

II IVV

y

Sc


0010203040506070809

1

0 02 04 06 08 1

C

y

III

III

Curve t Kc

I 04 20II 04 10III 02 20




= 1 and 119878119888


minus02

0

02

04

06

08

1

0 02 04 06 08 1

C

y

Curve Sc Kc

I 100 2II 617 1III 617 2





119888(ie 119878










Sl no 119905 119872 119878119888

119870119888

119873 119875119903

119863lowast

119870119901

1205910

1205911



Sl no 119872 119878119888

119870119888

119873 119863lowast

119870119901

1205910

1205911

1 05 022 2 2 2 100 1732053 09325152 2 022 2 2 2 100 166425 07954363 05 03 2 2 2 100 1732052 09325154 05 022 1 2 2 100 173205 09325155 05 022 2 4 2 100 1664279 08692876 05 022 2 2 4 100 2144507 13763827 05 022 2 2 2 05 142312 0524363


Sl no 119905 119878119888

119870119888

119875119903

119863lowast Nu

0Nu1





119888 119870119888 and 119863





119888





Sl no 119905 119878119888

119870119888

Sh0

Sh1

1 04 100 2 59757644 minus06128007882 02 100 2 6313751515 0158384443 04 617 2 7026366229 04705642814 02 100 0 567128182 minus0466307658


119888 and 119870




119888increases





Sl no 119905 119872 119878119888

119870c 119873 119875119903

119863lowast

119870119901

119876

1 04 05 022 2 2 18 2 100 01978362 02 05 022 2 2 18 2 100 01765723 04 2 022 2 2 18 2 100 00576554 04 05 03 2 2 18 2 100 05684755 04 05 022 1 2 18 2 100 00911936 04 05 022 2 4 18 2 100 01666677 04 05 022 2 2 2 2 100 02649398 04 05 022 2 2 18 4 100 03645839 04 05 022 2 2 18 2 05 0034241


Sl no 119872 119878119888

119870119888

119873 119863lowast

119870119901

119876

1 05 022 2 2 2 100 01190522 2 022 2 2 2 100 01042073 05 03 2 2 2 100 01190524 05 022 1 2 2 100 01190525 05 022 2 4 2 100 01950776 05 022 2 2 4 100 01223927 05 022 2 2 2 05 0094632







119901 In case of 119878

119888



119888and 119863


5 Conclusion











Appendices

A


erfc (x) =2

radic120587int

infin

119909

119890minus1205822

119889120582

119891 (1199091 1199092 1199093 1199094 1199095)

=1

2exp(

11990941199095

1199093

)


1199094

1199093

+ 1199091)


times erfc( 1199092

2radic1199095

minus radic(1199094

1199093

+ 1199091)1199095)

+ exp(1199092radic

1199094

1199093

+ 1199091)

times erfc( 1199092

2radic1199095

+ radic(1199094

1199093

+ 1199091)1199095)]

119892 (1199091 1199092 1199093 1199094 1199095)

=1

2radic

1199094

1199093

+ 1199091exp(

11990941199095

1199093

)


1199094

1199093

+ 1199091)

times erfc( 1199092

2radic1199095

+ radic(1199094

1199093

+ 1199091)1199095)


1199094

1199093

+ 1199091)

times erfc( 1199092

2radic1199095

minus radic(1199094

1199093

+ 1199091)1199095)]

minus2


1199092

3

41199095

+ 11990911199095))

(A1)

B


119878119888

119875119903minus 119878119888

+ 119875119903minus 119878119888]

1198602= minus

1 minus 119878119888

119878119888119870119888minus 119872 minus (1119870

119901)

times [119873 +


119901))

119878119888119870119888(1 minus 119875

119903) + (119872 + (1119870

119901)) (119875119903minus 119878119888)

]

1198603=

1 minus 119875119903

119872 + (1119870119901)

times [1 +


119901))

119878119888119870119888(1 minus 119875

119903) + (119872 + (1119870

119901)) (119875119903minus 119878119888)

]

1198604=

119863lowast119875119903(119875119903minus 119878119888)

119878119888119870119888(1 minus 119875

119903) + (119872 + (1119870

119901)) (119875119903minus 119878119888)

1198605=


119878119888119870119888minus 119872 minus (1119870

119901)

1198606=

119863lowast+ 1

119872 + (1119870119901)

1198607=

119870119888(119863lowast+ 1) minus (119872 + (1119870

119901)) (119873 + 1)

(119872 + (1119870119901)) (119870119888minus 119872 minus (1119870

119901))

1198608=


119870119888minus 119872 minus (1119870

119901)

1198609=

minus119878119888119870119888(119863lowast+ 1) + (119872 + (1119870

119901)) (119873 + 1)

(119872 + (1119870119901)) (119878119888119870119888minus 119872 minus (1119870

119901))

