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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 346281 13 pageshttpdxdoiorg1011552013346281

Research ArticleHeat and Mass Transfer with Free Convection MHD Flow Pasta Vertical Plate Embedded in a Porous Medium

Farhad Ali Ilyas Khan Sharidan Shafie and Norzieha Musthapa

Department of Mathematics Faculty of Science Universiti Teknologi Malaysia (UTM) Skudai 81310 Johor Bahru Malaysia

Correspondence should be addressed to Sharidan Shafie ridafieyahoocom

Received 7 November 2012 Accepted 11 April 2013

Academic Editor Zhijun Zhang

Copyright copy 2013 Farhad Ali 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

An analysis to investigate the combined effects of heat and mass transfer on free convection unsteady magnetohydrodynamic(MHD) flow of viscous fluid embedded in a porous medium is presentedThe flow in the fluid is induced due to uniformmotion ofthe plate The dimensionless coupled linear partial differential equations are solved by using Laplace transform method The solu-tions that have been obtained are expressed in simple forms in terms of elementary function exp(sdot) and complementary error func-tion erf 119888(sdot) They satisfy the governing equations all imposed initial and boundary conditions and can immediately be reduced totheir limiting solutionsThe influence of various embedded flowparameters such as theHartmannnumber permeability parameterGrashof number dimensionless time Prandtl number chemical reaction parameter Schmidt number and Soret number is analyzedgraphically Numerical solutions for skin friction Nusselt number and Sherwood number are also obtained in tabular forms

1 Introduction

The process of heat transfer or heat and mass transfertogether occurs simultaneously in a moving fluid and playsan important role in the design of chemical processing equip-ment nuclear reactors and formation and dispersion of fogA detailed discussion on this topic can be found in Raptis [1]Kim and Fedorov [2] El-Arabawy [3] Takhar et al [4] Alamet al [5] Chaudhary and Arpita [6] Ferdows et al [7] Rajeshet al [8] Rajesh and Varma [9] Bakr [10] and the referencestherein Dass et al [11] considered themass transfer effects onflow past an impulsively started infinite isothermal verticalplate with constant mass flux Muthucumaraswamy et al [12]presented an exact solution to the problem of flow past animpulsively started infinite vertical plate in the presence ofuniform heat and mass flux at the plate using Laplace trans-form technique

Recently the free convection flow of magnetohydrody-namic fluid has attractedmany researchers in view of its num-erous applications in geophysics astrophysics meteorologyaerodynamics magnetohydrodynamic power generators andpumps boundary layer controlenergy generators accelera-tors aerodynamics heating polymer technology petroleumindustry purification of crude oil and in material processing

such as extrusion metal forming continuous casting wireand glass fibre drawing Further the convective flow throughporousmediumhas applications in the field of chemical engi-neering for filtration and purification processes In petroleumtechnology it is used to study the movement of natural gasoil and water through oil channels or reservoirs and in thefield of agriculture engineering to study the undergroundwater resources (see eg Hayat and Abbas [13] Rahman andSattar [14] Kim [15] Kaviany [16] Vafai and Tien [17] Jhaand Apere [18] Mandal et al [19] Katagiri [20]) In view ofsuch applications Chaudhary and Jain [21] analyzed themag-netohydrodynamic free convection flow past an acceleratedsurface embedded in a porous medium and obtained theexact solutions for the velocity temperature and concen-tration fields using Laplace transform method Seth et al[22] investigated the unsteady MHD natural convection flowwith radiative heat transfer past an impulsively moving platewith ramped wall temperature Toki and Tokis [23] obtainedthe exact solutions for the unsteady free convection flowson a porous plate with time depending heating Toki [24]developed the analytical solutions for free convection andmass transfer flow near a moving vertical porous plate Das[25] developed the closed form solutions for the unsteadyMHD free convection flow with thermal radiation and mass

2 Mathematical Problems in Engineering

transfer over a moving vertical plate In this continuation theeffect of heatmass transfer on unsteadyMHD free convectionflow past a moving vertical plate in a porous medium wasinvestigated by Das and Jana [26] They considered theimpulsive uniform and oscillating motions of the platewith constant heat and mass diffusion and developed theexact solutions using Laplace transform technique RecentlyOsman et al [27] analyzed the thermal radiation and chem-ical reaction effects on unsteady MHD free convection flowthrough a porous plate embedded in a porous medium withheat sourcesink and the closed form solutions are obtainedKhan et al [28] and Sparrow and Cess [29] analyzed theeffects of Hall current and mass transfer on the unsteadyMHD free convection flow in a porous channel The motionin fluid is induced to the external pressure gradient andthe closed form solutions for the velocity temperature andconcentration fields are obtained

Motivated by the above investigations the present paperaims to study the combined heat and mass effects on theunsteady MHD free convection flow of an incompressibleviscous fluid passing through a porous medium The flow inthe fluid is caused due to the uniform motion of the plateExact solutions are derived for the velocity distributions tem-perature and concentration fields by using Laplace transformtechnique and presented graphically for small as well as largetimes To the best of authorsrsquo knowledge this problem hasnot been studied before and the reported results are newThepresent study is of course of great practical and technologicalimportance for example in astrophysical regimes the pres-ence of planetary debris cosmic dust and so forth and createsa suspended porous medium saturated with plasma fluidsCombined buoyancy-generated heat and mass transfer dueto temperature and concentration variations with unsteadyMHD free convection flow in fluid-saturated porous mediahas several important applications in a variety of engineeringprocesses including heat exchanger devices petroleum reser-voirs chemical catalytic reactors solar energy porous watercollector systems and ceramic materials

This paper is organized as follows A brief descriptionof the problem formulation is given in Section 2 The exactsolutions for the uniformly uniform motion of the plate arederived in Section 3 The graphical results and discussionare provided in Section 4 The conclusions of the paper aregiven in Section 5 whereas some future recommendations areincluded in Section 6

2 Description of the Problem Formulation

Let us consider the unsteady one dimensional flow ofan incompressible and electrically conducting viscous fluidcaused due to the uniform motion of the plate The 119909lowast-axisis taken along the plate in the vertical direction and 119910lowast-axisis taken normal to the plate The electrically conducting fluidoccupies the porous half space 119910lowast gt 0 A uniform magneticfield B

0is acting in the transverse direction to the flow

The magnetic Reynolds number is assumed to be small andtherefore the induced magnetic field is negligible comparedwith the applied magnetic field The applied magnetic fieldis also taken weak so that Hall and ion slip effects may be

neglected Initially both the plate and fluid are at the sametemperature 119879

lowast

infinand concentration 119862lowast

infin At time 119905 = 0+

the plate begins to slide in its own plane and acceleratesagainst the gravitational field with uniform acceleration in119909lowast-direction Then the temperature and concentration level

are raised to 119879lowast119908and 119862lowast

119908as shown in Figure 1

The Soret and thermal buoyancy effects are also consid-ered In addition to the above assumptions we assume thatthe internal dissipation is absent and the usual Boussinesqapproximation is taken into consideration Moreover thepressure gradient in the flow direction is compensated by thegradient of the hydrostatic pressure gradient of the fluid Asa result the governing equations of momentum energy andconcentration are derived as follows

120597119906lowast

120597119905lowast= ]

1205972119906lowast

120597119910lowast2minus12059011986120119906lowast

120588minus]119906lowast

119870lowast

+ 119892120573 (119879lowast

minus 119879lowast

infin) + 119892120573

lowast

(119862lowast

minus 119862lowast

infin)

(1)

120588119888119901

120597119879lowast

120597119905lowast= 119896

1205972119879lowast


119903

120597119910lowast (2)

120597119862lowast

120597119905lowast= 119863

1205972

119862lowast

120597119910lowast2+119863119870119879

119879119898

1205972

119879lowast

120597119910lowast2minus 119870lowast

119903(119862lowast

minus 119862lowast

infin) (3)

with the following initial and boundary conditions

119905lowast

le 0 119906lowast

= 0 119879lowast

= 119879lowast

infin 119862lowast

= 119862lowast

infin

forall119910lowast

ge 0

119905lowast

gt 0 119906lowast

= 119891 (119905lowast

) 119879lowast

= 119879lowast

119908 119862lowast

= 119862lowast

119908

at 119910lowast = 0

119906lowast

997888rarr 0 119879lowast

997888rarr 119879lowast

infin 119862lowast


infin

as 119910lowast 997888rarr infin

(4)

where 119891(119905lowast) is the uniform acceleration of the plate 119909lowast and119910lowast (m) are the distances along and perpendicular to theplate 119905lowast (s) is the time 119906lowast (msminus1) denote the fluid velocityin the 119909lowast-direction 119879lowast (K) temperature 119879lowast

infin(K) temper-

ature far from the plate 119879lowast119908(K) temperature at the wall

119862lowast (molmminus3) are the species concentration 119862lowast119908(molmminus3)

surface concentration 119862lowastinfin(molmminus3) species concentration

far from the surface 120573 (1K) the volumetric coefficient ofthermal expansion 120573lowast (molmminus3)minus1 or (m3molminus1) is thevolumetric coefficient of expansion for concentration ] =

120583120588 (m2 sminus1) the kinematic viscosity 120583 (kgmminus1 sminus1) viscosity120588 (kgmminus3) the fluid density 119888

119901(kgminus1 Kminus1) is the specific heat

capacity 119902lowast119903the radiative heat flux in 119909lowast-direction119863 (m2 sminus1)

is mass diffusivity 119896 (Wmminus1 Kminus1) is the thermal conductivityof the fluid 120590 (Smminus1) the electrical conductivity of the fluid119870lowast gt 0 (m2) is the permeability of the porous medium 119879

119898

(K) is the mean fluid temperature 119879lowastinfinis the free stream tem-

perature 119862lowastinfin

is the free stream concentration of the species119870119879

is the thermal-diffusion ratio and 119870lowast119903

the chemicalreaction constantThe radiative heat flux term for an optically

Mathematical Problems in Engineering 3

119892

119879lowastinfin

119862lowastinfin

Momentum boundary layer

Thermal boundary layer

Concentration boundary layer

Porous medium

119909lowast

119911lowast

119910lowast

119879lowast119908

119862lowast119908

B0

Figure 1 Flow geometry and physical coordinate system

thin fluid is simplified by making use of the Rosselandapproximation (Sparrow and Cess [29])

119902lowast

119903= minus

4120590lowast

3119896lowast120597119879lowast4

120597119910lowast (5)

where 120590lowast (Wmminus2 Kminus4) is the Stefan-Boltzmann constant and119896lowast (mminus1) is themean absorption coefficient It is assumed thatthe temperature differences within the flow are sufficientlysmall such that the term 119879lowast

4

is expressed as the linear func-tion of temperature Thus expanding 119879lowast

4

about 119879lowastinfin

usingTaylor series expansion and neglecting higher order terms weget

119879lowast4

≊ 4119879lowast3

infin119879lowast

minus 3119879lowast4

infin (6)

From (5) and (6) (2) reduces to the following form

120588119888119901

120597119879lowast

120597119905lowast= 119896

1205972119879lowast

120597119910lowast2+16120590lowast119879lowast

3

infin

3119896lowast1205972119879lowast

120597119910lowast2 (7)

3 Flow due to Uniform Motion of the Plate

For uniform motion of the plate we take 119891(119905lowast) = 119860119905lowast anddefine the following dimensionless variables

119906 =119906lowast

(]119860)13 119910 = 119910

lowast

(119860

]2)13

119905 = 119905lowast

(1198602

])

13

120579 =119879lowast minus 119879lowast

infin

119879lowast119908minus 119879lowastinfin


infin


(8)

where 119860 with dimension 1198711198792 denotes the uniform accelera-tion of the plate in119909-direction119906 is the dimensionless velocity119910 dimensionless coordinate perpendicular to the plate 119905 is thedimensionless time 120579 is the dimensionless temperature and120601 is the dimensionless species concentration

Hence the governing equations in dimensionless form are

120597119906

120597119905=1205972119906

1205971199102minus 119867119906 + Gr120579 + Gm120601

119906 (0 119905) = 119905 119906 (infin 119905) = 0 119905 gt 0

119906 (119910 0) = 0 119910 ge 0

(9)

119865lowast120597120579

120597119905=1205972120579

1205971199102

120579 (0 119905) = 1 120579 (infin 119905) = 0 119905 gt 0

120579 (119910 0) = 0 119910 ge 0

(10)

120597120601

120597119905=

1

Sc1205972120601

1205971199102+ Sr120597

2120579

1205971199102minus 120574120601

120601 (0 119905) = 1 120601 (infin 119905) = 0 119905 gt 0

120601 (119910 0) = 0 119910 ge 0

(11)

where

1

119870=

]43

119870lowast11986023 119872

2

=12059011986120]13

12058811986023

Sr =119863119870119879(119879lowast119908minus 119879lowastinfin)

119879119898] (119862lowast119908minus 119862lowastinfin) Pr =

120583119888119901

119896

Gr =119892120573 (119879lowast119908minus 119879lowastinfin)

119860 Gm =

119892120573lowast (119862lowast119908minus 119862lowastinfin)

119860

120574 =119870lowast119903]13

11986023

119867 = 1198722

+1

119870 119877 =

16120590lowast

119879lowast3

infin

3119896119896lowast

119865lowast

=Pr

1 + 119877 Sc = ]

119863

(12)


Here119872 is a magnetic parameter called Hartmann number119870is the dimensionless permeability Sc is Schmidt number 119877 isRadiation parameter Gr is Grashof number and Sr is Soretnumber The well-posed problems defined by (9)ndash(11) will besolved by using the Laplace transform technique Hence theproblem in the transformed plane is given as

1198892119906

1198891199102minus (119867 + 119902) 119906 + Gr120579 + Gm120601 = 0

119906 (0 119902) =1

1199022 119906 (infin 119902) = 0

1198892120579

1198891199102minus 119865lowast

119902120579 = 0

120579 (0 119902) =1

119902 120579 (infin 119902) = 0

1198892120601

1198891199102minus (120574 + 119902) Sc120601 + Sc Sr119889

2120579

1198891199102= 0

120601 (0 119902) =1

119902 120601 (infin 119902) = 0

(13)

where 119902 is the Laplace transformation parameterThe solutions of (13) in the transformed 119902-plane are given

by

119906 (119910 119902) =1

1199022exp(minus119910radic119902 + 119867) +

1198868

119902exp(minus119910radic119902 + 119867)

minus1198869

119902 + 1198860

exp(minus119910radic119902 + 119867)

+11988610

119902 minus 119867lowast1

exp(minus119910radic119902 + 119867)

+11988611

119902 minus 119867lowastexp(minus119910radic119902 + 119867)

+1198865


exp(minus119910radic119865lowast119902)

minus11988611

119902 minus 119867lowastexp(minus119910radic119865lowast119902) + 119886

4

119902exp(minus119910radic119865lowast119902)

minus1198866

119902exp(minus119910radic(119902 + 120574) Sc)

+1198869

119902 + 1198860

exp(minus119910radic(119902 + 120574) Sc)

minus1198867



(14)

120579 (119910 119902) =1

119902exp(minus119910radic119865lowast119902) (15)

120601 (119910 119902) =1


+119878lowast


exp (minus119910radic(119902 + 120574) Sc)

minus119878lowast


exp (minus119910radic119865lowast119902)

