international journal of heat and mass transfervafai.engr.ucr.edu/publications/2014a/ellahi.pdf ·...

International Journal of Heat and Mass Transfer 71 (2014) 706–719

Contents lists available at ScienceDirect

International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

Effects of heat and mass transfer on peristaltic flow in a non-uniformrectangular duct

0017-9310/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.12.038

⇑ Corresponding author at: Department of Mechanical Engineering, Bourns Hall,University of California Riverside, CA 92521, USA. Tel.: +1 951 275 6747.

E-mail addresses: [email protected], [email protected] (R. Ellahi).

R. Ellahi a,b,⇑, M. Mubashir Bhatti b, K. Vafai a

a Department of Mechanical Engineering, Bourns Hall, University of California Riverside, CA 92521, USAb Department of Mathematics and Statistics, FBAS, IIU, 44000 Islamabad, Pakistan

a r t i c l e i n f o

Article history:Received 22 August 2013Received in revised form 6 December 2013Accepted 13 December 2013

Keywords:Peristaltic flowHeat and mass transferRectangular ductAnalytical solution

a b s t r a c t

The model of bioheat transfer in tissues has attracted many researchers due to its application in thermo-therapy and human thermoregulation system. Currently bioheat is considered as a heat transfer inhuman body. In view of this the influence of heat and mass transfer on peristaltic flow in a non-uniformrectangular duct is studied under the consideration of long wavelength ð0� k!1Þ and low Reynoldsnumber ðRe! 0Þ. The flow is examined in wave frame of reference moving with the velocity c. Mathe-matical modeling is based upon the laws of mass, linear momentum, energy and concentration. Analysisis also presented for Prandtl number Pr, Eckert number E, Schmidt number Sc and Soret number Sr. Theinfluence of various emerging parameters of interest is seen for both two and three dimensional graphs.Numerical integration is used to analyze the novel features of volumetric flow rate, average volume flowrate, instantaneous flux and pressure gradient. The trapping bolus phenomena is also presented throughstream lines.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

It is now well established fact that most of the physiological flu-ids are non-Newtonian in character [1–10]. Several models havebeen proposed to explain such physiological fluids in order to findthe treatment of diagnostic problems that arise during the circula-tion in a human body. Practical applications and academic curios-ity in physiology have generated a lot of interest in studying theperistaltic motion in ducts. Peristaltic mechanism deals with thefluid transport that occurs by a progressive wave of area of con-traction or expansion along a length of tubes/channels. Peristaltictransport is found in living body such as movement of ovum in fe-male fallopian tube, swallowing food through esophagus, transportof lymph in lymphatic vessels vasomation of small blood vesselslike arterioles, venules, capillaries, transport of spermatozoa inducts efferentes of male reproductive tract etc. After the experi-mental work of Latham [11] on peristaltic transport, several theo-retical studies [12–20] have been undertaken by many researcherson peristaltic motion under one or more simplified assumptions ofsmall amplitude ratio, small wave number, low Reynolds numberand long wavelength etc.

Peristaltic flow with heat and mass transfer has many applica-tions in biomedical sciences and industry such as conduction in

tissues, heat convection due to blood flow from the pores of tissuesand radiation between environment and its surface, food process-ing and vasodilation. The processes of oxygenation and hemodial-ysis have also been visualized by considering peristaltic flows withheat transfer. Obviously there is a certain role of mass transfer inall these processes. Mass transfer is important phenomenon in dif-fusion process such as nutrients diffuse out from the blood toneighboring tissues. Mass transfer also occurs in many industrialprocesses like membrane separation process, reverse osmosis, dis-tillation process, combustion process and diffusion of chemicalimpurities. When the effects of heat and mass transfer are consid-ered simultaneously then the complicated relationships occur be-tween driving potentials and fluxes. The energy flux is inducedby temperature gradient. The composition gradients and mass fluxcan be produced by temperature gradient which is known as Soreteffect.

Investigations of heat and mass transfer in peristalsis yet havebeen considered by few researchers. For instance, The influenceof heat transfer and magnetic field on peristaltic transport of New-tonian fluid in a vertical annulus has been discussed by Mekheimerand elmaboud [21]. Ellahi et al. [22] investigated the series solu-tions for magnetohydrodynamic flow of non-Newtonian nanofluidand heat transfer in coaxial porous cylinder with slip conditions.Influence of heat and mass transfer on peristaltic flow of Eyring–Powell and Jeffrey’s fluids are discussed by Akbar and Nadeem[23,24]. Radhakrishnamacharya and Murty [25] studied the heattransfer on the peristaltic transport in a non-uniform channel.

http://crossmark.crossref.org/dialog/?doi=10.1016/j.ijheatmasstransfer.2013.12.038&domain=pdf

http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.12.038

mailto:[email protected]

mailto:[email protected]

http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.12.038

http://www.sciencedirect.com/science/journal/00179310

http://www.elsevier.com/locate/ijhmt

R. Ellahi et al. / International Journal of Heat and Mass Transfer 71 (2014) 706–719 707

Vajravelu et al. [26] reported the peristaltic flow and heat transferin a vertical annulus under long wavelength approximation. Acareful review of the literature reveals that a very little effortsare yet devoted to examine the peristaltic flow in a rectangularduct. Some relevant studies on the topic can be found from the listof references [27–29].