References






























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





Sl no 119905 119872 119878119888

119870119888

119873 119875119903

119863lowast

119870119901

1205910

1205911



Sl no 119872 119878119888

119870119888

119873 119863lowast

119870119901

1205910

1205911

1 05 022 2 2 2 100 1732053 09325152 2 022 2 2 2 100 166425 07954363 05 03 2 2 2 100 1732052 09325154 05 022 1 2 2 100 173205 09325155 05 022 2 4 2 100 1664279 08692876 05 022 2 2 4 100 2144507 13763827 05 022 2 2 2 05 142312 0524363


Sl no 119905 119878119888

119870119888

119875119903

119863lowast Nu

0Nu1





119888 119870119888 and 119863





119888





Sl no 119905 119878119888

119870119888

Sh0

Sh1

1 04 100 2 59757644 minus06128007882 02 100 2 6313751515 0158384443 04 617 2 7026366229 04705642814 02 100 0 567128182 minus0466307658


119888 and 119870




119888increases





Sl no 119905 119872 119878119888

119870c 119873 119875119903

119863lowast

119870119901

119876

1 04 05 022 2 2 18 2 100 01978362 02 05 022 2 2 18 2 100 01765723 04 2 022 2 2 18 2 100 00576554 04 05 03 2 2 18 2 100 05684755 04 05 022 1 2 18 2 100 00911936 04 05 022 2 4 18 2 100 01666677 04 05 022 2 2 2 2 100 02649398 04 05 022 2 2 18 4 100 03645839 04 05 022 2 2 18 2 05 0034241


Sl no 119872 119878119888

119870119888

119873 119863lowast

119870119901

119876

1 05 022 2 2 2 100 01190522 2 022 2 2 2 100 01042073 05 03 2 2 2 100 01190524 05 022 1 2 2 100 01190525 05 022 2 4 2 100 01950776 05 022 2 2 4 100 01223927 05 022 2 2 2 05 0094632







119901 In case of 119878

119888



119888and 119863


5 Conclusion











Appendices

A


erfc (x) =2

radic120587int

infin

119909

119890minus1205822

119889120582

119891 (1199091 1199092 1199093 1199094 1199095)

=1

2exp(

11990941199095

1199093

)


1199094

1199093

+ 1199091)


times erfc( 1199092

2radic1199095

minus radic(1199094

1199093

+ 1199091)1199095)

+ exp(1199092radic

1199094

1199093

+ 1199091)

times erfc( 1199092

2radic1199095

+ radic(1199094

1199093

+ 1199091)1199095)]

119892 (1199091 1199092 1199093 1199094 1199095)

=1

2radic

1199094

1199093

+ 1199091exp(

11990941199095

1199093

)


1199094

1199093

+ 1199091)

times erfc( 1199092

2radic1199095

+ radic(1199094

1199093

+ 1199091)1199095)


1199094

1199093

+ 1199091)

times erfc( 1199092

2radic1199095

minus radic(1199094

1199093

+ 1199091)1199095)]

minus2


1199092

3

41199095

+ 11990911199095))

(A1)

B


119878119888

119875119903minus 119878119888

+ 119875119903minus 119878119888]

1198602= minus

1 minus 119878119888

119878119888119870119888minus 119872 minus (1119870

119901)

times [119873 +


119901))

119878119888119870119888(1 minus 119875

119903) + (119872 + (1119870

119901)) (119875119903minus 119878119888)

]

1198603=

1 minus 119875119903

119872 + (1119870119901)

times [1 +


119901))

119878119888119870119888(1 minus 119875

119903) + (119872 + (1119870

119901)) (119875119903minus 119878119888)

]

1198604=

119863lowast119875119903(119875119903minus 119878119888)

119878119888119870119888(1 minus 119875

119903) + (119872 + (1119870

119901)) (119875119903minus 119878119888)

1198605=


119878119888119870119888minus 119872 minus (1119870

119901)

1198606=

119863lowast+ 1

119872 + (1119870119901)

1198607=

119870119888(119863lowast+ 1) minus (119872 + (1119870

119901)) (119873 + 1)

(119872 + (1119870119901)) (119870119888minus 119872 minus (1119870

119901))

1198608=


119870119888minus 119872 minus (1119870

119901)

1198609=

minus119878119888119870119888(119863lowast+ 1) + (119872 + (1119870

119901)) (119873 + 1)

(119872 + (1119870119901)) (119878119888119870119888minus 119872 minus (1119870

119901))

References






























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





Sl no 119905 119872 119878119888

119870c 119873 119875119903

119863lowast

119870119901

119876

1 04 05 022 2 2 18 2 100 01978362 02 05 022 2 2 18 2 100 01765723 04 2 022 2 2 18 2 100 00576554 04 05 03 2 2 18 2 100 05684755 04 05 022 1 2 18 2 100 00911936 04 05 022 2 4 18 2 100 01666677 04 05 022 2 2 2 2 100 02649398 04 05 022 2 2 18 4 100 03645839 04 05 022 2 2 18 2 05 0034241