(16)

where

Grlowast = Gr119865lowast minus 1

119867lowast

=119867

119865lowast minus 1

119867lowast

1=

120574Sc119865lowast minus Sc

119878lowast

=Sc Sr119865lowast

119865lowast minus 119878119888

1198860=120574Sc minus 119867Sc minus 1

1198861=

Gm119878lowast


1198862=

GmSc minus 1

1198863=Gm119878lowast

Sc minus 1 119886

4=Grlowast

119867lowast

1198865=

1198861


1198866=1198862

1198860

1198867=

1198863

119867lowast1+ 1198860

1198868= 1198866minus 1198864 119886

9= 1198866+ 1198867

11988610= 1198867minus 1198865 119886

11= 1198864+ 1198865

(17)

The inverse Laplace transform of (14)ndash(16) yields

119906 (119910 119905) = 1198681+ 11988681198682minus 11988691198683+ 119886101198684+ 119886111198685+ 11988651198686

minus 119886111198687+ 11988641198688minus 11988661198689+ 119886911986810minus 119886711986811

(18)

120579 (119910 119905) = 1198688 (19)

120601 (119910 119905) = 119878lowast

(11986811minus 1198686) + 1198689 (20)

with

1198681=1

2[(119905 minus

119910

2radic119867) 119890minus119910radic119867 erf 119888 (

119910

2radic119905minus radic119867119905)

+119890119910radic119867 erf 119888 (

119910

2radic119905+ radic119867119905)(119905 +

119910

2radic119867)]

1198682=1

2[119890minus119910radic119867 erf 119888 (

119910


+119890119910radic119867 erf 119888 (

119910

2radic119905+ radic119867119905)]

1198683=119890minus1198860119905

2[119890minus119910radic119867minus119886

0 erf 119888 (119910

2radic119905minus radic(119867 minus 119886

0) 119905)

+119890119910radic119867minus119886

0 erf 119888 (119910

2radic119905+ radic(119867 minus 119886

0) 119905)]

1198684=119890119867lowast

1119905

2[119890minus119910radic119867

lowast

1+119867 erf 119888 (

119910

2radic119905minus radic(119867lowast

1+ 119867) 119905)

+119890119910radic119867lowast

1+119867 erf 119888 (

119910

2radic119905+ radic(119867lowast

1+ 119867) 119905)]


1198685=119890119867lowast119905

2[119890minus119910radic119867

lowast+119867 erf 119888 (

119910

2radic119905minus radic(119867 + 119867lowast) 119905)

+119890119910radic119867lowast+119867 erf 119888 (

119910

2radic119905+ radic(119867 + 119867lowast) 119905)]

1198686=119890119867lowast

1119905

2[119890minus119910radic119865

lowast119867lowast

1 erf 119888 (119910radic119865lowast

2radic119905minus radic119867lowast

1119905)

+ 119890119910radic119865lowast119867lowast

1 erf 119888 (119910radic119865lowast

2radic119905+ radic119867lowast

1119905)]

1198687=119890119867lowast119905

2[119890minus119910radic119865

lowast119867lowast

erf 119888 (119910radic119865lowast

2radic119905minus radic119867lowast119905)

+119890119910radic119865lowast119867lowast


2radic119905+ radic119867lowast119905)]

1198688= erf 119888 (

119910radic119865lowast

2radic119905)

1198689=1

2[119890minus119910radic120574Sc erf 119888 (

119910radicSc2radic119905

minus radic120574119905)

+119890119910radic120574Sc erf 119888 (


+ radic120574119905)]

11986810=119890minus1198860119905

2[119890minus119910radicSc(120574minus119886

0) erf 119888 (


minus radic(120574 minus 1198860) 119905)

+ 119890119910radicSc(120574minus119886

0) erf 119888 (


+ radic(120574 minus 1198860) 119905)]

11986811=119890119867lowast

1119905

2[119890minus119910radicSc(119867lowast

1+120574) erf 119888 (


minus radic(119867lowast1+ 120574) 119905)

+ 119890119910radicSc(119867lowast

1+120574) erf 119888 (


+ radic(119867lowast1+ 120574) 119905)]

(21)

where erf 119888(119909) is the complementary error function It isimportant to note that the above solutions are valid for Pr = 1

and Sc = 1 The solutions for Pr = 1 and Sc = 1 can be easilyobtained by substituting Pr = Sc = 1 into (10) and (11) andrepeating the same process as discussed above

31 Skin-Friction The expression for skin-friction is given by

120591lowast

= minus120583120597119906lowast

120597119910lowast

10038161003816100381610038161003816100381610038161003816119910lowast=0 (22)

which in view of (8) reduces to

120591 = minus120597119906

120597119910 120591 =

120591lowast

1205881198802119900

(23)

Hence from (18) we get

120591 =1198868119890minus119867119905

radic120587119905minus11988611

radic119865lowast

radic120587119905+1198864

radic119865lowast

radic120587119905+1198865

radic119865lowast

radic120587119905minus1198866119890minus120574119905radicScradic120587119905

+119890minus119867119905radic119905

radic120587+119905radic119867

2(minus1 + erf (radic119867119905)) +

erf (radic119867119905)

2radic119867

+ 1198868

radic119867 erf (radic119867119905) + 119905radic119867

2(1 + erf (radic119867119905))

minus 1198866radicSc 120574 erf (radic120574119905) minus 119886

11119890119867lowast119905radic119867lowast119865lowast erf (radic119867lowast119905)

+ 1198865119890119867lowast

1119905radic119865lowast119867lowast

1erf (radic119867lowast

1119905)

minus1

211988611119890119867lowast119905

minus2119890minus119867119905minus119867

lowast119905

radic120587119905+ radic119867 +119867lowast

times (minus1 minus erf (radic(119867 + 119867lowast) 119905) + radic119867 +119867lowast)

times (1 minus erf (radic(119867 + 119867lowast) 119905))

minus1

211988610119890119867lowast

1119905

minus2119890minus(119867+119867

lowast

1)119905


1

times (minus1 minus erf (radic(119867 + 119867lowast1) 119905))

+radic119867 +119867lowast1(1 minus erf (radic(119867 + 119867lowast

1) 119905))

minus1

21198869119890minus1198860119905

21198901198860119905minus119867119905

radic120587119905minus radic119867 minus 119886

0

times (minus1 minus erf (radicminus1198860119905 + 119867119905))

minusradic119867 minus 1198860(1 minus erf (radicminus119886

0119905 + 119867119905))

minus1

21198869119890minus1198860119905

minus21198901198860119905minus120574119905radicSc

radic120587119905+ radic(minus119886

0+ 120574) Sc

times (minus1 minus erf (radic(minus1198860+ 120574) 119905))

+ radic(minus1198860+ 120574) Sc

times(1 minus erf (radic(minus1198860+ 120574) 119905))

minus1

21198867119890119867lowast

1119905

2119890minus(119867

lowast

1+120574)119905

radic120587119905minus radicScradic119867lowast

1+ 120574

times (minus1 minus erf (radic(119867lowast1+ 120574) 119905))


minus radicScradic119867lowast1+ 120574

times(1 minus erf (radic(119867lowast1+ 120574) 119905))

(24)

32 Nusselt Number The rate of heat transfer for the presentproblem is given as

Nu = minus119896120597119879lowast

120597119910lowast

10038161003816100381610038161003816100381610038161003816119910lowast=0

Nu = 120597120579

120597119910

10038161003816100381610038161003816100381610038161003816119910lowast=0

Nu = radic119865lowast

120587119905

(25)

33 Sherwood Number The rate of mass transfer is given by

Sh = 120597119862lowast

120597119910lowast

10038161003816100381610038161003816100381610038161003816119910lowast=0

Sh = minus1

radic120587119905119890minus120574119905

(minus119890120574119905radic119865lowast119878

lowast

+ radicSc + 119878lowastradicSc

minus119890119867lowast119905+120574119905

119878lowastradic120587119905119867lowast119865lowast erf (radic119867lowast119905))

+ 119890120574119905radic120574Sc120587119905 erf (radic120574119905) + 119890119867

lowast119905+120574119905

119878lowast

timesradic120587119905Scradic119867lowast + 120574 erf (radic(119867lowast + 120574) 119905)

(26)

It is important to note that solutions (18)ndash(20) satisfy allthe imposed boundary and initial conditions Further thesolutions obtained here are more general and the existingsolutions in the literature appeared as the limiting cases

(1) The present solutions given by (18)ndash(20) in theabsence of radiation effect and by taking the thermal-diffusion ratio (119870

119879) and the chemical reaction con-

stant (119870lowast119903) equal to zero reduce to the solutions of Das

and Jana [26] (see (42) (38) and (39))(2) The solutions (18)ndash(20) for the flow of optically thick

fluid in a nonporous medium with119870lowast119903= 119870119879= 0 give

the solutions of Das [25] (see (44) (31) and (32))

4 Graphical Results and Discussion

An exact analysis is presented to investigate the combinedeffects of heat mass transfer on the transient MHD free con-vective flow of an incompressible viscous fluid past a verticalplate moving with uniform motion and embedded in aporous medium The expressions for the velocity 119906 temper-ature 120579 and concentration 120601 are obtained by using Laplacetransform method In order to understand the physicalbehavior of the dimensionless parameters such as Hartmann

1

08

06

04

02

00 2 4 6 8 10 12 14

119910

119906

119872 = 0

119872 = 1119872 = 2119872 = 3

119905 = 1 119877 = 02 Pr = 071 Gr = 1119870 = 05 Gm = 1 120574 = 07 Sc = 2 Sr = 1

Figure 2 Velocity profiles for different values of119872

1

08

06

04

02

00 2 4 6 8 10 12 14

119910

119906

119870 = 05119870 = 10

119870 = 15119870 = 20

119905 = 1 119877 = 02 Pr = 071 Gr = 1119872 = 1 Gm = 1 120574 = 05 Sc = 2 Sr = 1

Figure 3 Velocity profiles for different values of 119870

number 119872 also called magnetic parameter permeabilityparameter 119870 Grashof number Gr dimensionless time 119905Prandtl number Pr radiation parameter 119877 chemical reactionparameter 120574 Schmidt number Sc and Soret number SrFigures 2ndash17 have been displayed for 119906 120579 and 120601

Figure 2 presents the velocity profile for different valuesof 119872 It is observed that the velocity and boundary layerthickness decreases upon increasing the Hartmann number119872 It is due to the fact that the application of transversemagnetic field results a resistive type force (called Lorentzforce) similar to drag force and upon increasing the values of119872 increases the drag force which leads to the deceleration ofthe flow Figure 3 is sketched in order to explore the variationsof permeability parameter 119870 It is found that the velocityincreases with increasing values of 119870 This is due to the factthat increasing values of 119870 reduces the drag force whichassists the fluid considerably to move fast The variation of


119905 = 1

119906

119910

25

2

15

1

05

00 2 4 6 8

Gm = 1119877 = 02 Pr = 071 Gr = 1119872 = 1 119870 = 2 120574 = 05 Sc = 06 Sr = 5

119905 = 15

119905 = 20

119905 = 25


12

1

08

06

04

02

0

119906

0 2 4 6 8 10 12 14119910

Gr = 0Gr = 1

Gr = 2Gr = 3

119905 = 1 119877 = 02119872 = 1 119870 = 2 120574 = 05 Sc = 2 Sr = 1

Pr = 071 Gm = 1

Figure 5 Velocity profiles for different values of Gr

1

08

06

04

02

0

119906

0 1 2 3 4 5 6119910

Sc = 022Sc = 03

Sc = 06Sc = 1

119905 = 1 119877 = 02 Pr = 071 Gm = 1119872 = 2 119870 = 1 120574 = 05 Gr = 2 Sr = 15

Figure 6 Velocity profiles for different values of Sc

175

15

125

1

075

05

025

00 5 10 15 20

119910

119906

Sr = 25Sr = 45

Sr = 65Sr = 8

119905 = 1 119877 = 02 Pr = 071 Gm = 1119872 = 2 119870 = 1 120574 = 05 Gr = 2 Sc = 15

Figure 7 Velocity profiles for different values of Sr

0 05 1 15 2119910

0

02

04

06

08

1

12

119906

Pr = 100Pr = 7

Pr = 1Pr = 071Pr = 0015

119905 = 1 119877 = 1 Gr = 1119872 = 1119870 = 4 Gm = 02 120574 = 15 Sc = 2 Sr = 01

Figure 8 Velocity profiles for different values of Pr

velocity for different values of dimensionless time 119905 is shownin Figure 4 It is noticed that velocity increases with increas-ing time Further this figure verifies the boundary conditionsof velocity given in (9) Initially velocity takes the values oftime and later for large values of 119910 and the velocity tends tozerowith increasing time It is observed fromFigure 5 that thefluid velocity increases with increasing Gr Figure 6 revealsthat velocity profiles decrease with the increase of Schmidtnumber Sc while an opposite phenomenon is observed incase of Soret number Sr as shown in Figure 7

Velocity temperature and concentration profiles forsome realistic values of Prandtl number Pr = 0015 071 10

70 100 which are important in the sense that they physicallycorrespond to mercury air electrolytic solution water andengine oil are shown in Figures 8ndash10 respectively From Fig-ure 8 it is found that the momentum boundary layer thick-ness increases for the fluids with Pr lt 1 and decreases for


1

08

06

04

02

0

119910

0 10 20 30 40 50

119905 = 06 119877 = 05

Pr = 0015Pr = 071Pr = 10

Pr = 70Pr = 100

120579

Figure 9 Temperature profiles for different values of Pr

12

1

08

06

04

02

0

119910

0 1 2 3 4 5 6

119905 = 1 119877 = 2 Sc = 1 Sr = 09

Pr = 0015Pr = 071Pr = 1

Pr = 7Pr = 100

120601

120574 = 02

Figure 10 Concentration profiles for different values of Pr

119905 = 01 Pr = 0711

08

06

04

02

00 05 1 15 2 25 3

119910

119877 = 05

119877 = 15

119877 = 30

119877 = 45

120579

Figure 11 Temperature profiles for different values of 119877

119905 = 03 119877 = 04 Sc = 09 Sr = 08 Pr = 10

0 05 1 15 2 25 3119910

1

08

06

04

02

0

120601

120574 = 0

120574 = 3

120574 = 6

120574 = 9

Figure 12 Concentration profiles for different values of 120574

119905 = 05 119877 = 04 Sr = 04 Pr = 071

Sc = 1Sc = 2

Sc = 3Sc = 4

1

08

06

04

02

0

120601

0 05 1 15 2 25 3119910

120574 = 02

Figure 13 Concentration profiles for different values of Sc

119905 = 03 119877 = 04 Sc = 03 Pr = 071

Sr = 0Sr = 1

Sr = 2Sr = 3

1

08

06

04

02

00 1 2 3 4

119910

120574 = 02

120601

Figure 14 Concentration profiles for different values of Sr


119910 = 5

119910 = 10

119910 = 15

119910 = 20

5

4

3

2

1

0

119906

0 1 2 3 4 5 6119905

Sr = 1119877 = 02 Pr = 071 Gm = 1119872 = 01 119870 = 2 120574 = 05 Gr = 2 Sc = 2


Pr = 071119877 = 31

08

06

04

02

00 2 4 6 8 10

119905

119910 = 0

119910 = 2

119910 = 3

119910 = 7

120579


Pr gt 1 The Prandtl number actually describes the relation-ship between momentum diffusivity and thermal diffusivityand hence controls the relative thickness of the momentumand thermal boundary layers When Pr is small that is Pr =0015 it is noticed that the heat diffuses very quickly com-pared to the velocity (momentum)Thismeans that for liquidmetals the thickness of the thermal boundary layer is muchbigger than the velocity boundary layer