To the best of our knowledge, no attempt is made to investi-gate the peristaltic flow in a non-uniform rectangular duct withheat and mass transfer so far. Such consideration is veryimportant since the heat transfer in human tissues involves com-plicated processes like heat transfer due to perfusion of arterial-venous blood through the pores of tissue, heat conduction intissues, external interactions and metabolic heat generation suchas electromagnetic radiation emitted from cell phones etc.Moreover, it is well known that heat and mass transfer problemin the presence of chemical reaction is very significant in the pro-cesses of geothermal reservoirs, drying, enhanced oil recovery,flow in a desert cooler, cooling of nuclear reactors thermal insu-lation and evaporation at the surface of a water body. These typesof flows involve many practical operations such as molecular dif-fusion of species in presence of chemical reaction within or atboundary. Heat and mass transfer effects are also encounteredin chemical industry like in the study of hot salty springs insea, in thermal recovery processes and in reservoirs. The resultsobtained for title problem reveal many interesting behaviors thatwarrant further study on heat and mass transfer problems withchemical reaction.

Fig. 1. Geometry o

Fig. 2. Velocity profile for different values of Q for fixed K� ¼ 0:9;

The flow modeling is based on continuity, momentum and en-ergy equations. These equations are first expressed in terms ofstream function and then solved in closed form when the longwavelength and small Reynolds number assumption hold. The ef-fects of magnetic field are taken into account. The obtained expres-sions are utilized to discuss the role of emerging parameters on theflow quantities. Numerical computations has been used to evaluatethe expression for pressure rise. Stream lines of various interestingparameters. Finally, the effect of various emerging parameters arediscussed through graphs and trapping phenomenon. This paper isarranged as follows. Section two presents the mathematicalformulation for the problem of interest. Section three deals withthe solution of the problem. Finally section four synthesis detailedcomputational results and discussion with the physical interpreta-tion of our findings.

2. Mathematical formulation

We consider the peristaltic flow of an incompressible viscusfluid in a non-uniform duct of rectangular cross section havingchannel width 2d and height 2aþ 2kX. We choose Cartesian coor-dinate system in such a way that X-axis is taken along the axialdirection, Y-axis is taken along the lateral direction and Z-axis isalong the vertical direction of channel. The flow geometry of theproblem [30] presented as Fig. 1.

Sinusoidal waves of long wavelength are assumed to travel withthe velocity c along the walls of channel. These waves are

f the problem.

b ¼ 1:2; / ¼ 0:6 (a) for 2-dimensional (b) for 3-dimensional.

Fig. 3. Velocity profile for different values of b for fixed K� ¼ 0:9; Q ¼ 2; / ¼ 0:6 (a) for 2-dimensional (b) for 3-dimensional.

Fig. 4. Temperature distribution for different values of Q for fixed K� ¼ 0:9; b ¼ 0:45; E ¼ 1; Pr ¼ 1; / ¼ 0:6 (a) for 2-dimensional (b) for 3-dimensional.

Fig. 5. Temperature distribution for different values of Pr at K� ¼ 0:9; b ¼ 0:45; E ¼ 1; / ¼ 0:6; Q ¼ 1 (a) for 2-dimensional (b) for 3-dimensional.

708 R. Ellahi et al. / International Journal of Heat and Mass Transfer 71 (2014) 706–719

physiologically from neuromuscular properties of any tubularsmooth muscle. From the physical point of view, this is taken careof by an elastic foundation in the mathematical model that beforeperistaltic motion starts, these walls have to support the hydro-static pressure P. The configuration of the wall surface is given by

Z ¼ HðX; tÞ ¼ �a� kX � b sin2pkðX � ctÞ

� �; ð1Þ

where a and b are amplitudes of the waves, k is wavelength, t istime and X represents the direction of wave propagation. The walls

Fig. 6. Temperature distribution for different values of E at K� ¼ 0:9; b ¼ 0:45; Pr ¼ 1; / ¼ 0:6; Q ¼ 1 (a) for 2-dimensional (b) for 3-dimensional.

Fig. 7. Temperature distribution for different values of b at K� ¼ 0:9; b ¼ 0:45; E ¼ 1; Pr ¼ 1; / ¼ 0:6; Q ¼ 1 (a) for 2-dimensional (b) for 3-dimensional.

Fig. 8. Concentration distribution for different values of b at K� ¼ 0:9; Pr ¼ 2; E ¼ 1; Sc ¼ 1; Sr ¼ 1; Q ¼ 1; / ¼ 0:6 (a) for 2-dimensional (b) for 3-dimensional.


parallel to XZ-plane are not interrupted so here peristaltic wave mo-tion do not contribute. Moreover, the lateral velocity is zero as thereis no change in lateral direction of the duct cross section.

In laboratory frame ðX;YÞ the governing equations, which de-scribe the flow of an incompressible viscous fluid with heat andmass transfer are given as follows:

@U@Xþ @W@Z¼ 0; ð2Þ

q@U@tþ U

@U@XþW

@U@Z

� �¼ � @P

@Zþ @

2U

@X2 þ@2U

@Y2 þ@2U

@Z2 ; ð3Þ

0 ¼ � @P@Yþ @

2V

@X2 þ@2V

@Y2 þ@2V

@Z2 ; ð4Þ

q@W@tþ U

@W@XþW

@W@Z

� �¼ � @P

@Zþ @

2W

@X2 þ@2W

@Y2 þ@2W

@Z2 ; ð5Þ

Fig. 9. Concentration distribution for different values of Sr at K� ¼ 0:9; Pr ¼ 2; E ¼ 1; b ¼ 0:45; Sc ¼ 1; Q ¼ 1; / ¼ 0:6 (a) for 2-dimensional (b) for 3-dimensional.