Sl no 119872 119878119888

119870119888

119873 119863lowast

119870119901

119876

1 05 022 2 2 2 100 01190522 2 022 2 2 2 100 01042073 05 03 2 2 2 100 01190524 05 022 1 2 2 100 01190525 05 022 2 4 2 100 01950776 05 022 2 2 4 100 01223927 05 022 2 2 2 05 0094632







119901 In case of 119878

119888



119888and 119863


5 Conclusion











Appendices

A


erfc (x) =2

radic120587int

infin

119909

119890minus1205822

119889120582

119891 (1199091 1199092 1199093 1199094 1199095)

=1

2exp(

11990941199095

1199093

)


1199094

1199093

+ 1199091)


times erfc( 1199092

2radic1199095

minus radic(1199094

1199093

+ 1199091)1199095)

+ exp(1199092radic

1199094

1199093

+ 1199091)

times erfc( 1199092

2radic1199095

+ radic(1199094

1199093

+ 1199091)1199095)]

119892 (1199091 1199092 1199093 1199094 1199095)

=1

2radic

1199094

1199093

+ 1199091exp(

11990941199095

1199093

)


1199094

1199093

+ 1199091)

times erfc( 1199092

2radic1199095

+ radic(1199094

1199093

+ 1199091)1199095)


1199094

1199093

+ 1199091)

times erfc( 1199092

2radic1199095

minus radic(1199094

1199093

+ 1199091)1199095)]

minus2


1199092

3

41199095

+ 11990911199095))

(A1)

B


119878119888

119875119903minus 119878119888

+ 119875119903minus 119878119888]

1198602= minus

1 minus 119878119888

119878119888119870119888minus 119872 minus (1119870

119901)

times [119873 +


119901))

119878119888119870119888(1 minus 119875

119903) + (119872 + (1119870

119901)) (119875119903minus 119878119888)

]

1198603=

1 minus 119875119903

119872 + (1119870119901)

times [1 +


119901))

119878119888119870119888(1 minus 119875

119903) + (119872 + (1119870

119901)) (119875119903minus 119878119888)

]

1198604=

119863lowast119875119903(119875119903minus 119878119888)

119878119888119870119888(1 minus 119875

119903) + (119872 + (1119870

119901)) (119875119903minus 119878119888)

1198605=


119878119888119870119888minus 119872 minus (1119870

119901)

1198606=

119863lowast+ 1

119872 + (1119870119901)

1198607=

119870119888(119863lowast+ 1) minus (119872 + (1119870

119901)) (119873 + 1)

(119872 + (1119870119901)) (119870119888minus 119872 minus (1119870

119901))

1198608=


119870119888minus 119872 minus (1119870

119901)

1198609=

minus119878119888119870119888(119863lowast+ 1) + (119872 + (1119870

119901)) (119873 + 1)

(119872 + (1119870119901)) (119878119888119870119888minus 119872 minus (1119870

119901))

References






























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




times erfc( 1199092

2radic1199095

minus radic(1199094

1199093

+ 1199091)1199095)

+ exp(1199092radic

1199094

1199093

+ 1199091)

times erfc( 1199092

2radic1199095

+ radic(1199094

1199093

+ 1199091)1199095)]

119892 (1199091 1199092 1199093 1199094 1199095)

=1

2radic

1199094

1199093

+ 1199091exp(

11990941199095

1199093

)


1199094

1199093

+ 1199091)

times erfc( 1199092

2radic1199095

+ radic(1199094

1199093

+ 1199091)1199095)


1199094

1199093

+ 1199091)

times erfc( 1199092

2radic1199095

minus radic(1199094

1199093

+ 1199091)1199095)]

minus2


1199092

3

41199095

+ 11990911199095))

(A1)

B


119878119888

119875119903minus 119878119888

+ 119875119903minus 119878119888]

1198602= minus

1 minus 119878119888

119878119888119870119888minus 119872 minus (1119870

119901)

times [119873 +


119901))

119878119888119870119888(1 minus 119875

119903) + (119872 + (1119870

119901)) (119875119903minus 119878119888)

]

1198603=

1 minus 119875119903

119872 + (1119870119901)

times [1 +


119901))

119878119888119870119888(1 minus 119875

119903) + (119872 + (1119870

119901)) (119875119903minus 119878119888)

]

1198604=

119863lowast119875119903(119875119903minus 119878119888)

119878119888119870119888(1 minus 119875

119903) + (119872 + (1119870

119901)) (119875119903minus 119878119888)

1198605=


119878119888119870119888minus 119872 minus (1119870

119901)

1198606=

119863lowast+ 1

119872 + (1119870119901)

1198607=

119870119888(119863lowast+ 1) minus (119872 + (1119870

119901)) (119873 + 1)

(119872 + (1119870119901)) (119870119888minus 119872 minus (1119870

119901))

1198608=


119870119888minus 119872 minus (1119870

119901)

1198609=

minus119878119888119870119888(119863lowast+ 1) + (119872 + (1119870

119901)) (119873 + 1)

(119872 + (1119870119901)) (119878119888119870119888minus 119872 minus (1119870

119901))

References






























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



research article free convection heat and mass transfer...

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