In Figure 9 we observe that the temperature decreaseswith increasing values of Prandtl number Pr It is alsoobserved that the thermal boundary layer thickness is maxi-mum near the plate and decreases with increasing distancesfrom the leading edge and finally approaches to zero Further-more it is noticed that the thermal boundary layer formercury which corresponds to Pr = 0015 is greater thanthose for air electrolytic solution water and engine oilIt is justified due to the fact that thermal conductivity offluid decreases with increasing Prandtl number Pr and hence

119877 = 04 Γ = 07 Sc = 03 Sr = 05 Pr = 071

0 1 2 3 4 5 6119905

1

08

06

04

02

0

119910 = 0

119910 = 1

119910 = 2119910 = 5

120601


decreases the thermal boundary layer thickness and thetemperature profiles We observed from Figure 10 that theconcentration of the fluid increases for large values of Prandtlnumber Pr

The effects of radiation parameter 119877 on the temperatureprofiles are shown in Figure 11 It is found that the tempera-ture profiles 120579 being as a decreasing function of 119877 deceleratethe flow and reduce the fluid velocity Such an effect mayalso be expected as increasing radiation parameter 119877 makesthe fluid thick and ultimately causes the temperature and thethermal boundary layer thickness to decrease The influenceof 120574 Sc and Sr on the concentration profiles 120601 is shown inFigures 12ndash14 It is depicted from Figures 12 and 13 that theincreasing values of 120574 and Sc lead to fall in the concentrationprofiles Figure 14 depicts that the concentration profilesincrease when Soret number Sr is increased Furthermore weobserve that in the absence of Soret effects the concentrationprofile tends to a steady state in terms of 119910 this may be seenfrom (11) When Soret effects are present then at large timesthe solutal solution consists of this steady-state solution andan evolving ldquoparticular integralrdquo due to the presence of thetemperature term

An important aspect of the unsteady problem is that itdescribes the flow situation for small times (119905 ≪ 1) as well aslarge times (119905 rarr infin) Therefore the present solutions forvelocity distributions temperature and concentration pro-files are displayed for both small and large times (see Figures15ndash17) The velocity versus time graph for different values ofindependent variable119910 is plotted in Figure 15 It is found fromFigure 15 that the velocity decreases as independent variable119910 increases Further it is interesting to note that initiallywhen 119905 = 0 the fluid velocity is zero which is also true fromthe initial condition given in (9) However it is observedthat as time increases the velocity increases and after sometime of initiation this transition stops and the fluid motionbecomes independent of time and hence the solutions arecalled steady-state solutions This transition is smooth as wecan see from the graph On the other hand from the velocity


119905 = 04 119877 = 1 Gr = 1 Gm = 1

(IV)

(III)

(II)

(I)

119906

04

03

02

01

0

119910

0 05 1 15 2 25 3 35

Curves Pr 119872 119870 Curves Pr 119872 119870

(I) 071 2 01

(II) 7 2 01

(III) 071 4 01

(IV) 071 2 04

120574 = 2 Sc = 06 Sr = 1

Figure 18 Combined effect of various parameters on velocityprofiles

versus time graph for different values of the independentvariable 119910 (see Figure 15) it is found that the velocity at 119910 = 0is maximum and continuously decreases for large values of 119910It is further noted from this figure that for large values of 119910that is when 119910 rarr infin the velocity profile approaches to zeroA similar behavior was also expected in view of the bound-ary conditions given in (9) Hence this figure shows the cor-rectness for the obtained analytical result given by (18)

Similarly the next two Figures 16 and 17 are plottedto describe the transient and steady-state solutions whichinclude the effects of heating and mass diffusion It is clearfrom Figure 16 that the dimensionless temperature 120579 has itsmaximum value unity at 119910 = 0 and then decreasing forfurther large values of 119910 and ultimately approaches to zero Asimilar behavior was also expected due the fact that the tem-perature profile is 1 for 119910 = 0 and for large values of 119910 itsvalue approaches to 0 which is mathematically true in viewof the boundary conditions given in (10) From Figure 17 itis depicted that the variation of time on the concentrationprofile presents similar results as for the temperature profilein qualitative sense However these results are not the samequantitatively

A very important phenomenon to see the combinedeffects of the embedded flow parameters on the velocity tem-perature and concentration profiles is analyzed in Figures 18ndash21 Figure 18 is plotted to observe the combined effects of Pr119872 and119870 on velocity in case of cooling of the plate (Gr gt 0) asshown by Curves IndashIV Curves I amp II are sketched to displaythe effects of Pr on velocity The values of Prandtl numberare chosen as Pr = 071 (air) and Pr = 7 (water) whichare the most encountered fluids in nature and frequentlyused in engineering and industry We can from the compari-son of Curves I amp II that velocity decreases upon increasingPrandtl number Pr Curves I amp III present the influence ofHartmann number 119872 on velocity profiles It is clear from

(I)

(II)

(III)

(IV)

(V)

119906

02

015

01

005

0

119910

0 05 1 15 2 25 3

119905 = 04 119877 = 1 Pr = 071 119872 = 2 120574 = 2 119870 = 01

Curves Gr Gm Sc Sr Curves Gr Gm Sc Sr

(I)

(II)

(III)

(IV)

(V)

05

1

05

1

1

2

022

022

022

1

1

1

05

05

1

1

06

022

1

2

Figure 19 Combined effect of various parameters on velocity pro-files

1

08

06

04

02

00 5 10 15 20

119910

(I)

(II)

(III)

(IV)

071 05 2

1 05 2

(I)

(II)

Curves Pr 119877 119905

071 1 2

071 05 3

(III)

(IV)

Curves Pr 119877 119905

120579

Figure 20 Combined effect of various parameters on temperatureprofiles

these curves that velocity decreases when 119872 is increasedThe effect of permeability parameter 119870 on the velocity isquite different to that of 119872 This fact is shown from thecomparison of Curves I amp IV Figure 19 is plotted to showthe effects of Grashof number Gr modified Grashof numberGm Schmidth number Sc and Soret number Sr on velocityprofiles Curves I amp II show that velocity increases when Gris increased It is observed that the effect of Gm on velocityis the same as Gr This fact is shown from the comparison ofCurves I amp III The effect of Sc on velocity is shown from thecomparison of Curves I amp IV Here we choose the Schmidthnumber values as Sc = 022 and Sc = 06 which physically


1

08

06

04

02

0

(I)

(II)

(III)

(IV)

(V)

Curves Pr Sc Sr Curves Pr Sc Sr

(I)

(II)

(III)

(IV)

(V)

071

06

05

01

7

06

05

01

071

1

05

01 071 06

101

071

06

05 04

0 1 2 3 4 5 6119910

120601

120574 120574

Figure 21 Combined effect of various parameters on concentrationprofiles

correspond to Helium and water vapours respectively Fromthese curves it is clear that velocity decreases when Sc isincreased The effect of Soret number Sr on velocity is quiteopposite to that of Sc

Figure 20 is plotted to show the effects of Prandtl numberPr radiation parameter 119877 and time 119905 on the temperatureprofiles The comparison of Curves I amp II shows the effects ofPr on the temperature profiles Twodifferent values of Prandtlnumber Pr namely Pr = 071 and Pr = 1 correspondingto air and electrolyte are chosen It is observed that tem-perature decreases with increasing Pr Furthermore the tem-perature profiles for increasing values of radiation parameter119877 indicate an increasing behavior as shown in Curves I amp IIIA behavior was expected because the radiation parameter 119877signifies the relative contribution of conduction heat trans-fer to thermal radiation transfer The effect of time 119905 on tem-perature is the same as observed for radiation This fact isshown from the comparison of Curves I amp IV Graphicalresults of concentration profiles for different values of Prandtlnumber Pr Schmidth number Sc Soret number Sr andchemical reaction parameter 120574 are shown in Figure 21 Com-parison of Curves I amp II shows that concentration profilesincrease for the increasing values of Pr The effect of Sc onthe concentration is shown from the comparison of CurvesI amp III Here we choose real values for Schmidth number asSc = 06 and Sc = 1 which physically correspond to watervapours and methanol It is observed that an increase in Scdecreases the concentration The effect of Soret number Sron the concentration is seen from the comparison of CurvesI amp IV It is observed that concentration increases when Srincreases The effect of chemical reaction parameter 120574 on theconcentration is quite opposite to that of SrThis fact is shownfrom the comparison of Curves I amp V

The numerical values of the skin friction (120591) Nusseltnumber (Nu) and Sherwood number (Sh) are computed in

Table 1 The effects of various parameters on skin friction (120591) when119905 = 1 119877 = 02 120574 = 07

Pr 119872 119870 Gr Sc Sr Gm 120591

071 1 05 1 2 1 1 127

1 1 05 1 2 1 1 130

071 2 05 1 2 1 1 205

071 1 1 1 2 1 1 094

071 1 05 2 2 1 1 085

071 1 05 1 3 1 1 131

071 1 05 1 2 3 1 125

071 1 05 1 2 1 2 096

Table 2 The effects of various parameters on Nusselt number (Nu)when 119905 = 1

Pr 119877 Nu071 02 043

1 02 051

071 04 040

Table 3 The effects of various parameters on Sherwood number(Sh) when 119905 = 1 119877 = 01Gr = 1119872 = 119870 = 1Gm = 2

Pr 120574 Sc Sr Sh071 1 06 2 136

1 1 06 2 084

071 2 06 2 19

071 1 1 2 096

071 1 06 3 164

Tables 1ndash3 In all these tables it is noted that the comparisonof each parameter ismade with first row in the correspondingtable It is found fromTable 1 that the effect of each parameteron the skin friction shows quite opposite effect to that ofthe velocity of the fluid For instance when we increase themagnetic parameter 119872 the skin friction increases as weobserved previously velocity decreases It is observed fromTable 2 that Nusselt number increases with increasing valuesof Prandtl number Pr whereas it decreases when the radia-tion parameter119877 is increased FromTable 3 we observed thatSherwood number goes on increasing with increasing 120574 andSc but the trend reverses for large values of Pr and Sr

5 Conclusions

The exact solutions for the unsteady free convection MHDflow of an incompressible viscous fluid passing through aporous medium and heat and mass transfer are developed byusing Laplace transform method for the uniform motion ofthe plateThe solutions that have been obtained are displayedfor both small and large times which describe the motionof the fluid for some time after its initiation After that timethe transient part disappears and the motion of the fluid isdescribed by the steady-state solutions which are indepen-dent of initial conditions The effects of different parameterssuch as Grashof number Gr Hartmann number 119872 porosity


parameter 119870 Prandtl number Pr radiation parameter 119877Schmidt number Sc Soret number Sr and chemical reactionparameter 120574 on the velocity distributions temperature andconcentration profiles are discussed Themain conclusions ofthe problem are listed below

(i) The effects ofHartmann number and porosity param-eter on velocity are opposite

(ii) The velocity increases with increasing values of119870 Grand 119905 whereas it decreases for larger values of119872 andPr gt 1

(iii) The temperature and thermal boundary layer de-crease owing to the increase in the values of 119877 andPr

(iv) The fluid concentration decreases with increasingvalues of 120574 and Sc whereas it increases when Sr andPr are increased

6 Future Recommendations

Convective heat transfer is a mechanism of heat transferoccurring because of bulk motion of fluids and it is one ofthe major modes of heat transfer and is also a major modeof mass transfer in fluids Convective heat and mass transfertakes place through both diffusionmdashthe random Brownianmotion of individual particles in the fluidmdashand advection inwhichmatter or heat is transported by the larger-scalemotionof currents in the fluid Due to its role in heat transfer naturalconvection plays a role in the structure of Earthrsquos atmosphereits oceans and its mantle Natural convection also plays a rolein stellar physics Motivated by the investigations especiallythose they considered the exact analysis of the heat and masstransfer phenomenon (see for example Seth et al [22] Toki[24] Das and Jana [26] Osman et al [27] Khan et al [28]and Sparrow and Cess [29]) and the extensive applications ofnon-Newtonian fluids in the industrial manufacturing sectorit is of great interest to extend the present work for non-Newtonian fluids Of course in non-Newtonian fluids thefluids of second grade and Maxwell form the simplest fluidmodels where the present analysis can be extended Howeverthe present study can also by analyzed for Oldroyd-B andBurger fluidsThere are also cylindrical and spherical coordi-nate systems where such type of investigations are scarce Ofcourse we can extend this work for such type of geometricalconfigurations

Acknowledgment

The authors would like to acknowledge the Research Man-agement Centre UTM for the financial support through votenumbers 4F109 and 02H80 for this research

References

[1] A Raptis ldquoFlow of a micropolar fluid past a continuously mov-ing plate by the presence of radiationrdquo International Journal ofHeat and Mass Transfer vol 41 no 18 pp 2865ndash2866 1998

[2] Y J Kim and A G Fedorov ldquoTransient mixed radiative convec-tion flow of a micropolar fluid past a moving semi-infinitevertical porous platerdquo International Journal of Heat and MassTransfer vol 46 no 10 pp 1751ndash1758 2003

[3] H A M El-Arabawy ldquoEffect of suctioninjection on the flow ofa micropolar fluid past a continuously moving plate in the pre-sence of radiationrdquo International Journal of Heat and MassTransfer vol 46 no 8 pp 1471ndash1477 2003

[4] H S Takhar S Roy and G Nath ldquoUnsteady free convectionflow over an infinite vertical porous plate due to the combinedeffects of thermal and mass diffusion magnetic field and Hallcurrentsrdquo Heat and Mass Transfer vol 39 no 10 pp 825ndash8342003

[5] M S Alam M M Rahman and M A Samad ldquoDufour andSoret effects on unsteadyMHD free convection andmass trans-fer flow past a vertical porous plate in a porous mediumrdquo Non-linear Analysis Modelling and Control vol 11 no 3 pp 217ndash2262006

[6] R C Chaudhary and J Arpita ldquoCombined heat andmass trans-fer effect on MHD free convection flow past an oscillating plateembedded in porous mediumrdquo Romanian Journal of Physicsvol 52 no 5ndash7 pp 505ndash524 2007

[7] M Ferdows K Kaino and J C Crepeau ldquoMHD free convectionand mass transfer flow in a porous media with simultaneousrotating fluidrdquo International Journal of Dynamics of Fluids vol4 no 1 pp 69ndash82 2008

[8] V Rajesh S Vijaya and k varma ldquoHeat Source effects onMHD flow past an exponentially accelerated vertical plate withvariable temperature through a porous mediumrdquo InternationalJournal of AppliedMathematics andMechanics vol 6 no 12 pp68ndash78 2010

[9] V Rajesh and S V K Varma ldquoRadiation effects on MHD flowthrough a porousmediumwith variable temperature or variablemass diffusionrdquo Journal of Applied Mathematics and Mechanicsvol 6 no 1 pp 39ndash57 2010

[10] A A Bakr ldquoEffects of chemical reaction on MHD free convec-tion andmass transfer flowof amicropolar fluidwith oscillatoryplate velocity and constant heat source in a rotating frame ofreferencerdquoCommunications inNonlinear Science andNumericalSimulation vol 16 no 2 pp 698ndash710 2011

[11] U N Das S N Ray and V M Soundalgekar ldquoMass transfereffects on flow past an impulsively started infinite vertical platewith constant mass fluxmdashan exact solutionrdquo Heat and MassTransfer vol 31 no 3 pp 163ndash167 1996

[12] R Muthucumaraswamy P Ganesan and V M SoundalgekarldquoHeat and mass transfer effects on flow past an impulsivelystarted vertical platerdquo Acta Mechanica vol 146 no 1-2 pp 1ndash8 2001