Fig. 10. Concentration distribution for different values of Sc at K� ¼ 0:9; Pr ¼ 2; E ¼ 1; b ¼ 0:45; Sr ¼ 1; Q ¼ 1; / ¼ 0:6 (a) for 2-dimensional (b) for 3-dimensional.

Fig. 11. Concentration distribution for different values of Q at K� ¼ 0:9; Pr ¼ 2; E ¼ 1; b ¼ 0:45; Sc ¼ 1; Sr ¼ 1; / ¼ 0:6 (a) for 2-dimensional (b) for 3-dimensional.


f@T@tþ U

@T@XþW

@T@Z

� �¼ j

q@2T

@X2 þ@2T

@Y2 þ@2T

@Z2

!

þ m 2@U@X

� �2

þ @W@Z

� �2" #

þ @U@Y

� �2

þ @W@Y

� �2

þ @U@Zþ @W@X

� �2( )

;

ð6Þ

@C@tþU

@C@XþW

@C@Z¼D

@2C

@X2þ@2C

@Y2þ@2C

@Z2

!þDKT

Tm

@2T

@X2þ@2T

@Y2þ@2T

@Z2

!;

ð7Þ

in which q is the fluid density, f is the specific heat at constantvolume, m is the kinematic viscosity of the fluid, j is the thermal

Fig. 12. Concentration distribution for different values of Pr at K� ¼ 0:9; E ¼ 1; b ¼ 0:45; Sc ¼ 1; Sr ¼ 1; Q ¼ 1; / ¼ 0:6 (a) for 2-dimensional (b) for 3-dimensional.

Fig. 13. Concentration distribution for different values of E at K� ¼ 0:9; Pr ¼ 2; b ¼ 0:45; Sc ¼ 1; Sr ¼ 1; Q ¼ 1; / ¼ 0:6 (a) for 2-dimensional (b) for 3-dimensional.

Fig. 14. Variation of dp=dx with x for different values of b at / ¼ 0:7; K� ¼ 0:45;Q ¼ 0:05. Fig. 15. Variation of dp=dx with x for different values of K� at / ¼ 0:65; b ¼ 1;

Q ¼ 0:01.


conductivity of the fluid, D is the coefficient of mass diffusivity, Tm isthe mean temperature, KT is the thermal diffusion ratio, T and C aretemperature and concentration of the fluid, respectively.

Under the assumptions that the channel length is an integralmultiple of the wave length k and the pressure difference acrossthe ends of the channel is a constant, the flow is inherently unstea-dy in the laboratory frame ðX;YÞ and become steady in the waveframe ðx; yÞ which is moving with velocity ‘c’ along the wave. Thetransformation between these two frames is given by

x ¼ X � ct; y ¼ Y; z ¼ Z; u ¼ U � c; w ¼W ; pðx; zÞ ¼ PðX; Z; tÞ:ð8Þ

where U;V ; P are the velocity components, pressure in the labora-tory frame and u;v ;p are the velocity components, pressure in thewave frame, respectively.

The variables are rendered dimensionless by using the follow-ing quantities

Fig. 16. Variation of dp=dx with x for different values of / at K� ¼ 0:65; b ¼ 1;Q ¼ 0:01.

Fig. 17. Variation of dp=dx with x for different values of Q at K� ¼ 0:5; b ¼ 0:9;/ ¼ 0:7.

Fig. 18. Variation of Dp with Q for different values of b at K� ¼ 0:5; / ¼ 0:7.

Fig. 19. Variation of Dp with Q for different values of / at b ¼ 0:5; K� ¼ 0:5.

Fig. 20. Variation of Dp with Q for different values of K� at / ¼ 0:5; b ¼ 0:5.


x ¼ xk; y ¼ y

d; z ¼ z

a; u ¼ u

c; w ¼ w

cd; t ¼ ct

k;

h ¼ Ha; p ¼ a2p

lck; Re ¼ qac

l; d ¼ a

k; / ¼ b

a; K� ¼ k

a;

b ¼ ad; h ¼ T � T0

T1 � T0; u ¼ C � C0

C1 � C0ð9Þ

Using the above non-dimensional quantities in Eqs. (2)–(7), theequations governing the flow after dropping the bars become

@u@xþ @w@z¼ 0; ð10Þ

Re d u@u@xþw

@u@z

� �¼ � @p

@xþ d

@2u@x2 þ b2 @

2u@y2 þ

@2u@z2 ; ð11Þ

0 ¼ � @p@yþ d2 @

2v@x2 þ d2 @

2v@y2 þ d

@2v@z2 ; ð12Þ

Re d2 u@w@xþw

@w@z

� �¼ � @p

@zþ d2 @

2w@x2 þ db2 @

2w@y2 þ d2 @

2w@z2 ; ð13Þ

Re d u@h@xþdw

@h@z

� �¼� k

fmqd2 @

2h@x2þb2 @

2h@y2þ@

2h@z2

!

þ c2

fðT1�T0Þd2 @w

@z

� �2

þb2 @u@y

� �2

þ2d2 @u@x

� �2

þ2d2 @w@y

� �2

þ @u@zþd2 @w

@x

� �2" #

;

ð14Þ

Re d u@u@xþwd

@u@z

� �¼ �Dq

ld2 @

2u@x2 þ b2 @

2u@y2 þ

@2u@z2

!

þ qDKTðT1 � T0ÞlTmðC1 � C0Þ

d2 @2h@x2 þ b2 @

2h@y2 þ

@2h@z2

!; ð15Þ

in which

Pr ¼ fmqk; E ¼ c2

fðT1 � T0Þ; Sr ¼ qDKTðT1 � T0Þ

lTmðC1 � C0Þ; Sc

¼ lDq

; ð16Þ

where Re is the Reynolds number, d is the dimensionless wavenumber, Pr is the Prandtl number, E is the Eckert number, Sc andSr are the Schmidt and Soret numbers, respectively.