[13] T Hayat and Z Abbas ldquoHeat transfer analysis on the MHDflow of a second grade fluid in a channel with porous mediumrdquoChaos Solitons and Fractals vol 38 no 2 pp 556ndash567 2008

[14] M M Rahman and M A Sattar ldquoMagnetohydrodynamic con-vective flow of a micropolar fluid past a continuously movingvertical porous plate in the presence of heat generationabsorp-tionrdquo Journal of Heat Transfer vol 128 no 2 pp 142ndash152 2006

[15] Y J Kim ldquoUnsteadyMHD convection flow of polar fluids past averticalmoving porous plate in a porousmediumrdquo InternationalJournal of Heat andMass Transfer vol 44 no 15 pp 2791ndash27992001

[16] M Kaviany ldquoBoundary-layer treatment of forced convectionheat transfer from a semi-infinite flat plate embedded in porous


mediardquo Journal of Heat Transfer vol 109 no 2 pp 345ndash3491987

[17] K Vafai and C L Tien ldquoBoundary and inertia effects on flowand heat transfer in porousmediardquo International Journal of Heatand Mass Transfer vol 24 no 2 pp 195ndash203 1981

[18] B K Jha and C A Apere ldquoCombined effect of hall and ion-slipcurrents on unsteady MHD couette flows in a rotating systemrdquoJournal of the Physical Society of Japan vol 79 Article ID 1044012010

[19] G Mandal K K Mandal and G Choudhury ldquoOn combinedeffects of coriolis force and hall current on steadyMHD couetteflow and hear transferrdquo Journal of the Physical Society of Japanvol 51 no 1982 2010

[20] M Katagiri ldquoFlow formation in Couette motion in magne-tohydro-dynamicsrdquo Journal of the Physical Society of Japan vol17 pp 393ndash396 1962

[21] R C Chaudhary and J Arpita ldquoAn exact solution of magne-tohydrodynamic convection flow past an accelerated surfaceembedded in a porous mediumrdquo International Journal of Heatand Mass Transfer vol 53 no 7-8 pp 1609ndash1611 2010

[22] G S Seth M S Ansari and R Nandkeolyar ldquoMHD naturalconvection flow with radiative heat transfer past an impulsivelymoving plate with ramped wall temperaturerdquo Heat Mass andTransfer vol 47 pp 551ndash561 2011

[23] C J Toki and J N Tokis ldquoExact solutions for the unsteadyfree convection flows on a porous plate with time-dependentheatingrdquo Zeitschrift fur AngewandteMathematik undMechanikvol 87 no 1 pp 4ndash13 2007

[24] C J Toki ldquoFree convection and mass transfer flow near a mov-ing vertical porous plate an analytical solutionrdquo Journal ofAppliedMechanics Transactions ASME vol 75 no 1 Article ID0110141 2008

[25] K Das ldquoExact solution of MHD free convection flow and masstransfer near amoving vertical plate in presence of thermal rad-iationrdquo African Journal of Mathematical Physics vol 8 pp 29ndash41 2010

[26] K Das and S Jana ldquoHeat and mass transfer effects on unsteadyMHD free convection flow near a moving vertical plate in por-ous mediumrdquo Bulletin of Society of Mathematicians vol 17 pp15ndash32 2010

[27] A N A Osman S M Abo-Dahab and R A Mohamed ldquoAna-lytical solution of thermal radiation and chemical reactioneffects on unsteady MHD convection through porous mediawith heat sourcesinkrdquo Mathematical Problems in Engineeringvol 2011 Article ID 205181 18 pages 2011

[28] I Khan F Ali S Shafie and N Mustapha ldquoEffects of Hall cur-rent and mass transfer on the unsteady MHD flow in a porouschannelrdquo Journal of the Physical Society of Japan vol 80 no 6Article ID 064401 pp 1ndash6 2011

[29] E M Sparrow and R D Cess Radiation Heat Transfer Hemi-sphere Washington DC USA 1978

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Algebra

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Stochastic AnalysisInternational Journal of


transfer over a moving vertical plate In this continuation theeffect of heatmass transfer on unsteadyMHD free convectionflow past a moving vertical plate in a porous medium wasinvestigated by Das and Jana [26] They considered theimpulsive uniform and oscillating motions of the platewith constant heat and mass diffusion and developed theexact solutions using Laplace transform technique RecentlyOsman et al [27] analyzed the thermal radiation and chem-ical reaction effects on unsteady MHD free convection flowthrough a porous plate embedded in a porous medium withheat sourcesink and the closed form solutions are obtainedKhan et al [28] and Sparrow and Cess [29] analyzed theeffects of Hall current and mass transfer on the unsteadyMHD free convection flow in a porous channel The motionin fluid is induced to the external pressure gradient andthe closed form solutions for the velocity temperature andconcentration fields are obtained

Motivated by the above investigations the present paperaims to study the combined heat and mass effects on theunsteady MHD free convection flow of an incompressibleviscous fluid passing through a porous medium The flow inthe fluid is caused due to the uniform motion of the plateExact solutions are derived for the velocity distributions tem-perature and concentration fields by using Laplace transformtechnique and presented graphically for small as well as largetimes To the best of authorsrsquo knowledge this problem hasnot been studied before and the reported results are newThepresent study is of course of great practical and technologicalimportance for example in astrophysical regimes the pres-ence of planetary debris cosmic dust and so forth and createsa suspended porous medium saturated with plasma fluidsCombined buoyancy-generated heat and mass transfer dueto temperature and concentration variations with unsteadyMHD free convection flow in fluid-saturated porous mediahas several important applications in a variety of engineeringprocesses including heat exchanger devices petroleum reser-voirs chemical catalytic reactors solar energy porous watercollector systems and ceramic materials

This paper is organized as follows A brief descriptionof the problem formulation is given in Section 2 The exactsolutions for the uniformly uniform motion of the plate arederived in Section 3 The graphical results and discussionare provided in Section 4 The conclusions of the paper aregiven in Section 5 whereas some future recommendations areincluded in Section 6

2 Description of the Problem Formulation

Let us consider the unsteady one dimensional flow ofan incompressible and electrically conducting viscous fluidcaused due to the uniform motion of the plate The 119909lowast-axisis taken along the plate in the vertical direction and 119910lowast-axisis taken normal to the plate The electrically conducting fluidoccupies the porous half space 119910lowast gt 0 A uniform magneticfield B

0is acting in the transverse direction to the flow

The magnetic Reynolds number is assumed to be small andtherefore the induced magnetic field is negligible comparedwith the applied magnetic field The applied magnetic fieldis also taken weak so that Hall and ion slip effects may be

neglected Initially both the plate and fluid are at the sametemperature 119879

lowast

infinand concentration 119862lowast

infin At time 119905 = 0+

the plate begins to slide in its own plane and acceleratesagainst the gravitational field with uniform acceleration in119909lowast-direction Then the temperature and concentration level

are raised to 119879lowast119908and 119862lowast

119908as shown in Figure 1

The Soret and thermal buoyancy effects are also consid-ered In addition to the above assumptions we assume thatthe internal dissipation is absent and the usual Boussinesqapproximation is taken into consideration Moreover thepressure gradient in the flow direction is compensated by thegradient of the hydrostatic pressure gradient of the fluid Asa result the governing equations of momentum energy andconcentration are derived as follows

120597119906lowast

120597119905lowast= ]

1205972119906lowast


120588minus]119906lowast

119870lowast

+ 119892120573 (119879lowast

minus 119879lowast

infin) + 119892120573

lowast

(119862lowast

minus 119862lowast

infin)

(1)

120588119888119901

120597119879lowast

120597119905lowast= 119896

1205972119879lowast


119903

120597119910lowast (2)

120597119862lowast

120597119905lowast= 119863

1205972

119862lowast

120597119910lowast2+119863119870119879

119879119898

1205972

119879lowast


119903(119862lowast

minus 119862lowast

infin) (3)

with the following initial and boundary conditions

119905lowast

le 0 119906lowast

= 0 119879lowast

= 119879lowast

infin 119862lowast

= 119862lowast

infin

forall119910lowast

ge 0

119905lowast

gt 0 119906lowast

= 119891 (119905lowast

) 119879lowast

= 119879lowast

119908 119862lowast

= 119862lowast

119908

at 119910lowast = 0

119906lowast

997888rarr 0 119879lowast


infin 119862lowast


infin

as 119910lowast 997888rarr infin

(4)

where 119891(119905lowast) is the uniform acceleration of the plate 119909lowast and119910lowast (m) are the distances along and perpendicular to theplate 119905lowast (s) is the time 119906lowast (msminus1) denote the fluid velocityin the 119909lowast-direction 119879lowast (K) temperature 119879lowast

infin(K) temper-

ature far from the plate 119879lowast119908(K) temperature at the wall

119862lowast (molmminus3) are the species concentration 119862lowast119908(molmminus3)

surface concentration 119862lowastinfin(molmminus3) species concentration

far from the surface 120573 (1K) the volumetric coefficient ofthermal expansion 120573lowast (molmminus3)minus1 or (m3molminus1) is thevolumetric coefficient of expansion for concentration ] =

120583120588 (m2 sminus1) the kinematic viscosity 120583 (kgmminus1 sminus1) viscosity120588 (kgmminus3) the fluid density 119888

119901(kgminus1 Kminus1) is the specific heat

capacity 119902lowast119903the radiative heat flux in 119909lowast-direction119863 (m2 sminus1)

is mass diffusivity 119896 (Wmminus1 Kminus1) is the thermal conductivityof the fluid 120590 (Smminus1) the electrical conductivity of the fluid119870lowast gt 0 (m2) is the permeability of the porous medium 119879

119898

(K) is the mean fluid temperature 119879lowastinfinis the free stream tem-

perature 119862lowastinfin

is the free stream concentration of the species119870119879

is the thermal-diffusion ratio and 119870lowast119903

the chemicalreaction constantThe radiative heat flux term for an optically


119892

119879lowastinfin

119862lowastinfin




Porous medium

119909lowast

119911lowast

119910lowast

119879lowast119908

119862lowast119908

B0



119902lowast

119903= minus

4120590lowast

3119896lowast120597119879lowast4

120597119910lowast (5)


4


4



119879lowast4

≊ 4119879lowast3

infin119879lowast


infin (6)


120588119888119901

120597119879lowast

120597119905lowast= 119896

1205972119879lowast


3

infin

3119896lowast1205972119879lowast

120597119910lowast2 (7)



119906 =119906lowast

(]119860)13 119910 = 119910

lowast

(119860

]2)13

119905 = 119905lowast

(1198602

])

13


infin



infin


(8)



120597119906

120597119905=1205972119906

1205971199102minus 119867119906 + Gr120579 + Gm120601

119906 (0 119905) = 119905 119906 (infin 119905) = 0 119905 gt 0

119906 (119910 0) = 0 119910 ge 0

(9)

119865lowast120597120579

120597119905=1205972120579

1205971199102

120579 (0 119905) = 1 120579 (infin 119905) = 0 119905 gt 0

120579 (119910 0) = 0 119910 ge 0

(10)

120597120601

120597119905=

1

Sc1205972120601

1205971199102+ Sr120597

2120579

1205971199102minus 120574120601

120601 (0 119905) = 1 120601 (infin 119905) = 0 119905 gt 0

120601 (119910 0) = 0 119910 ge 0

(11)

where

1

119870=

]43

119870lowast11986023 119872

2

=12059011986120]13

12058811986023



120583119888119901

119896


119860 Gm =


119860

120574 =119870lowast119903]13

11986023

119867 = 1198722

+1

119870 119877 =

16120590lowast

119879lowast3

infin

3119896119896lowast

119865lowast

=Pr

1 + 119877 Sc = ]

119863

(12)



1198892119906

1198891199102minus (119867 + 119902) 119906 + Gr120579 + Gm120601 = 0

119906 (0 119902) =1

1199022 119906 (infin 119902) = 0

1198892120579

1198891199102minus 119865lowast

119902120579 = 0

120579 (0 119902) =1

119902 120579 (infin 119902) = 0

1198892120601

1198891199102minus (120574 + 119902) Sc120601 + Sc Sr119889

2120579

1198891199102= 0

120601 (0 119902) =1

119902 120601 (infin 119902) = 0

(13)


by

119906 (119910 119902) =1

1199022exp(minus119910radic119902 + 119867) +

1198868

119902exp(minus119910radic119902 + 119867)

minus1198869

119902 + 1198860

exp(minus119910radic119902 + 119867)

+11988610


exp(minus119910radic119902 + 119867)

+11988611


+1198865



minus11988611


4


minus1198866


+1198869

119902 + 1198860


minus1198867



(14)

120579 (119910 119902) =1


120601 (119910 119902) =1


+119878lowast



minus119878lowast



(16)

where


119867lowast

=119867


119867lowast

1=


119878lowast

=Sc Sr119865lowast

119865lowast minus 119878119888

1198860=120574Sc minus 119867Sc minus 1

1198861=

Gm119878lowast


1198862=

GmSc minus 1

1198863=Gm119878lowast

Sc minus 1 119886

4=Grlowast

119867lowast

1198865=

1198861


1198866=1198862

1198860

1198867=

1198863

119867lowast1+ 1198860

1198868= 1198866minus 1198864 119886

9= 1198866+ 1198867

11988610= 1198867minus 1198865 119886

11= 1198864+ 1198865

(17)


119906 (119910 119905) = 1198681+ 11988681198682minus 11988691198683+ 119886101198684+ 119886111198685+ 11988651198686


(18)

120579 (119910 119905) = 1198688 (19)

120601 (119910 119905) = 119878lowast

(11986811minus 1198686) + 1198689 (20)

with

1198681=1

2[(119905 minus

119910


119910


+119890119910radic119867 erf 119888 (

119910

2radic119905+ radic119867119905)(119905 +

119910

2radic119867)]

1198682=1

2[119890minus119910radic119867 erf 119888 (

119910


+119890119910radic119867 erf 119888 (

119910

2radic119905+ radic119867119905)]

1198683=119890minus1198860119905


0 erf 119888 (119910


0) 119905)

+119890119910radic119867minus119886

0 erf 119888 (119910


0) 119905)]

1198684=119890119867lowast

1119905

2[119890minus119910radic119867

lowast

1+119867 erf 119888 (

119910


1+ 119867) 119905)

+119890119910radic119867lowast

1+119867 erf 119888 (

119910


1+ 119867) 119905)]


1198685=119890119867lowast119905

2[119890minus119910radic119867

lowast+119867 erf 119888 (

119910


+119890119910radic119867lowast+119867 erf 119888 (

119910


1198686=119890119867lowast

1119905

2[119890minus119910radic119865

lowast119867lowast

1 erf 119888 (119910radic119865lowast


1119905)


1 erf 119888 (119910radic119865lowast


1119905)]

1198687=119890119867lowast119905

2[119890minus119910radic119865

lowast119867lowast






1198688= erf 119888 (


2radic119905)

1198689=1




+119890119910radic120574Sc erf 119888 (


+ radic120574119905)]

11986810=119890minus1198860119905


0) erf 119888 (



+ 119890119910radicSc(120574minus119886

0) erf 119888 (


+ radic(120574 minus 1198860) 119905)]

11986811=119890119867lowast

1119905


1+120574) erf 119888 (



+ 119890119910radicSc(119867lowast

1+120574) erf 119888 (


+ radic(119867lowast1+ 120574) 119905)]

(21)