Using long wave length and low Reynolds number approxima-tion Eqs. (11)–(15)reduce to

dpdx¼ b2 @

2u@y2 þ

@2u@z2 ; ð17Þ

Fig. 21. Stream lines for different values of b, (a) for b ¼ 0:1, (b) for b ¼ 0:4, (c) for b ¼ 0:6, (d) for b ¼ 0:8.The other parameters are K� ¼ 0:4; / ¼ 0:45; Q ¼ 0:01; y ¼ 0:9.


�PrE b2 @u@y

� �2

þ @u@z

� �2" #

¼ b2 @2h@y2 þ

@2h@z2 ; ð18Þ

0 ¼ 1Sc

b2 @2u@y2 þ

@2u@z2

!þ Sr b2 @

2h@y2 þ

@2h@z2

!: ð19Þ

The expression of stream function satisfying Eq. (2) are defined asU ¼ @W=@Z;W ¼ �@W=@X.

The corresponding non-dimensional boundary conditions are

uðx; y; zÞ ¼ �1 at y ¼ �1;

hðx; y; zÞ ¼ c1 at y ¼ 1 and hðx; y; zÞ ¼ c3 at z ¼ �1;

uðx; y; zÞ ¼ c2 at y ¼ 1 and uðx; y; zÞ ¼ c4 at z ¼ �1;

ð20Þ

uðx; y; zÞ ¼ �1 at z ¼ �hðxÞ

hðx; y; zÞ ¼ 1 at z ¼ �hðxÞ and hðx; y; zÞ ¼ 0 at z ¼ hðxÞ;

uðx; y; zÞ ¼ 1 at z ¼ �hðxÞ and uðx; y; zÞ ¼ 0 at z ¼ hðxÞ;

ð21Þ

where 0 6 / 6 1. For straight duct / ¼ 0 whereas / ¼ 1 corre-sponds to total occlusion.

3. Solution of the problem

The analytical solutions of non-homogenous partial differentialgiven in Eqs. (17) to (19) satisfying the boundary conditions (20)and (21) are

uðx; y; zÞ ¼ �1þ 12

dpdxðz2 � h2Þ þ

X1m¼1

Am cosffiffiffiffiffiffikm

pz cosh

ffiffiffiffiffiffikmp

by;

ð22Þ


hðx;y;zÞ¼X1m¼1

Cm cosffiffiffiffiffiffikmp

z

sec hffiffiffiffikmp

b yþDm cos


z

csc hffiffiffiffikmp

b yþ 1

24hkm

12hkmþ2Eh5ðdp=dxÞ2Prkm�12zkm�2Ehðdp=dxÞ2Prz4km

�þ3A2

mEhPrkm cos2hffiffiffiffiffiffikm

p�3A2

mEhPrkm cos2zffiffiffiffiffiffikm

pþ96AmEhðdp=dxÞPrcosh

ffiffiffiffiffiffikm

pcosh


by�96AmEhðdp=dxÞ

Prcoszffiffiffiffiffiffikm

pcosh


byþ6A2

mEh3Prk2mðcosh


byÞ

2

�6A2mEhPrz2k2

m coshffiffiffiffiffiffikmp

by

� �2

þ48AmEh2ðdp=dxÞPrffiffiffiffiffiffikm

p

coshffiffiffiffiffiffikmp

bysinh


p�48AmEhðdp=dxÞPrz


pcosh


b

sinzffiffiffiffiffiffikm

pþ6A2

mEh3Prk2m sinh


by

� �2

�6A2mEhPrk2

mz2 sinhffiffiffiffiffiffikmp

by

� �2!;

ð23Þ

uðx;y;zÞ¼X1m¼1

Em

secffiffiffiffiffiffikmp

zsechffiffiffiffikmp

b yþFm cos


z

csc hffiffiffiffikmp

b yþ 1

24hkm12hkm

�2Eh5ðdp=dxÞ2PrScSrkm�12zkm�2Ehðdp=dxÞ2ScSrPrz4km

�3A2mEhPrkmScSrcos2h


pþ3A2

mEhScSrPrkm

�cos2zffiffiffiffiffiffikm

p�96AmEhðdp=dxÞPrScSrcosh


by

�coshffiffiffiffiffiffikm

pþ96AmEhðdp=dxÞPrScSrcosh


by

�coszffiffiffiffiffiffikm

p�6A2

mEh3PrScSrk2m cosh


by

� �2

þ6A2mEhz2PrScSrk2

m coshffiffiffiffiffiffikmp

by

� �2

�48AmEh2ðdp=dxÞPrScSrffiffiffiffiffiffikm

pcosh


bysinh

�ffiffiffiffiffiffikm

pþ48AmEhðdp=dxÞPrScSrz


pcosh


bysinh

�ffiffiffiffiffiffikm

p�6A2

mEh3PrScSrk2m sinh


by

� �2

þ6A2mEhz2PrScSrk2

m sinhffiffiffiffiffiffikmp

by

� �2

; ð24Þ

where

Am ¼X1m¼1

1

h cos hffiffiffiffikmp

b

�2hdpdx

cos hffiffiffiffiffiffikmp

kmþ 2

dpdx

sin hffiffiffiffiffiffikmp

k3=2m

!;

Cm ¼X1m¼1

16hk5=2

m

1

coshffiffiffiffikmp

b

� 48Ehðdp=dxÞ2Prffiffiffiffiffiffikm

pcos


ph

þ 8Eh3ðdp=dxÞ2Prk3=2m cos


phþ 12A2

mEPrk2m cosh


by

� �2

� ðhffiffiffiffiffiffikm

pcos


ph� sin


phÞ þ 48Ehðdp=dxÞ2Pr sin


ph

� 24Eh2kmðdp=dxÞ2Pr sinffiffiffiffiffiffikm

p� 12k2

m sinffiffiffiffiffiffikm

pþ 12c1k

2m

� sinffiffiffiffiffiffikm

ph12c3k

2m sin


phþ 3A2

mEPrk2m sin


ph

þ 6AmEðdp=dxÞPrkm cos hffiffiffiffiffiffikmp

byð4h


pþ 2h


pcos 2


ph

� 3 sin 2ffiffiffiffiffiffikm

phÞ � A2

mEPrk2m sin 3


phþ 12A2

mEPrk2m

� sinhffiffiffiffiffiffikmp

by

� �2

ðhffiffiffiffiffiffikm

pcos


ph� sin


phÞ;

Dm ¼X1m¼1

ðc1 � c3Þ sinffiffiffiffiffiffikmp

h csc hffiffiffiffikmp

b

hffiffiffiffiffiffikmp ;

Em¼X1m¼1

� 2sinffiffiffiffiffiffikmp

h

hffiffiffiffiffiffikmp

coshffiffiffiffikmp

b

þ 2c2 sinffiffiffiffiffiffikmp

h


coshffiffiffiffikmp

b

þ 2c4 sinffiffiffiffiffiffikmp

h


coshffiffiffiffikmp

b

þ3Eh3ðdp=dxÞ2PrScSr sinffiffiffiffiffiffikmp

h

3hffiffiffiffiffiffikmp

coshffiffiffiffikmp

b

þ3A2mEPrScSr cos


2hsinffiffiffiffiffiffikmp

h

2hffiffiffiffiffiffikmp

coshffiffiffiffikmp

b

þA2mEPrScSr cosh


by

� �2

�sinffiffiffiffiffiffikmp

h

coshffiffiffiffikmp

b

þ8AmEPrScSrðdp=dxÞcosh

ffiffiffiffikmp

b yðsinffiffiffiffiffiffikmp

hÞ2

km coshffiffiffiffikmp

b

� 1hkm

A2mEPrScSr

coshffiffiffiffikmp

b


by

� �2

ð2hffiffiffiffiffiffikm

pcos


ph

þðh2km�2Þsinffiffiffiffiffiffikm

phÞ�Eðdp=dxÞ2PrScSr

6coshffiffiffiffikmp

b

� 8hðh2km�6Þcosffiffiffiffiffiffikmp

h

k2m

þ2ð24�12h2kmþh4k2mÞsin


h

k5=2m

!

þAmEPrScSrðdp=dxÞhk3=2

m coshffiffiffiffikmp

b


byð2h


pcos2


ph�sin2


phÞ

þAm8Eðdp=dxÞPrScSr

hk3=2m

coshffiffiffiffikmp


hÞkm cosh

ffiffiffiffikmp

b

�Am4Eðdp=dxÞPrScSr cosh

ffiffiffiffikmp


hÞhk3=2

m coshffiffiffiffikmp

b

2hffiffiffiffiffiffikm

p

þsin2ffiffiffiffiffiffikm

ph�AmEðdp=dxÞPrScSr

h12ffiffiffiffiffiffikmp

coshffiffiffiffikmp

b

ð3sinffiffiffiffiffiffikm

phþsin3


phÞ

þA2mEhPrScSr


p sinffiffiffiffiffiffikmp

h sinhffiffiffiffikmp

b y� �2

coshffiffiffiffikmp

b

0B@

1CA

� A2mEPrScSr


coshffiffiffiffikmp

b

sinhffiffiffiffiffiffikmp

by

� �2

ð2hffiffiffiffiffiffikm

pcos


ph

þðh2km�2Þsinffiffiffiffiffiffikm

phÞ;

Fm ¼X1m¼1

ðc2 � c4Þ sinffiffiffiffiffiffikmp

h


sinffiffiffiffikmp

b

:

The volumetric flow rate q in a rectangular duct can be calcu-lated as

q ¼Z 1

0

Z hðxÞ

0uðx; y; zÞdydz: ð28Þ

The instantaneous flux is given by

�Q ¼Z 1

0

Z hðxÞ

0ðuþ 1Þdydz ¼ qþ hðxÞ: ð29Þ

The average volume flow rate over one period ðT ¼ k=cÞ of the peri-staltic wave is defined as

Fig. 22. Stream lines for different values of /, (a) for / ¼ 0:3, (b) for / ¼ 0:4, (c) for / ¼ 0:5, (d) for / ¼ 0:6. The other parameters are K� ¼ 0:4; b ¼ 0:2; Q ¼ 0:01; y ¼ 0:9.


Q ¼ 1T

Z T

0

�Q dt ¼ qþ 1: ð30Þ

The pressure gradient dp=dx is obtained after solving Eqs. (29) and(30).

dpdx¼ ð3ð1� hðxÞ � QÞk2

mð2hðxÞffiffiffiffiffiffikm

pþ sin 2hðxÞ


pÞÞ

= 2h4k5=2m þ h3k2

m sin 2hðxÞffiffiffiffiffiffikm

p� 12b sin2 hðxÞ

��


ptanh


bþ 6hðxÞb


psin 2hðxÞ


ptanh


b

�: ð31Þ

Also, the non-dimensional pressure rise Dp is evaluated numericallyby using the following expression

Dp ¼Z 1

0

dpdx

dx: ð32Þ

Numerical integration for the integral given in Eq. (32) is performedusing software built-in Mathematica.