120591lowast

= minus120583120597119906lowast

120597119910lowast

10038161003816100381610038161003816100381610038161003816119910lowast=0 (22)


120591 = minus120597119906

120597119910 120591 =

120591lowast

1205881198802119900

(23)


120591 =1198868119890minus119867119905

radic120587119905minus11988611

radic119865lowast

radic120587119905+1198864

radic119865lowast

radic120587119905+1198865

radic119865lowast


+119890minus119867119905radic119905

radic120587+119905radic119867

2(minus1 + erf (radic119867119905)) +

erf (radic119867119905)

2radic119867

+ 1198868


2(1 + erf (radic119867119905))



+ 1198865119890119867lowast



1119905)

minus1

211988611119890119867lowast119905


lowast119905




minus1

211988610119890119867lowast

1119905

minus2119890minus(119867+119867

lowast

1)119905


1



1) 119905))

minus1

21198869119890minus1198860119905

21198901198860119905minus119867119905


0



0119905 + 119867119905))

minus1

21198869119890minus1198860119905



0+ 120574) Sc


+ radic(minus1198860+ 120574) Sc


minus1

21198867119890119867lowast

1119905

2119890minus(119867

lowast

1+120574)119905


1+ 120574





(24)


Nu = minus119896120597119879lowast

120597119910lowast

10038161003816100381610038161003816100381610038161003816119910lowast=0

Nu = 120597120579

120597119910

10038161003816100381610038161003816100381610038161003816119910lowast=0


120587119905

(25)


Sh = 120597119862lowast

120597119910lowast

10038161003816100381610038161003816100381610038161003816119910lowast=0

Sh = minus1

radic120587119905119890minus120574119905


lowast


minus119890119867lowast119905+120574119905



lowast119905+120574119905

119878lowast


(26)










1

08

06

04

02

00 2 4 6 8 10 12 14

119910

119906

119872 = 0

119872 = 1119872 = 2119872 = 3

119905 = 1 119877 = 02 Pr = 071 Gr = 1119870 = 05 Gm = 1 120574 = 07 Sc = 2 Sr = 1


1

08

06

04

02

00 2 4 6 8 10 12 14

119910

119906

119870 = 05119870 = 10

119870 = 15119870 = 20

119905 = 1 119877 = 02 Pr = 071 Gr = 1119872 = 1 Gm = 1 120574 = 05 Sc = 2 Sr = 1





119905 = 1

119906

119910

25

2

15

1

05

00 2 4 6 8

Gm = 1119877 = 02 Pr = 071 Gr = 1119872 = 1 119870 = 2 120574 = 05 Sc = 06 Sr = 5

119905 = 15

119905 = 20

119905 = 25


12

1

08

06

04

02

0

119906

0 2 4 6 8 10 12 14119910

Gr = 0Gr = 1

Gr = 2Gr = 3

119905 = 1 119877 = 02119872 = 1 119870 = 2 120574 = 05 Sc = 2 Sr = 1

Pr = 071 Gm = 1


1

08

06

04

02

0

119906

0 1 2 3 4 5 6119910

Sc = 022Sc = 03

Sc = 06Sc = 1

119905 = 1 119877 = 02 Pr = 071 Gm = 1119872 = 2 119870 = 1 120574 = 05 Gr = 2 Sr = 15


175

15

125

1

075

05

025

00 5 10 15 20

119910

119906

Sr = 25Sr = 45

Sr = 65Sr = 8

119905 = 1 119877 = 02 Pr = 071 Gm = 1119872 = 2 119870 = 1 120574 = 05 Gr = 2 Sc = 15


0 05 1 15 2119910

0

02

04

06

08

1

12

119906

Pr = 100Pr = 7

Pr = 1Pr = 071Pr = 0015

119905 = 1 119877 = 1 Gr = 1119872 = 1119870 = 4 Gm = 02 120574 = 15 Sc = 2 Sr = 01






1

08

06

04

02

0

119910

0 10 20 30 40 50

119905 = 06 119877 = 05

Pr = 0015Pr = 071Pr = 10

Pr = 70Pr = 100

120579


12

1

08

06

04

02

0

119910

0 1 2 3 4 5 6

119905 = 1 119877 = 2 Sc = 1 Sr = 09

Pr = 0015Pr = 071Pr = 1

Pr = 7Pr = 100

120601

120574 = 02


119905 = 01 Pr = 0711

08

06

04

02

00 05 1 15 2 25 3

119910

119877 = 05

119877 = 15

119877 = 30

119877 = 45

120579


119905 = 03 119877 = 04 Sc = 09 Sr = 08 Pr = 10

0 05 1 15 2 25 3119910

1

08

06

04

02

0

120601

120574 = 0

120574 = 3

120574 = 6

120574 = 9


119905 = 05 119877 = 04 Sr = 04 Pr = 071

Sc = 1Sc = 2

Sc = 3Sc = 4

1

08

06

04

02

0

120601

0 05 1 15 2 25 3119910

120574 = 02


119905 = 03 119877 = 04 Sc = 03 Pr = 071

Sr = 0Sr = 1

Sr = 2Sr = 3

1

08

06

04

02

00 1 2 3 4

119910

120574 = 02

120601



119910 = 5

119910 = 10

119910 = 15

119910 = 20

5

4

3

2

1

0

119906

0 1 2 3 4 5 6119905

Sr = 1119877 = 02 Pr = 071 Gm = 1119872 = 01 119870 = 2 120574 = 05 Gr = 2 Sc = 2


Pr = 071119877 = 31

08

06

04

02

00 2 4 6 8 10

119905

119910 = 0

119910 = 2

119910 = 3

119910 = 7

120579




119877 = 04 Γ = 07 Sc = 03 Sr = 05 Pr = 071

0 1 2 3 4 5 6119905

1

08

06

04

02

0

119910 = 0

119910 = 1

119910 = 2119910 = 5

120601






119905 = 04 119877 = 1 Gr = 1 Gm = 1

(IV)

(III)

(II)

(I)

119906

04

03

02

01

0

119910

0 05 1 15 2 25 3 35


(I) 071 2 01

(II) 7 2 01

(III) 071 4 01

(IV) 071 2 04

120574 = 2 Sc = 06 Sr = 1





(I)

(II)

(III)

(IV)

(V)

119906

02

015

01

005

0

119910

0 05 1 15 2 25 3

119905 = 04 119877 = 1 Pr = 071 119872 = 2 120574 = 2 119870 = 01


(I)

(II)

(III)

(IV)

(V)

05

1

05

1

1

2

022

022

022

1

1

1

05

05

1

1

06

022

1

2


1

08

06

04

02

00 5 10 15 20

119910

(I)

(II)

(III)

(IV)

071 05 2

1 05 2

(I)

(II)

Curves Pr 119877 119905

071 1 2

071 05 3

(III)

(IV)

Curves Pr 119877 119905

120579




1

08

06

04

02

0

(I)

(II)

(III)

(IV)

(V)


(I)

(II)

(III)

(IV)

(V)

071

06

05

01

7

06

05

01

071

1

05

01 071 06

101

071

06

05 04

0 1 2 3 4 5 6119910

120601

120574 120574






Pr 119872 119870 Gr Sc Sr Gm 120591

071 1 05 1 2 1 1 127

1 1 05 1 2 1 1 130

071 2 05 1 2 1 1 205

071 1 1 1 2 1 1 094

071 1 05 2 2 1 1 085

071 1 05 1 3 1 1 131

071 1 05 1 2 3 1 125

071 1 05 1 2 1 2 096


Pr 119877 Nu071 02 043

1 02 051

071 04 040


Pr 120574 Sc Sr Sh071 1 06 2 136

1 1 06 2 084

071 2 06 2 19

071 1 1 2 096

071 1 06 3 164


5 Conclusions










Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119892

119879lowastinfin

119862lowastinfin




Porous medium

119909lowast

119911lowast

119910lowast

119879lowast119908

119862lowast119908

B0



119902lowast

119903= minus

4120590lowast

3119896lowast120597119879lowast4

120597119910lowast (5)


4


4



119879lowast4

≊ 4119879lowast3

infin119879lowast


infin (6)


120588119888119901

120597119879lowast

120597119905lowast= 119896

1205972119879lowast


3

infin

3119896lowast1205972119879lowast

120597119910lowast2 (7)



119906 =119906lowast

(]119860)13 119910 = 119910

lowast

(119860

]2)13

119905 = 119905lowast

(1198602

])

13


infin



infin


(8)



120597119906

120597119905=1205972119906

1205971199102minus 119867119906 + Gr120579 + Gm120601

119906 (0 119905) = 119905 119906 (infin 119905) = 0 119905 gt 0

119906 (119910 0) = 0 119910 ge 0

(9)

119865lowast120597120579

120597119905=1205972120579

1205971199102

120579 (0 119905) = 1 120579 (infin 119905) = 0 119905 gt 0

120579 (119910 0) = 0 119910 ge 0

(10)

120597120601

120597119905=

1

Sc1205972120601

1205971199102+ Sr120597

2120579

1205971199102minus 120574120601

120601 (0 119905) = 1 120601 (infin 119905) = 0 119905 gt 0

120601 (119910 0) = 0 119910 ge 0

(11)

where

1

119870=

]43

119870lowast11986023 119872

2

=12059011986120]13

12058811986023



120583119888119901

119896


119860 Gm =


119860

120574 =119870lowast119903]13

11986023

119867 = 1198722

+1

119870 119877 =

16120590lowast

119879lowast3

infin

3119896119896lowast

119865lowast

=Pr

1 + 119877 Sc = ]

119863

(12)



1198892119906

1198891199102minus (119867 + 119902) 119906 + Gr120579 + Gm120601 = 0

119906 (0 119902) =1

1199022 119906 (infin 119902) = 0

1198892120579

1198891199102minus 119865lowast

119902120579 = 0

120579 (0 119902) =1

119902 120579 (infin 119902) = 0

1198892120601

1198891199102minus (120574 + 119902) Sc120601 + Sc Sr119889

2120579

1198891199102= 0

120601 (0 119902) =1

119902 120601 (infin 119902) = 0

(13)


by

119906 (119910 119902) =1

1199022exp(minus119910radic119902 + 119867) +

1198868

119902exp(minus119910radic119902 + 119867)

minus1198869

119902 + 1198860

exp(minus119910radic119902 + 119867)

+11988610


exp(minus119910radic119902 + 119867)

+11988611


+1198865



minus11988611


4


minus1198866


+1198869

119902 + 1198860


minus1198867



(14)

120579 (119910 119902) =1


120601 (119910 119902) =1


+119878lowast



minus119878lowast



(16)

where


119867lowast

=119867


119867lowast

1=


119878lowast

=Sc Sr119865lowast

119865lowast minus 119878119888

1198860=120574Sc minus 119867Sc minus 1

1198861=

Gm119878lowast


1198862=

GmSc minus 1

1198863=Gm119878lowast

Sc minus 1 119886

4=Grlowast

119867lowast

1198865=

1198861


1198866=1198862

1198860

1198867=

1198863

119867lowast1+ 1198860

1198868= 1198866minus 1198864 119886

9= 1198866+ 1198867

11988610= 1198867minus 1198865 119886

11= 1198864+ 1198865

(17)


119906 (119910 119905) = 1198681+ 11988681198682minus 11988691198683+ 119886101198684+ 119886111198685+ 11988651198686


(18)

120579 (119910 119905) = 1198688 (19)

120601 (119910 119905) = 119878lowast

(11986811minus 1198686) + 1198689 (20)

with

1198681=1

2[(119905 minus

119910


119910


+119890119910radic119867 erf 119888 (

119910

2radic119905+ radic119867119905)(119905 +

119910

2radic119867)]

1198682=1

2[119890minus119910radic119867 erf 119888 (

119910


+119890119910radic119867 erf 119888 (

119910

2radic119905+ radic119867119905)]

1198683=119890minus1198860119905


0 erf 119888 (119910


0) 119905)

+119890119910radic119867minus119886

0 erf 119888 (119910


0) 119905)]

1198684=119890119867lowast

1119905

2[119890minus119910radic119867

lowast

1+119867 erf 119888 (

119910


1+ 119867) 119905)

+119890119910radic119867lowast

1+119867 erf 119888 (

119910


1+ 119867) 119905)]


1198685=119890119867lowast119905

2[119890minus119910radic119867

lowast+119867 erf 119888 (

119910


+119890119910radic119867lowast+119867 erf 119888 (

119910


1198686=119890119867lowast

1119905

2[119890minus119910radic119865

lowast119867lowast

1 erf 119888 (119910radic119865lowast


1119905)


1 erf 119888 (119910radic119865lowast


1119905)]

1198687=119890119867lowast119905

2[119890minus119910radic119865

lowast119867lowast






1198688= erf 119888 (


2radic119905)

1198689=1




+119890119910radic120574Sc erf 119888 (


+ radic120574119905)]

11986810=119890minus1198860119905


0) erf 119888 (



+ 119890119910radicSc(120574minus119886

0) erf 119888 (


+ radic(120574 minus 1198860) 119905)]

11986811=119890119867lowast

1119905


1+120574) erf 119888 (



+ 119890119910radicSc(119867lowast

1+120574) erf 119888 (


+ radic(119867lowast1+ 120574) 119905)]

(21)




120591lowast

= minus120583120597119906lowast

120597119910lowast

10038161003816100381610038161003816100381610038161003816119910lowast=0 (22)


120591 = minus120597119906

120597119910 120591 =

120591lowast

1205881198802119900

(23)


120591 =1198868119890minus119867119905

radic120587119905minus11988611

radic119865lowast

radic120587119905+1198864

radic119865lowast

radic120587119905+1198865

radic119865lowast


+119890minus119867119905radic119905

radic120587+119905radic119867

2(minus1 + erf (radic119867119905)) +

erf (radic119867119905)

2radic119867

+ 1198868


2(1 + erf (radic119867119905))



+ 1198865119890119867lowast



1119905)

minus1

211988611119890119867lowast119905


lowast119905




minus1

211988610119890119867lowast

1119905

minus2119890minus(119867+119867

lowast

1)119905


1



1) 119905))

minus1

21198869119890minus1198860119905

21198901198860119905minus119867119905


0



0119905 + 119867119905))

minus1

21198869119890minus1198860119905



0+ 120574) Sc


+ radic(minus1198860+ 120574) Sc


minus1

21198867119890119867lowast

1119905

2119890minus(119867

lowast

1+120574)119905


1+ 120574





(24)


Nu = minus119896120597119879lowast

120597119910lowast

10038161003816100381610038161003816100381610038161003816119910lowast=0

Nu = 120597120579

120597119910

10038161003816100381610038161003816100381610038161003816119910lowast=0


120587119905

(25)


Sh = 120597119862lowast

120597119910lowast

10038161003816100381610038161003816100381610038161003816119910lowast=0

Sh = minus1

radic120587119905119890minus120574119905


lowast


minus119890119867lowast119905+120574119905



lowast119905+120574119905

119878lowast


(26)










1

08

06

04

02

00 2 4 6 8 10 12 14

119910

119906

119872 = 0

119872 = 1119872 = 2119872 = 3

119905 = 1 119877 = 02 Pr = 071 Gr = 1119870 = 05 Gm = 1 120574 = 07 Sc = 2 Sr = 1


1

08

06

04

02

00 2 4 6 8 10 12 14

119910

119906

119870 = 05119870 = 10

119870 = 15119870 = 20

119905 = 1 119877 = 02 Pr = 071 Gr = 1119872 = 1 Gm = 1 120574 = 05 Sc = 2 Sr = 1





119905 = 1

119906

119910

25

2

15

1

05

00 2 4 6 8

Gm = 1119877 = 02 Pr = 071 Gr = 1119872 = 1 119870 = 2 120574 = 05 Sc = 06 Sr = 5

119905 = 15

119905 = 20

119905 = 25


12

1

08

06

04

02

0

119906

0 2 4 6 8 10 12 14119910

Gr = 0Gr = 1

Gr = 2Gr = 3

119905 = 1 119877 = 02119872 = 1 119870 = 2 120574 = 05 Sc = 2 Sr = 1

Pr = 071 Gm = 1


1

08

06

04

02

0

119906

0 1 2 3 4 5 6119910

Sc = 022Sc = 03

Sc = 06Sc = 1

119905 = 1 119877 = 02 Pr = 071 Gm = 1119872 = 2 119870 = 1 120574 = 05 Gr = 2 Sr = 15


175

15

125

1

075

05

025

00 5 10 15 20

119910

119906

Sr = 25Sr = 45

Sr = 65Sr = 8

119905 = 1 119877 = 02 Pr = 071 Gm = 1119872 = 2 119870 = 1 120574 = 05 Gr = 2 Sc = 15


0 05 1 15 2119910

0

02

04

06

08

1

12

119906

Pr = 100Pr = 7

Pr = 1Pr = 071Pr = 0015

119905 = 1 119877 = 1 Gr = 1119872 = 1119870 = 4 Gm = 02 120574 = 15 Sc = 2 Sr = 01






1

08

06

04

02

0

119910

0 10 20 30 40 50

119905 = 06 119877 = 05

Pr = 0015Pr = 071Pr = 10

Pr = 70Pr = 100

120579


12

1

08

06

04

02

0

119910

0 1 2 3 4 5 6

119905 = 1 119877 = 2 Sc = 1 Sr = 09

Pr = 0015Pr = 071Pr = 1

Pr = 7Pr = 100

120601

120574 = 02


119905 = 01 Pr = 0711

08

06

04

02

00 05 1 15 2 25 3

119910

119877 = 05

119877 = 15

119877 = 30

119877 = 45

120579


119905 = 03 119877 = 04 Sc = 09 Sr = 08 Pr = 10

0 05 1 15 2 25 3119910

1

08

06

04

02

0

120601

120574 = 0

120574 = 3

120574 = 6

120574 = 9


119905 = 05 119877 = 04 Sr = 04 Pr = 071

Sc = 1Sc = 2

Sc = 3Sc = 4

1

08

06

04

02

0

120601

0 05 1 15 2 25 3119910

120574 = 02


119905 = 03 119877 = 04 Sc = 03 Pr = 071

Sr = 0Sr = 1

Sr = 2Sr = 3

1

08

06

04

02

00 1 2 3 4

119910

120574 = 02

120601



119910 = 5

119910 = 10

119910 = 15

119910 = 20

5

4

3

2

1

0

119906

0 1 2 3 4 5 6119905

Sr = 1119877 = 02 Pr = 071 Gm = 1119872 = 01 119870 = 2 120574 = 05 Gr = 2 Sc = 2


Pr = 071119877 = 31

08

06

04

02

00 2 4 6 8 10

119905

119910 = 0

119910 = 2

119910 = 3

119910 = 7

120579




119877 = 04 Γ = 07 Sc = 03 Sr = 05 Pr = 071

0 1 2 3 4 5 6119905

1

08

06

04

02

0

119910 = 0

119910 = 1

119910 = 2119910 = 5

120601






119905 = 04 119877 = 1 Gr = 1 Gm = 1

(IV)

(III)

(II)

(I)

119906

04

03

02

01

0

119910

0 05 1 15 2 25 3 35


(I) 071 2 01

(II) 7 2 01

(III) 071 4 01

(IV) 071 2 04

120574 = 2 Sc = 06 Sr = 1





(I)

(II)

(III)

(IV)

(V)

119906

02

015

01

005

0

119910

0 05 1 15 2 25 3

119905 = 04 119877 = 1 Pr = 071 119872 = 2 120574 = 2 119870 = 01


(I)

(II)

(III)

(IV)

(V)

05

1

05

1

1

2

022

022

022

1

1

1

05

05

1

1

06

022

1

2


1

08

06

04

02

00 5 10 15 20

119910

(I)

(II)

(III)

(IV)

071 05 2

1 05 2

(I)

(II)

Curves Pr 119877 119905

071 1 2

071 05 3

(III)

(IV)

Curves Pr 119877 119905

120579




1

08

06

04

02

0

(I)

(II)

(III)

(IV)

(V)


(I)

(II)

(III)

(IV)

(V)

071

06

05

01

7

06

05

01

071

1

05

01 071 06

101

071

06

05 04

0 1 2 3 4 5 6119910

120601

120574 120574






Pr 119872 119870 Gr Sc Sr Gm 120591

071 1 05 1 2 1 1 127

1 1 05 1 2 1 1 130

071 2 05 1 2 1 1 205

071 1 1 1 2 1 1 094

071 1 05 2 2 1 1 085

071 1 05 1 3 1 1 131

071 1 05 1 2 3 1 125

071 1 05 1 2 1 2 096


Pr 119877 Nu071 02 043

1 02 051

071 04 040


Pr 120574 Sc Sr Sh071 1 06 2 136

1 1 06 2 084

071 2 06 2 19

071 1 1 2 096

071 1 06 3 164


5 Conclusions










Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198892119906

1198891199102minus (119867 + 119902) 119906 + Gr120579 + Gm120601 = 0

119906 (0 119902) =1

1199022 119906 (infin 119902) = 0

1198892120579

1198891199102minus 119865lowast

119902120579 = 0

120579 (0 119902) =1

119902 120579 (infin 119902) = 0

1198892120601

1198891199102minus (120574 + 119902) Sc120601 + Sc Sr119889

2120579

1198891199102= 0

120601 (0 119902) =1

119902 120601 (infin 119902) = 0

(13)


by

119906 (119910 119902) =1

1199022exp(minus119910radic119902 + 119867) +

1198868

119902exp(minus119910radic119902 + 119867)

minus1198869

119902 + 1198860

exp(minus119910radic119902 + 119867)

+11988610


exp(minus119910radic119902 + 119867)

+11988611


+1198865



minus11988611


4


minus1198866


+1198869

119902 + 1198860


minus1198867



(14)

120579 (119910 119902) =1


120601 (119910 119902) =1


+119878lowast



minus119878lowast



(16)

where


119867lowast

=119867


119867lowast

1=


119878lowast

=Sc Sr119865lowast

119865lowast minus 119878119888

1198860=120574Sc minus 119867Sc minus 1

1198861=

Gm119878lowast


1198862=

GmSc minus 1

1198863=Gm119878lowast

Sc minus 1 119886

4=Grlowast

119867lowast

1198865=

1198861


1198866=1198862

1198860

1198867=

1198863

119867lowast1+ 1198860

1198868= 1198866minus 1198864 119886

9= 1198866+ 1198867

11988610= 1198867minus 1198865 119886

11= 1198864+ 1198865

(17)


119906 (119910 119905) = 1198681+ 11988681198682minus 11988691198683+ 119886101198684+ 119886111198685+ 11988651198686


(18)

120579 (119910 119905) = 1198688 (19)

120601 (119910 119905) = 119878lowast

(11986811minus 1198686) + 1198689 (20)

with

1198681=1

2[(119905 minus

119910


119910


+119890119910radic119867 erf 119888 (

119910

2radic119905+ radic119867119905)(119905 +

119910

2radic119867)]

1198682=1

2[119890minus119910radic119867 erf 119888 (

119910


+119890119910radic119867 erf 119888 (

119910

2radic119905+ radic119867119905)]

1198683=119890minus1198860119905


0 erf 119888 (119910


0) 119905)

+119890119910radic119867minus119886

0 erf 119888 (119910


0) 119905)]

1198684=119890119867lowast

1119905

2[119890minus119910radic119867

lowast

1+119867 erf 119888 (

119910


1+ 119867) 119905)

+119890119910radic119867lowast

1+119867 erf 119888 (

119910


1+ 119867) 119905)]


1198685=119890119867lowast119905

2[119890minus119910radic119867

lowast+119867 erf 119888 (

119910


+119890119910radic119867lowast+119867 erf 119888 (

119910


1198686=119890119867lowast

1119905

2[119890minus119910radic119865

lowast119867lowast

1 erf 119888 (119910radic119865lowast


1119905)


1 erf 119888 (119910radic119865lowast


1119905)]

1198687=119890119867lowast119905

2[119890minus119910radic119865

lowast119867lowast






1198688= erf 119888 (


2radic119905)

1198689=1




+119890119910radic120574Sc erf 119888 (


+ radic120574119905)]

11986810=119890minus1198860119905


0) erf 119888 (



+ 119890119910radicSc(120574minus119886

0) erf 119888 (


+ radic(120574 minus 1198860) 119905)]

11986811=119890119867lowast

1119905


1+120574) erf 119888 (



+ 119890119910radicSc(119867lowast

1+120574) erf 119888 (


+ radic(119867lowast1+ 120574) 119905)]

(21)




120591lowast

= minus120583120597119906lowast

120597119910lowast

10038161003816100381610038161003816100381610038161003816119910lowast=0 (22)


120591 = minus120597119906

120597119910 120591 =

120591lowast

1205881198802119900

(23)


120591 =1198868119890minus119867119905

radic120587119905minus11988611

radic119865lowast

radic120587119905+1198864

radic119865lowast

radic120587119905+1198865

radic119865lowast


+119890minus119867119905radic119905

radic120587+119905radic119867

2(minus1 + erf (radic119867119905)) +

erf (radic119867119905)

2radic119867

+ 1198868


2(1 + erf (radic119867119905))



+ 1198865119890119867lowast



1119905)

minus1

211988611119890119867lowast119905


lowast119905




minus1

211988610119890119867lowast

1119905

minus2119890minus(119867+119867

lowast

1)119905


1



1) 119905))

minus1

21198869119890minus1198860119905

21198901198860119905minus119867119905


0



0119905 + 119867119905))

minus1

21198869119890minus1198860119905



0+ 120574) Sc


+ radic(minus1198860+ 120574) Sc


minus1

21198867119890119867lowast

1119905

2119890minus(119867

lowast

1+120574)119905


1+ 120574





(24)


Nu = minus119896120597119879lowast

120597119910lowast

10038161003816100381610038161003816100381610038161003816119910lowast=0

Nu = 120597120579

120597119910

10038161003816100381610038161003816100381610038161003816119910lowast=0


120587119905

(25)


Sh = 120597119862lowast

120597119910lowast

10038161003816100381610038161003816100381610038161003816119910lowast=0

Sh = minus1

radic120587119905119890minus120574119905


lowast


minus119890119867lowast119905+120574119905



lowast119905+120574119905

119878lowast


(26)










1

08

06

04

02

00 2 4 6 8 10 12 14

119910

119906

119872 = 0

119872 = 1119872 = 2119872 = 3

119905 = 1 119877 = 02 Pr = 071 Gr = 1119870 = 05 Gm = 1 120574 = 07 Sc = 2 Sr = 1


1

08

06

04

02

00 2 4 6 8 10 12 14

119910

119906

119870 = 05119870 = 10

119870 = 15119870 = 20

119905 = 1 119877 = 02 Pr = 071 Gr = 1119872 = 1 Gm = 1 120574 = 05 Sc = 2 Sr = 1





119905 = 1

119906

119910

25

2

15

1

05

00 2 4 6 8

Gm = 1119877 = 02 Pr = 071 Gr = 1119872 = 1 119870 = 2 120574 = 05 Sc = 06 Sr = 5

119905 = 15

119905 = 20

119905 = 25


12

1

08

06

04

02

0

119906

0 2 4 6 8 10 12 14119910

Gr = 0Gr = 1

Gr = 2Gr = 3

119905 = 1 119877 = 02119872 = 1 119870 = 2 120574 = 05 Sc = 2 Sr = 1

Pr = 071 Gm = 1


1

08

06

04

02

0

119906

0 1 2 3 4 5 6119910

Sc = 022Sc = 03

Sc = 06Sc = 1

119905 = 1 119877 = 02 Pr = 071 Gm = 1119872 = 2 119870 = 1 120574 = 05 Gr = 2 Sr = 15


175

15

125

1

075

05

025

00 5 10 15 20

119910

119906

Sr = 25Sr = 45

Sr = 65Sr = 8

119905 = 1 119877 = 02 Pr = 071 Gm = 1119872 = 2 119870 = 1 120574 = 05 Gr = 2 Sc = 15


0 05 1 15 2119910

0

02

04

06

08

1

12

119906

Pr = 100Pr = 7

Pr = 1Pr = 071Pr = 0015

119905 = 1 119877 = 1 Gr = 1119872 = 1119870 = 4 Gm = 02 120574 = 15 Sc = 2 Sr = 01






1

08

06

04

02

0

119910

0 10 20 30 40 50

119905 = 06 119877 = 05

Pr = 0015Pr = 071Pr = 10

Pr = 70Pr = 100

120579


12

1

08

06

04

02

0

119910

0 1 2 3 4 5 6

119905 = 1 119877 = 2 Sc = 1 Sr = 09

Pr = 0015Pr = 071Pr = 1

Pr = 7Pr = 100

120601

120574 = 02


119905 = 01 Pr = 0711

08

06

04

02

00 05 1 15 2 25 3

119910

119877 = 05

119877 = 15

119877 = 30

119877 = 45

120579


119905 = 03 119877 = 04 Sc = 09 Sr = 08 Pr = 10

0 05 1 15 2 25 3119910

1

08

06

04

02

0

120601

120574 = 0

120574 = 3

120574 = 6

120574 = 9


119905 = 05 119877 = 04 Sr = 04 Pr = 071

Sc = 1Sc = 2

Sc = 3Sc = 4

1

08

06

04

02

0

120601

0 05 1 15 2 25 3119910

120574 = 02


119905 = 03 119877 = 04 Sc = 03 Pr = 071

Sr = 0Sr = 1

Sr = 2Sr = 3

1

08

06

04

02

00 1 2 3 4

119910

120574 = 02

120601



119910 = 5

119910 = 10

119910 = 15

119910 = 20

5

4

3

2

1

0

119906

0 1 2 3 4 5 6119905

Sr = 1119877 = 02 Pr = 071 Gm = 1119872 = 01 119870 = 2 120574 = 05 Gr = 2 Sc = 2


Pr = 071119877 = 31

08

06

04

02

00 2 4 6 8 10

119905

119910 = 0

119910 = 2

119910 = 3

119910 = 7

120579




119877 = 04 Γ = 07 Sc = 03 Sr = 05 Pr = 071

0 1 2 3 4 5 6119905

1

08

06

04

02

0

119910 = 0

119910 = 1

119910 = 2119910 = 5

120601






119905 = 04 119877 = 1 Gr = 1 Gm = 1

(IV)

(III)

(II)

(I)

119906

04

03

02

01

0

119910

0 05 1 15 2 25 3 35


(I) 071 2 01

(II) 7 2 01

(III) 071 4 01

(IV) 071 2 04

120574 = 2 Sc = 06 Sr = 1





(I)

(II)

(III)

(IV)

(V)

119906

02

015

01

005

0

119910

0 05 1 15 2 25 3

119905 = 04 119877 = 1 Pr = 071 119872 = 2 120574 = 2 119870 = 01


(I)

(II)

(III)

(IV)

(V)