4. Results and discussion

In fact the aim of this section is in three folds. Firstly thebehavior of parameters involved in the expressions of velocity u,temperature h and mass concentration u. Secondly and thirdlythe pumping characteristics and trapping mechanism respectivelyare taken into account. For this purpose Figs. 2–24 are sketched tomeasure the features of all parameters. In particular, the variationsof Reynolds number, Prandtl number, Eckert number, Schmidt andSoret numbers are examined.

Figs. 2 and 3 illustrate the variations of velocity u. Fig. 2 showsthat by increasing flow rate Q, velocity filed increases. From Fig. 3 itcan be seen that with increase of aspect ratio b, the velocity field

Fig. 23. Stream lines for different values of Q, (a) for Q ¼ 0:01, (b) for Q ¼ 0:03, (c) for Q ¼ 0:07, (d) for Q ¼ 0:1. The other parameters are K� ¼ 0:4; b ¼ 0:1;/ ¼ 0:45; y ¼ 0:9.


decreases and maximum velocity is obtained in the centre of chan-nel. Figs. 4–7 describe the variation of temperature profile. FromFig. 4, it can be observed that with the increase of flow rate Q, tem-perature profile increases. It can also noticed from Fig. 5 that tem-perature profile decreases by decreasing the Prandtl Number Pr.Fig. 6 shows that with the increase of Eckert number E, the temper-ature profile increases. Fig. 7 illustrates the effects of aspect ratio bon temperature profile. From this figure it can be observed thattemperature near the walls has same for all values of aspect ratiob but in the region z 2 ½�0:5;0:5�, the temperature profile increasesby increasing the value of b. Figs. 8–13 are plotted to study thebehavior of Eckert number E, Schmidt number Sc, Soret numberSr, flow rate Q, aspect ratio b and Prandtl number Pr on concen-trated profile u. Concentration distribution for different values ofb is shown in Fig. 8. It is depicted that concentration decreases withthe increase of b. It also reveals from Fig. 9 that concentration

profile decreases by increasing the Soret number Sr. Figs. (10)and (11) describe the simultaneous effects of Schmidt number Scand flow rate Q on concentration. It is seen that concentration de-creases with the increase of Schmidt number Sc and flow rate Q.Fig. 12 displays against the various values of Prandtl number Pr.It is depicted that concentration profile is minimum for the smallvalues of Pr. Effects of Eckert number on concentration are shownin Fig. 13. It is observed that the concentration profile decreaseswith the increase of Eckert number.

Figs. 14–17 illustrates the variation of dp=dx versus x. It revealsfrom Fig. 14 with the increase of aspect ratio b in the narrow part ofthe channel x 2 ½0:3;0:7�, the pressure gradient increases while inthe wider parts of the channel x 2 ½0;0:3� and x 2 ½0:7;1�, thepressure gradient is decreasing. Fig. 15 depicts that the pressuregradient decreases for large values of K� . Fig. 16 shows that whenamplitude ratio /, increases the pressure gradient increases in

Fig. 24. Stream lines for different values of K� , (a) for K� ¼ 0:2, (b) for K� ¼ 0:3, (c) for K� ¼ 0:4, (d) for K� ¼ 0:5. The other parameters are b ¼ 0:2; / ¼ 0:4; Q ¼ 0:01; y ¼ 0:9.

Table 1The convergence of analytical series solutionsgiven in Eq. (23) is achieved at 10th-order ofapproximations, that is Am ! 0 as m!1.

m!1 Am ! 0

1 �1:524432 0:001133 0:000014 1:87871� 10�7

5 4:06248� 10�9

6 1:02262� 10�10

7 2:84733� 10�12

8 8:51862� 10�14

9 2:6895� 10�15

10 8:85389� 10�17

Fig. 25. Velocity profile for rectangular duct ðb – 0Þ and rectangular channelðb ¼ 0Þ.


Fig. 26. Concentration profile for rectangular duct ðb – 0Þ and rectangular channelðb ¼ 0Þ.

Fig. 27. Temperature profile for rectangular duct ðb – 0Þ and rectangular channelðb ¼ 0Þ.

Table 2Difference between the rectangular duct ðb – 0Þand the rectangular channel ðb ¼ 0Þ.

hðxÞ uðx; y; zÞ uðx; y; zÞ uðx; y; zÞ uðx; y; zÞ hðx; y; zÞ hðx; y; zÞb ¼ 0 b – 0 b ¼ 0 b – 0 b ¼ 0 b – 0

�0:85 �1 �1 1 1 1 1�0:68 �0:4600 �0:5232 0:4572 0:2820 1:3428 1:5167�0:51 �0:0399 �0:1731 0:1471 �0:1094 1:4528 1:7072�0:34 0:2600 0:0650 �0:0308 �0:3164 1:4308 1:7134�0:17 0:4400 0:2029 �0:1488 �0:4401 1:3488 1:63660 0:5000 0:2480 �0:2500 �0:5413 1:2500 1:53760:17 0:4400 0:2029 �0:3488 �0:6401 1:1488 1:43660:34 0:2600 0:0650 �0:4308 �0:7164 1:0308 1:31340:51 �0:0399 �0:1731 �0:4528 �0:7094 0:8528 1:10720:68 �0:4600 �0:5232 �0:3428 �0:5179 0:5428 0:71670:85 �1 �1 0 0 0 0


narrow part of the channel whereas decreasing behavior is ob-served in the wider parts of the channel.