05

1

05

1

1

2

022

022

022

1

1

1

05

05

1

1

06

022

1

2


1

08

06

04

02

00 5 10 15 20

119910

(I)

(II)

(III)

(IV)

071 05 2

1 05 2

(I)

(II)

Curves Pr 119877 119905

071 1 2

071 05 3

(III)

(IV)

Curves Pr 119877 119905

120579




1

08

06

04

02

0

(I)

(II)

(III)

(IV)

(V)


(I)

(II)

(III)

(IV)

(V)

071

06

05

01

7

06

05

01

071

1

05

01 071 06

101

071

06

05 04

0 1 2 3 4 5 6119910

120601

120574 120574






Pr 119872 119870 Gr Sc Sr Gm 120591

071 1 05 1 2 1 1 127

1 1 05 1 2 1 1 130

071 2 05 1 2 1 1 205

071 1 1 1 2 1 1 094

071 1 05 2 2 1 1 085

071 1 05 1 3 1 1 131

071 1 05 1 2 3 1 125

071 1 05 1 2 1 2 096


Pr 119877 Nu071 02 043

1 02 051

071 04 040


Pr 120574 Sc Sr Sh071 1 06 2 136

1 1 06 2 084

071 2 06 2 19

071 1 1 2 096

071 1 06 3 164


5 Conclusions










Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1198685=119890119867lowast119905

2[119890minus119910radic119867

lowast+119867 erf 119888 (

119910


+119890119910radic119867lowast+119867 erf 119888 (

119910


1198686=119890119867lowast

1119905

2[119890minus119910radic119865

lowast119867lowast

1 erf 119888 (119910radic119865lowast


1119905)


1 erf 119888 (119910radic119865lowast


1119905)]

1198687=119890119867lowast119905

2[119890minus119910radic119865

lowast119867lowast






1198688= erf 119888 (


2radic119905)

1198689=1




+119890119910radic120574Sc erf 119888 (


+ radic120574119905)]

11986810=119890minus1198860119905


0) erf 119888 (



+ 119890119910radicSc(120574minus119886

0) erf 119888 (


+ radic(120574 minus 1198860) 119905)]

11986811=119890119867lowast

1119905


1+120574) erf 119888 (



+ 119890119910radicSc(119867lowast

1+120574) erf 119888 (


+ radic(119867lowast1+ 120574) 119905)]

(21)




120591lowast

= minus120583120597119906lowast

120597119910lowast

10038161003816100381610038161003816100381610038161003816119910lowast=0 (22)


120591 = minus120597119906

120597119910 120591 =

120591lowast

1205881198802119900

(23)


120591 =1198868119890minus119867119905

radic120587119905minus11988611

radic119865lowast

radic120587119905+1198864

radic119865lowast

radic120587119905+1198865

radic119865lowast


+119890minus119867119905radic119905

radic120587+119905radic119867

2(minus1 + erf (radic119867119905)) +

erf (radic119867119905)

2radic119867

+ 1198868


2(1 + erf (radic119867119905))



+ 1198865119890119867lowast



1119905)

minus1

211988611119890119867lowast119905


lowast119905




minus1

211988610119890119867lowast

1119905

minus2119890minus(119867+119867

lowast

1)119905


1



1) 119905))

minus1

21198869119890minus1198860119905

21198901198860119905minus119867119905


0



0119905 + 119867119905))

minus1

21198869119890minus1198860119905



0+ 120574) Sc


+ radic(minus1198860+ 120574) Sc


minus1

21198867119890119867lowast

1119905

2119890minus(119867

lowast

1+120574)119905


1+ 120574





(24)


Nu = minus119896120597119879lowast

120597119910lowast

10038161003816100381610038161003816100381610038161003816119910lowast=0

Nu = 120597120579

120597119910

10038161003816100381610038161003816100381610038161003816119910lowast=0


120587119905

(25)


Sh = 120597119862lowast

120597119910lowast

10038161003816100381610038161003816100381610038161003816119910lowast=0

Sh = minus1

radic120587119905119890minus120574119905


lowast


minus119890119867lowast119905+120574119905



lowast119905+120574119905

119878lowast


(26)










1

08

06

04

02

00 2 4 6 8 10 12 14

119910

119906

119872 = 0

119872 = 1119872 = 2119872 = 3

119905 = 1 119877 = 02 Pr = 071 Gr = 1119870 = 05 Gm = 1 120574 = 07 Sc = 2 Sr = 1


1

08

06

04

02

00 2 4 6 8 10 12 14

119910

119906

119870 = 05119870 = 10

119870 = 15119870 = 20

119905 = 1 119877 = 02 Pr = 071 Gr = 1119872 = 1 Gm = 1 120574 = 05 Sc = 2 Sr = 1





119905 = 1

119906

119910

25

2

15

1

05

00 2 4 6 8

Gm = 1119877 = 02 Pr = 071 Gr = 1119872 = 1 119870 = 2 120574 = 05 Sc = 06 Sr = 5

119905 = 15

119905 = 20

119905 = 25


12

1

08

06

04

02

0

119906

0 2 4 6 8 10 12 14119910

Gr = 0Gr = 1

Gr = 2Gr = 3

119905 = 1 119877 = 02119872 = 1 119870 = 2 120574 = 05 Sc = 2 Sr = 1

Pr = 071 Gm = 1


1

08

06

04

02

0

119906

0 1 2 3 4 5 6119910

Sc = 022Sc = 03

Sc = 06Sc = 1

119905 = 1 119877 = 02 Pr = 071 Gm = 1119872 = 2 119870 = 1 120574 = 05 Gr = 2 Sr = 15


175

15

125

1

075

05

025

00 5 10 15 20

119910

119906

Sr = 25Sr = 45

Sr = 65Sr = 8

119905 = 1 119877 = 02 Pr = 071 Gm = 1119872 = 2 119870 = 1 120574 = 05 Gr = 2 Sc = 15


0 05 1 15 2119910

0

02

04

06

08

1

12

119906

Pr = 100Pr = 7

Pr = 1Pr = 071Pr = 0015

119905 = 1 119877 = 1 Gr = 1119872 = 1119870 = 4 Gm = 02 120574 = 15 Sc = 2 Sr = 01






1

08

06

04

02

0

119910

0 10 20 30 40 50

119905 = 06 119877 = 05

Pr = 0015Pr = 071Pr = 10

Pr = 70Pr = 100

120579


12

1

08

06

04

02

0

119910

0 1 2 3 4 5 6

119905 = 1 119877 = 2 Sc = 1 Sr = 09

Pr = 0015Pr = 071Pr = 1

Pr = 7Pr = 100

120601

120574 = 02


119905 = 01 Pr = 0711

08

06

04

02

00 05 1 15 2 25 3

119910

119877 = 05

119877 = 15

119877 = 30

119877 = 45

120579


119905 = 03 119877 = 04 Sc = 09 Sr = 08 Pr = 10

0 05 1 15 2 25 3119910

1

08

06

04

02

0

120601

120574 = 0

120574 = 3

120574 = 6

120574 = 9


119905 = 05 119877 = 04 Sr = 04 Pr = 071

Sc = 1Sc = 2

Sc = 3Sc = 4

1

08

06

04

02

0

120601

0 05 1 15 2 25 3119910

120574 = 02


119905 = 03 119877 = 04 Sc = 03 Pr = 071

Sr = 0Sr = 1

Sr = 2Sr = 3

1

08

06

04

02

00 1 2 3 4

119910

120574 = 02

120601



119910 = 5

119910 = 10

119910 = 15

119910 = 20

5

4

3

2

1

0

119906

0 1 2 3 4 5 6119905

Sr = 1119877 = 02 Pr = 071 Gm = 1119872 = 01 119870 = 2 120574 = 05 Gr = 2 Sc = 2


Pr = 071119877 = 31

08

06

04

02

00 2 4 6 8 10

119905

119910 = 0

119910 = 2

119910 = 3

119910 = 7

120579




119877 = 04 Γ = 07 Sc = 03 Sr = 05 Pr = 071

0 1 2 3 4 5 6119905

1

08

06

04

02

0

119910 = 0

119910 = 1

119910 = 2119910 = 5

120601






119905 = 04 119877 = 1 Gr = 1 Gm = 1

(IV)

(III)

(II)

(I)

119906

04

03

02

01

0

119910

0 05 1 15 2 25 3 35


(I) 071 2 01

(II) 7 2 01

(III) 071 4 01

(IV) 071 2 04

120574 = 2 Sc = 06 Sr = 1





(I)

(II)

(III)

(IV)

(V)

119906

02

015

01

005

0

119910

0 05 1 15 2 25 3

119905 = 04 119877 = 1 Pr = 071 119872 = 2 120574 = 2 119870 = 01


(I)

(II)

(III)

(IV)

(V)

05

1

05

1

1

2

022

022

022

1

1

1

05

05

1

1

06

022

1

2


1

08

06

04

02

00 5 10 15 20

119910

(I)

(II)

(III)

(IV)

071 05 2

1 05 2

(I)

(II)

Curves Pr 119877 119905

071 1 2

071 05 3

(III)

(IV)

Curves Pr 119877 119905

120579




1

08

06

04

02

0

(I)

(II)

(III)

(IV)

(V)


(I)

(II)

(III)

(IV)

(V)

071

06

05

01

7

06

05

01

071

1

05

01 071 06

101

071

06

05 04

0 1 2 3 4 5 6119910

120601

120574 120574






Pr 119872 119870 Gr Sc Sr Gm 120591

071 1 05 1 2 1 1 127

1 1 05 1 2 1 1 130

071 2 05 1 2 1 1 205

071 1 1 1 2 1 1 094

071 1 05 2 2 1 1 085

071 1 05 1 3 1 1 131

071 1 05 1 2 3 1 125

071 1 05 1 2 1 2 096


Pr 119877 Nu071 02 043

1 02 051

071 04 040


Pr 120574 Sc Sr Sh071 1 06 2 136

1 1 06 2 084

071 2 06 2 19

071 1 1 2 096

071 1 06 3 164


5 Conclusions










Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












(24)


Nu = minus119896120597119879lowast

120597119910lowast

10038161003816100381610038161003816100381610038161003816119910lowast=0

Nu = 120597120579

120597119910

10038161003816100381610038161003816100381610038161003816119910lowast=0


120587119905

(25)


Sh = 120597119862lowast

120597119910lowast

10038161003816100381610038161003816100381610038161003816119910lowast=0

Sh = minus1

radic120587119905119890minus120574119905


lowast


minus119890119867lowast119905+120574119905



lowast119905+120574119905

119878lowast


(26)










1

08

06

04

02

00 2 4 6 8 10 12 14

119910

119906

119872 = 0

119872 = 1119872 = 2119872 = 3

119905 = 1 119877 = 02 Pr = 071 Gr = 1119870 = 05 Gm = 1 120574 = 07 Sc = 2 Sr = 1


1

08

06

04

02

00 2 4 6 8 10 12 14

119910

119906

119870 = 05119870 = 10

119870 = 15119870 = 20

119905 = 1 119877 = 02 Pr = 071 Gr = 1119872 = 1 Gm = 1 120574 = 05 Sc = 2 Sr = 1





119905 = 1

119906

119910

25

2

15

1

05

00 2 4 6 8

Gm = 1119877 = 02 Pr = 071 Gr = 1119872 = 1 119870 = 2 120574 = 05 Sc = 06 Sr = 5

119905 = 15

119905 = 20

119905 = 25


12

1

08

06

04

02

0

119906

0 2 4 6 8 10 12 14119910

Gr = 0Gr = 1

Gr = 2Gr = 3

119905 = 1 119877 = 02119872 = 1 119870 = 2 120574 = 05 Sc = 2 Sr = 1

Pr = 071 Gm = 1


1

08

06

04

02

0

119906

0 1 2 3 4 5 6119910

Sc = 022Sc = 03

Sc = 06Sc = 1

119905 = 1 119877 = 02 Pr = 071 Gm = 1119872 = 2 119870 = 1 120574 = 05 Gr = 2 Sr = 15


175

15

125

1

075

05

025

00 5 10 15 20

119910

119906

Sr = 25Sr = 45

Sr = 65Sr = 8

119905 = 1 119877 = 02 Pr = 071 Gm = 1119872 = 2 119870 = 1 120574 = 05 Gr = 2 Sc = 15


0 05 1 15 2119910

0

02

04

06

08

1

12

119906

Pr = 100Pr = 7

Pr = 1Pr = 071Pr = 0015

119905 = 1 119877 = 1 Gr = 1119872 = 1119870 = 4 Gm = 02 120574 = 15 Sc = 2 Sr = 01






1

08

06

04

02

0

119910

0 10 20 30 40 50

119905 = 06 119877 = 05

Pr = 0015Pr = 071Pr = 10

Pr = 70Pr = 100

120579


12

1

08

06

04

02

0

119910

0 1 2 3 4 5 6

119905 = 1 119877 = 2 Sc = 1 Sr = 09

Pr = 0015Pr = 071Pr = 1

Pr = 7Pr = 100

120601

120574 = 02


119905 = 01 Pr = 0711

08

06

04

02

00 05 1 15 2 25 3

119910

119877 = 05

119877 = 15

119877 = 30

119877 = 45

120579


119905 = 03 119877 = 04 Sc = 09 Sr = 08 Pr = 10

0 05 1 15 2 25 3119910

1

08

06

04

02

0

120601

120574 = 0

120574 = 3

120574 = 6

120574 = 9


119905 = 05 119877 = 04 Sr = 04 Pr = 071

Sc = 1Sc = 2

Sc = 3Sc = 4

1

08

06

04

02

0

120601

0 05 1 15 2 25 3119910

120574 = 02


119905 = 03 119877 = 04 Sc = 03 Pr = 071

Sr = 0Sr = 1

Sr = 2Sr = 3

1

08

06

04

02

00 1 2 3 4

119910

120574 = 02

120601



119910 = 5

119910 = 10

119910 = 15

119910 = 20

5

4

3

2

1

0

119906

0 1 2 3 4 5 6119905

Sr = 1119877 = 02 Pr = 071 Gm = 1119872 = 01 119870 = 2 120574 = 05 Gr = 2 Sc = 2


Pr = 071119877 = 31

08

06

04

02

00 2 4 6 8 10

119905

119910 = 0

119910 = 2

119910 = 3

119910 = 7

120579




119877 = 04 Γ = 07 Sc = 03 Sr = 05 Pr = 071

0 1 2 3 4 5 6119905

1

08

06

04

02

0

119910 = 0

119910 = 1

119910 = 2119910 = 5

120601






119905 = 04 119877 = 1 Gr = 1 Gm = 1

(IV)

(III)

(II)

(I)

119906

04

03

02

01

0

119910

0 05 1 15 2 25 3 35


(I) 071 2 01

(II) 7 2 01

(III) 071 4 01

(IV) 071 2 04

120574 = 2 Sc = 06 Sr = 1





(I)

(II)

(III)

(IV)

(V)

119906

02

015

01

005

0

119910

0 05 1 15 2 25 3

119905 = 04 119877 = 1 Pr = 071 119872 = 2 120574 = 2 119870 = 01


(I)

(II)

(III)

(IV)

(V)

05

1

05

1

1

2

022

022

022

1

1

1

05

05

1

1

06

022

1

2


1

08

06

04

02

00 5 10 15 20

119910

(I)

(II)

(III)

(IV)

071 05 2

1 05 2

(I)

(II)

Curves Pr 119877 119905

071 1 2

071 05 3

(III)

(IV)

Curves Pr 119877 119905

120579




1

08

06

04

02

0

(I)

(II)

(III)

(IV)

(V)


(I)

(II)

(III)

(IV)

(V)

071

06

05

01

7

06

05

01

071

1

05

01 071 06

101

071

06

05 04

0 1 2 3 4 5 6119910

120601

120574 120574






Pr 119872 119870 Gr Sc Sr Gm 120591

071 1 05 1 2 1 1 127

1 1 05 1 2 1 1 130

071 2 05 1 2 1 1 205

071 1 1 1 2 1 1 094

071 1 05 2 2 1 1 085

071 1 05 1 3 1 1 131

071 1 05 1 2 3 1 125

071 1 05 1 2 1 2 096


Pr 119877 Nu071 02 043

1 02 051

071 04 040


Pr 120574 Sc Sr Sh071 1 06 2 136

1 1 06 2 084

071 2 06 2 19

071 1 1 2 096

071 1 06 3 164


5 Conclusions










Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119905 = 1

119906

119910

25

2

15

1

05

00 2 4 6 8

Gm = 1119877 = 02 Pr = 071 Gr = 1119872 = 1 119870 = 2 120574 = 05 Sc = 06 Sr = 5

119905 = 15

119905 = 20

119905 = 25


12

1

08

06

04

02

0

119906

0 2 4 6 8 10 12 14119910

Gr = 0Gr = 1

Gr = 2Gr = 3

119905 = 1 119877 = 02119872 = 1 119870 = 2 120574 = 05 Sc = 2 Sr = 1

Pr = 071 Gm = 1


1

08

06

04

02

0

119906

0 1 2 3 4 5 6119910

Sc = 022Sc = 03

Sc = 06Sc = 1

119905 = 1 119877 = 02 Pr = 071 Gm = 1119872 = 2 119870 = 1 120574 = 05 Gr = 2 Sr = 15


175

15

125

1

075

05

025

00 5 10 15 20

119910

119906

Sr = 25Sr = 45

Sr = 65Sr = 8

119905 = 1 119877 = 02 Pr = 071 Gm = 1119872 = 2 119870 = 1 120574 = 05 Gr = 2 Sc = 15


0 05 1 15 2119910

0

02

04

06

08

1

12

119906

Pr = 100Pr = 7

Pr = 1Pr = 071Pr = 0015

119905 = 1 119877 = 1 Gr = 1119872 = 1119870 = 4 Gm = 02 120574 = 15 Sc = 2 Sr = 01






1

08

06

04

02

0

119910

0 10 20 30 40 50

119905 = 06 119877 = 05

Pr = 0015Pr = 071Pr = 10

Pr = 70Pr = 100

120579


12

1

08

06

04

02

0

119910

0 1 2 3 4 5 6

119905 = 1 119877 = 2 Sc = 1 Sr = 09

Pr = 0015Pr = 071Pr = 1

Pr = 7Pr = 100

120601

120574 = 02


119905 = 01 Pr = 0711

08

06

04

02

00 05 1 15 2 25 3

119910

119877 = 05

119877 = 15

119877 = 30

119877 = 45

120579


119905 = 03 119877 = 04 Sc = 09 Sr = 08 Pr = 10

0 05 1 15 2 25 3119910

1

08

06

04

02

0

120601

120574 = 0

120574 = 3

120574 = 6

120574 = 9


119905 = 05 119877 = 04 Sr = 04 Pr = 071

Sc = 1Sc = 2

Sc = 3Sc = 4

1

08

06

04

02

0

120601

0 05 1 15 2 25 3119910

120574 = 02


119905 = 03 119877 = 04 Sc = 03 Pr = 071

Sr = 0Sr = 1

Sr = 2Sr = 3

1

08

06

04

02

00 1 2 3 4

119910

120574 = 02

120601



119910 = 5

119910 = 10

119910 = 15

119910 = 20

5

4

3

2

1

0

119906

0 1 2 3 4 5 6119905

Sr = 1119877 = 02 Pr = 071 Gm = 1119872 = 01 119870 = 2 120574 = 05 Gr = 2 Sc = 2


Pr = 071119877 = 31

08

06

04

02

00 2 4 6 8 10

119905

119910 = 0

119910 = 2

119910 = 3

119910 = 7

120579




119877 = 04 Γ = 07 Sc = 03 Sr = 05 Pr = 071

0 1 2 3 4 5 6119905

1

08

06

04

02

0

119910 = 0

119910 = 1

119910 = 2119910 = 5

120601






119905 = 04 119877 = 1 Gr = 1 Gm = 1

(IV)

(III)

(II)

(I)

119906

04

03

02

01

0

119910

0 05 1 15 2 25 3 35


(I) 071 2 01

(II) 7 2 01

(III) 071 4 01

(IV) 071 2 04

120574 = 2 Sc = 06 Sr = 1





(I)

(II)

(III)

(IV)

(V)

119906

02

015

01

005

0

119910

0 05 1 15 2 25 3

119905 = 04 119877 = 1 Pr = 071 119872 = 2 120574 = 2 119870 = 01


(I)

(II)

(III)

(IV)

(V)

05

1

05

1

1

2

022

022

022

1

1

1

05

05

1

1

06

022

1

2


1

08

06

04

02

00 5 10 15 20

119910

(I)

(II)

(III)

(IV)

071 05 2

1 05 2

(I)

(II)

Curves Pr 119877 119905

071 1 2

071 05 3

(III)

(IV)

Curves Pr 119877 119905

120579




1

08

06

04

02

0

(I)

(II)

(III)

(IV)

(V)


(I)

(II)

(III)

(IV)

(V)

071

06

05

01

7

06

05

01

071

1

05

01 071 06

101

071

06

05 04

0 1 2 3 4 5 6119910

120601

120574 120574






Pr 119872 119870 Gr Sc Sr Gm 120591

071 1 05 1 2 1 1 127

1 1 05 1 2 1 1 130

071 2 05 1 2 1 1 205

071 1 1 1 2 1 1 094

071 1 05 2 2 1 1 085

071 1 05 1 3 1 1 131

071 1 05 1 2 3 1 125

071 1 05 1 2 1 2 096


Pr 119877 Nu071 02 043

1 02 051

071 04 040


Pr 120574 Sc Sr Sh071 1 06 2 136

1 1 06 2 084

071 2 06 2 19

071 1 1 2 096

071 1 06 3 164


5 Conclusions










Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

08

06

04

02

0

119910

0 10 20 30 40 50

119905 = 06 119877 = 05

Pr = 0015Pr = 071Pr = 10

Pr = 70Pr = 100

120579


12

1

08

06

04

02

0

119910

0 1 2 3 4 5 6

119905 = 1 119877 = 2 Sc = 1 Sr = 09

Pr = 0015Pr = 071Pr = 1

Pr = 7Pr = 100

120601

120574 = 02


119905 = 01 Pr = 0711

08

06

04

02

00 05 1 15 2 25 3

119910

119877 = 05

119877 = 15

119877 = 30

119877 = 45

120579


119905 = 03 119877 = 04 Sc = 09 Sr = 08 Pr = 10

0 05 1 15 2 25 3119910

1

08

06

04

02

0

120601

120574 = 0

120574 = 3

120574 = 6

120574 = 9


119905 = 05 119877 = 04 Sr = 04 Pr = 071

Sc = 1Sc = 2

Sc = 3Sc = 4

1

08

06

04

02

0

120601

0 05 1 15 2 25 3119910

120574 = 02


119905 = 03 119877 = 04 Sc = 03 Pr = 071

Sr = 0Sr = 1

Sr = 2Sr = 3

1

08

06

04

02

00 1 2 3 4

119910

120574 = 02

120601



119910 = 5

119910 = 10

119910 = 15

119910 = 20

5

4

3

2

1

0

119906

0 1 2 3 4 5 6119905

Sr = 1119877 = 02 Pr = 071 Gm = 1119872 = 01 119870 = 2 120574 = 05 Gr = 2 Sc = 2


Pr = 071119877 = 31

08

06

04

02

00 2 4 6 8 10

119905

119910 = 0

119910 = 2

119910 = 3

119910 = 7

120579




119877 = 04 Γ = 07 Sc = 03 Sr = 05 Pr = 071

0 1 2 3 4 5 6119905

1

08

06

04

02

0

119910 = 0

119910 = 1

119910 = 2119910 = 5

120601






119905 = 04 119877 = 1 Gr = 1 Gm = 1

(IV)

(III)

(II)

(I)

119906

04

03

02

01

0

119910

0 05 1 15 2 25 3 35


(I) 071 2 01

(II) 7 2 01

(III) 071 4 01

(IV) 071 2 04

120574 = 2 Sc = 06 Sr = 1





(I)

(II)

(III)

(IV)

(V)

119906

02

015

01

005

0

119910

0 05 1 15 2 25 3

119905 = 04 119877 = 1 Pr = 071 119872 = 2 120574 = 2 119870 = 01


(I)

(II)

(III)

(IV)

(V)

05

1

05

1

1

2

022

022

022

1

1

1

05

05

1

1

06

022

1

2


1

08

06

04

02

00 5 10 15 20

119910

(I)

(II)

(III)

(IV)

071 05 2

1 05 2

(I)

(II)

Curves Pr 119877 119905

071 1 2

071 05 3

(III)

(IV)

Curves Pr 119877 119905

120579




1

08

06

04

02

0

(I)

(II)

(III)

(IV)

(V)


(I)

(II)

(III)

(IV)

(V)

071

06

05

01

7

06

05

01

071

1

05

01 071 06

101

071

06

05 04

0 1 2 3 4 5 6119910

120601

120574 120574






Pr 119872 119870 Gr Sc Sr Gm 120591

071 1 05 1 2 1 1 127

1 1 05 1 2 1 1 130

071 2 05 1 2 1 1 205

071 1 1 1 2 1 1 094

071 1 05 2 2 1 1 085

071 1 05 1 3 1 1 131

071 1 05 1 2 3 1 125

071 1 05 1 2 1 2 096


Pr 119877 Nu071 02 043

1 02 051

071 04 040


Pr 120574 Sc Sr Sh071 1 06 2 136

1 1 06 2 084

071 2 06 2 19

071 1 1 2 096

071 1 06 3 164


5 Conclusions










Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119910 = 5

119910 = 10

119910 = 15

119910 = 20

5

4

3

2

1

0

119906

0 1 2 3 4 5 6119905

Sr = 1119877 = 02 Pr = 071 Gm = 1119872 = 01 119870 = 2 120574 = 05 Gr = 2 Sc = 2


Pr = 071119877 = 31

08

06

04

02

00 2 4 6 8 10

119905

119910 = 0

119910 = 2

119910 = 3

119910 = 7

120579




119877 = 04 Γ = 07 Sc = 03 Sr = 05 Pr = 071

0 1 2 3 4 5 6119905

1

08

06

04

02

0

119910 = 0

119910 = 1

119910 = 2119910 = 5

120601






119905 = 04 119877 = 1 Gr = 1 Gm = 1

(IV)

(III)

(II)

(I)

119906

04

03

02

01

0

119910

0 05 1 15 2 25 3 35


(I) 071 2 01

(II) 7 2 01

(III) 071 4 01

(IV) 071 2 04

120574 = 2 Sc = 06 Sr = 1





(I)

(II)

(III)

(IV)

(V)

119906

02

015

01

005

0

119910

0 05 1 15 2 25 3

119905 = 04 119877 = 1 Pr = 071 119872 = 2 120574 = 2 119870 = 01


(I)

(II)

(III)

(IV)

(V)

05

1

05

1

1

2

022

022

022

1

1

1

05

05

1

1

06

022

1

2


1

08

06

04

02

00 5 10 15 20

119910

(I)

(II)

(III)

(IV)

071 05 2

1 05 2

(I)

(II)

Curves Pr 119877 119905

071 1 2

071 05 3

(III)

(IV)

Curves Pr 119877 119905

120579




1

08

06

04

02

0

(I)

(II)

(III)

(IV)

(V)


(I)

(II)

(III)

(IV)

(V)

071

06

05

01

7

06

05

01

071

1

05

01 071 06

101

071

06

05 04

0 1 2 3 4 5 6119910

120601

120574 120574






Pr 119872 119870 Gr Sc Sr Gm 120591

071 1 05 1 2 1 1 127

1 1 05 1 2 1 1 130

071 2 05 1 2 1 1 205

071 1 1 1 2 1 1 094

071 1 05 2 2 1 1 085

071 1 05 1 3 1 1 131

071 1 05 1 2 3 1 125

071 1 05 1 2 1 2 096


Pr 119877 Nu071 02 043

1 02 051

071 04 040


Pr 120574 Sc Sr Sh071 1 06 2 136

1 1 06 2 084

071 2 06 2 19

071 1 1 2 096

071 1 06 3 164


5 Conclusions










Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119905 = 04 119877 = 1 Gr = 1 Gm = 1

(IV)

(III)

(II)

(I)

119906

04

03

02

01

0

119910

0 05 1 15 2 25 3 35


(I) 071 2 01

(II) 7 2 01

(III) 071 4 01

(IV) 071 2 04

120574 = 2 Sc = 06 Sr = 1





(I)

(II)

(III)

(IV)

(V)

119906

02

015

01

005

0

119910

0 05 1 15 2 25 3

119905 = 04 119877 = 1 Pr = 071 119872 = 2 120574 = 2 119870 = 01


(I)

(II)

(III)

(IV)

(V)

05

1

05

1

1

2

022

022

022

1

1

1

05

05

1

1

06

022

1

2


1

08

06

04

02

00 5 10 15 20

119910

(I)

(II)

(III)

(IV)

071 05 2

1 05 2

(I)

(II)

Curves Pr 119877 119905

071 1 2

071 05 3

(III)

(IV)

Curves Pr 119877 119905

120579




1

08

06

04

02

0

(I)

(II)

(III)

(IV)

(V)


(I)

(II)

(III)

(IV)

(V)

071

06

05

01

7

06

05

01

071

1

05

01 071 06

101

071

06

05 04

0 1 2 3 4 5 6119910

120601

120574 120574






Pr 119872 119870 Gr Sc Sr Gm 120591

071 1 05 1 2 1 1 127

1 1 05 1 2 1 1 130

071 2 05 1 2 1 1 205

071 1 1 1 2 1 1 094

071 1 05 2 2 1 1 085

071 1 05 1 3 1 1 131

071 1 05 1 2 3 1 125

071 1 05 1 2 1 2 096


Pr 119877 Nu071 02 043

1 02 051

071 04 040


Pr 120574 Sc Sr Sh071 1 06 2 136

1 1 06 2 084

071 2 06 2 19

071 1 1 2 096

071 1 06 3 164


5 Conclusions










Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

08

06

04

02

0

(I)

(II)

(III)

(IV)

(V)


(I)

(II)

(III)

(IV)

(V)

071

06

05

01

7

06

05

01

071

1

05

01 071 06

101

071

06

05 04

0 1 2 3 4 5 6119910

120601

120574 120574






Pr 119872 119870 Gr Sc Sr Gm 120591

071 1 05 1 2 1 1 127

1 1 05 1 2 1 1 130

071 2 05 1 2 1 1 205

071 1 1 1 2 1 1 094

071 1 05 2 2 1 1 085

071 1 05 1 3 1 1 131

071 1 05 1 2 3 1 125

071 1 05 1 2 1 2 096


Pr 119877 Nu071 02 043

1 02 051

071 04 040


Pr 120574 Sc Sr Sh071 1 06 2 136

1 1 06 2 084

071 2 06 2 19

071 1 1 2 096

071 1 06 3 164


5 Conclusions










Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















Acknowledgment


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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