The effects pressure gradient versus flow rate are depicted inFig. 17. It shows that flow rate decreases by an increase in the pres-sure gradient.

Figs. 18–20 present the variations of pressure rise Dp againstthe flow rate Q. Fig. 18 depicts that in retrograde pumpingðDp > 0; Q < 0Þ region, the pumping rate increases by the increaseof aspect ratio b but gives opposite behavior in copumpingðDp < 0;Q > 0Þ region. It can be noticed from Fig. 19 that whenamplitude ratio / increases then in retrograde pumpingðDp > 0; Q < 0Þ region the pumping rate increases while in cop-umping region ðDp < 0; Q > 0Þ when Q 2 ½0;0:4� its behavior re-main same whereas in copumping region when Q 2 ½0:4;2�attitude of Dp is quite opposite. Influence of non uniform parame-ter K� is shown in Fig. 20. It is evident from results that an increasein K� pumping rate increases in retrograde pumping and copum-ping regions ðDp > 0; Q < 0Þ and ðDp < 0; Q > 0Þ respectively

when Q 2 ½0;0:5�. An opposite effect is seen in copumpingðDp < 0; Q > 0Þ region when Q 2 ½0:5;2�. In peristaltic motiontrapping is another interesting phenomena. Basically it is formula-tion of an internally circulating bolus of fluid by closed streamlines. The trapped bolus pushed a head along a peristaltic waves.In the wave frame, streamlines under certain conditions split totrap a bolus which moves as a whole with the speed of the wave.The formation of an internally circulating bolus of the fluid byclosed streamline is called trapping. The bolus defined as a volumeof fluid bounded by a closed streamlines in the wave frame istransported at the wave. For this purpose, the trapping phenome-non is presented by plotting stream lines and are shown in Figs. 21–24 for various parameters. The stream lines for various values of bis sketched in Fig. 21. It is depicted that with the rise of aspect ratiob, the size of trapping bolus decreases but the size of bolus in-creases. Fig. 22 displays for amplitude ration /, it is noted thatthe size of the bolus becomes smaller and trapping bolus increaseswith the increase of /. The stream lines for different values of flowrate Q are sketched in Fig. 3, It is seen that the trapping bolus re-duces with an increase of Q. Fig. 24, illustrates the effects of K�.It is measured here that when the parameter K increases thenthe size of bolus increases and as a result trapping bolus reducein numbers.

Moreover, one can easily observe that the convergence of ana-lytical series solutions given in Eq. (23) depends on Am. It is foundthat the convergence of series solutions is achieved at 10th�orderof approximations (Table 1). In Figs. 25–27, we present the effectsof rectangular channel ðb ¼ 0Þ and rectangular duct ðb – 0Þ for nonuniform channel on velocity, concentration and temperature pro-files. It is found that the velocity and concentration layers in caseof rectangular duct ðb – 0Þ are smaller than rectangular channelðb ¼ 0Þ. However, the temperature layer in rectangular duct is lar-ger than the rectangular channel. Furthermore, the results for rect-angular channel can be recovered by taking b ¼ 0. It is noticed thatwhen b ¼ 1 the rectangular duct becomes a square duct. Finally acomparison is also given in Table 2 in order to show the differenceof presented study with the existing literature.

5. Concluding remarks

In this paper, peristaltic flow in a non-uniform rectangular ducthas been studied. The influence of heat transfer is taken into ac-count. Analysis for Prandtl Number Pr, Eckert number E, Schmidtnumber Sc and Soret number Sr is presented. The features of theflow characteristics are analyzed in detail by plotting graphs andnumerical tables. It is found that as m!1)Am ! 0 that is theconvergence of the solutions is achieved at 10th-order of approxi-mations. Finally a comparison with the existing literature is alsomade. It is observed that the results for rectangular channel canbe recovered by taking b ¼ 0. It is also noticed that when b ¼ 1the rectangular duct becomes a square duct. The results obtainedfor the flow of peristaltic fluid in a non uniform duct reveal manyinteresting behaviors that warrant further study on the non-New-tonian fluid phenomena, especially the shear-thinning phenomena.Shear-thinning reduces the wall shear stress.

References

[1] K.R. Rajagopal, Thermodynamics and stability of non-Newtonian fluids (Ph.D.thesis), University of Minnesota, 1978.

[2] K.R. Rajagopal, Longitudinal and torsional oscillations of a rod in a non-Newtonian fluid, Acta Mech. 49 (1983) 282–285.

[3] K.R. Rajagopal, A.S. Gupta, An exact solution for the flow of a non-Newtonianfluid past an infinite plate, Meccanica 19 (1984) 158–160.

[4] K.R. Rajagopal, A note on unsteady unidirectional flows of a non-Newtonianfluid, Int. J. Non Linear Mech. 17 (1982) 369–373.

http://refhub.elsevier.com/S0017-9310(13)01082-X/h0015







[5] C. Fetecau, C. Fetecau, Starting solutions for the motion of a second grade fluiddue to longitudinal and torsional oscillations of a circular cylinder, Int. J. Eng.Sci. 44 (2006) 788–796.

[6] C. Fetecau, C. Fetecau, On some axial Couette flows of non-Newtonian fluids, Z.Angew. Math. Phys. (ZAMP) 56 (2005) 1098–1106.

[7] M. Hameed, S. Nadeem, Unsteady MHD flow of a non-Newtonian fluid on aporous plate, J. Math. Anal. Appl. 325 (2007) 724–733.

[8] R. Ellahi, The effects of MHD and temperature dependent viscosity on the flowof non-Newtonian nanofluid in a pipe: analytical solutions, Appl. Math. Model.37 (2013) 1451–1457.

[9] R. Ellahi, Exact solutions of flows of an Oldroyd 8-constant fluid with nonlinearslip conditions, Zeitschrift Fur Naturforschung A 65 (2010) 1081–1086.

[10] S. Nadeem, R. Mehmood, N.S. Akbar, Non-orthogonal stagnation point flow of anon-Newtonian fluid towards a stretching surface with heat transfer, Int. J.Heat Mass Transfer 55 (2012) 3964–3970.

[11] T.W. Latham, Fluid motions in a peristaltic pump (M.S. thesis), MIT, 1966.[12] S. Akram, A. Ghafoor, S. Nadeem, Mixed convective heat and Mass transfer on a

peristaltic flow of a non-Newtonian fluid in a vertical asymmetric channel,Heat Trans. Asian Res. 41 (2012) 613–633.

[13] Kh. S. Mekheimer, Non-linear peristaltic transport of magnetohydrodynamicflow in an inclined planar channel, Arab. J. Sci. Eng. 28 (2003) 183–201.

[14] R. Ellahi, A. Riaz, S. Nadeem, M. Mushtaq, Series solutions ofmagnetohydrodynamic peristaltic flow of a Jeffrey fluid in eccentriccylinders, J. Appl. Math. Inform. Sci. 7 (2013) 1441–1449.

[15] Ambreen. A. Khan, R. Ellahi, K. Vafai, Peristaltic transport of Jeffrey fluid withvariable viscosity through a porous medium in an asymmetric channel, Adv.Math. Phys. (1012) (2012) 1–15. Article ID 169642.

[16] Kh. S. Mekheimer, Y.A. Elmaboud, A.I.A. Lateef, Peristaltic transport through aneccentric cylinders: mathematical model, Appl. Bionics Biomech. 10 (2013)19–27.

[17] Y.A. Elmaboud, Kh. S. Mekheimer, A.I.A. Lateef, Thermo properties of a couplestress fluid in an asymmetric channel with peristalsis, J. Heat Transfer 135 (4)(2013), http://dx.doi.org/10.1115/1.4023127.

[18] Kh. S. Mekheimer, S.R. Komy, S.I. Abdelsalam, Simultaneous effects of magneticfield and space porosity on compressible Maxwell fluid transport induced by a

surface acoustic wave in a micro-channel, Chin. Phys. B 22 (2013), http://dx.doi.org/10.1088/1674-1056/22/12/124702.

[19] Kh. S. Mekheimer, Y.A. Elmaboud, A.I.A. Lateef, Particulate suspension flowinduced by sinusoidal peristaltic waves through eccentric cylinders: threadannular, Int. J. Biomath. 6 (2013) 1350026.

[20] Kh. S. Mekheimer, A.M. Salem, A.Z. Zaheer, Peristatcally induced MHD slip flowin a porous medium due to a surface acoustic wavy wall, J. Egypt. Math. (2013)1–9.

[21] Kh. S. Mekheimer, Y. Abd elmaboud, The influence of heat transfer andmagnetic field on peristaltic transport of a Newtonian fluid in a verticalannulus: application of an endoscope, Phys. Lett. A 372 (2008) 1657–1665.

[22] R. Ellahi, A. Zeeshan, K. Vafai, H.U. Rehman, Series solutions formagnetohydrodynamic flow of non-Newtonian nanofluid and heat transferin coaxial porous cylinder with slip conditions, J. Nanoeng. Nanosyst. 225(2011) 123–132.

[23] N.S. Akbar, S. Nadeem, Thermal and velocity slip effects on the peristaltic flowof a six constant Jeffrey’s fluid model, Int. J. Heat Mass Transfer 55 (2012)3964–3970.

[24] N.S. Akbar, S. Nadeem, Characteristics of heating scheme and mass transfer onthe peristaltic flow for an Eyring–Powell fluid in an endoscope, Int. J. HeatMass Transfer 52 (2012) 375–383.

[25] G. Radhakrishnamacharya, V.R. Murty, Heat transfer to peristaltic transport ina non-uniform channel, Defence Sci. J. 43 (1993) 275–280.

[26] K. Vajravelu, G. Radhakrishnamacharya, V.R. Murty, Peristaltic flow and heattransfer in a vertical annulus, with long wave approximation, Int. J. NonlinearMech. 42 (2007) 754–759.

[27] S. Nadeem, Safia Akram, T. Hayat, Awatif A Hendi, Peristaltic flow of a Carreaufluid in a rectangular duct, J. Fluids Eng. 134 (2012) 041201.

[28] S. Nadeem, Safia Akram, Pristaltic flow of a Jeffrey fluid in a rectangular duct,Nonlinear Anal. Real world Appl. 11 (2010) 4238–4247.

[29] S. Nadeem, Safia Akram, T. Hayat, Awatif A Hendi, Peristaltic flow of a Carreaufluid in a rectangular duct, J. Fluids Eng. 134 (2012) 041201.

[30] Kh. S. Mekheimer, S.Z.A. Husseny, A.I.A. Lateef, Effect of lateral walls onperistaltic flow through an asymmetric rectangular duct, Appl. BionicsBiomech. 8 (2011) 1–14.






























http://dx.doi.org/10.1115/1.4023127

http://dx.doi.org/10.1088/1674-1056/22/12/124702

http://dx.doi.org/10.1088/1674-1056/22/12/124702


































international journal of heat and mass transfervafai.engr.ucr.edu/publications/2014a/ellahi.pdf ·...

Documents