Heat Transfer in Peristaltic Flow of Viscoelastic Fluid in ... · Heat transfer in peristaltic flow of viscoelastic fluid 1587 whereδis the wave number,Re is the Reynolds number,

Applied Mathematical Sciences, Vol. 4, 2010, no. 32, 1583 - 1606

Heat Transfer in Peristaltic Flow of Viscoelastic Fluid

in an Asymmetric Channel

1A. M. Sobh, 2S. S. Al Azab and 3H. H. Madi

1Department of Mathematics, Faculty of Applied Science, Al Aqsa University Gaza, P.O. Box 4051, Palestine

[email protected]

2Head of Mathematics Department, Faculty of Science for Girls Ain Shams University, Egypt.

3Department of Mathematics, College of Science and Technology, Khan Younis, Palestine E-mail:[email protected]

Abstract In this paper, we carry out a study of peristaltic flow of Oldroyd fluid in an asymmetric channel in the presence of heat transfer. The governing equations of motion and energy are simplified using along wave length approximation. A closed form solution for the axial velocity and the temperature is obtained using perturbation method. Furthermore, The effect of various parameters of interest on axial velocity, temperature and heat transfer coefficient are discussed numerically and explained graphically. Mathematics Subject Classification: 76Z05 Keywords: Asymmetric channel; Viscoelastic fluid; Peristalsis; Heat transfer 1 Introduction The peristaltic flow of biofluids in different geometries has many application in mathematics, biology and engineering. The initial mathematical models of peristalsis obtained by a train of sinusoidal waves in an infinitely long symmetric channel or tube have been investigated by Shapiro et al. [2] and Fung and Yih [17].

1584 A. M. Sobh, S. S. Al Azab and H. H. Madi After these studies, many investigations were done to understand the peristaltic action for Newtonian and non Newtonian fluids in different situations. The importance of the study of peristaltic transport in an asymmetric channel has been brought out by Eytan and Elad [13] with an application in intra uterine fluid flow in anon-pregnant uterus. After this study, some investigations were done to understand the mechanism of peristalsis in asymmetric channels. Mishra and Rao [10] have investigated the flow in an asymmetric channel generated by peristaltic waves propagating on the walls with different amplitudes and phases. Rao and Mishra [5] discussed the non-linear and curvature effects on the peristaltic flow of a viscous fluid in asymmetric channel when the ratio of channel width to the wavelength is small. Ebaid [1] studied the effects of magnetic field and wall slip condition on the peristaltic transport of a Newtonian fluid in an asymmetric channel. Also Kothandapani and Srinivas [9] discussed the non-linear peristaltic transport of a Newtonian fluid in an inclined asymmetric channel through a porous medium. Furthermore, Haroun [8] studied the effect of wall compliance on peristaltic transport of a Newtonian fluid in an asymmetric channel . In their paper, Hayat et al.[16] investigated the peristaltic mechanism of a Maxwell fluid in asymmetric channel and Subba Reddy et al. [11] investigated the peristaltic motion of a power-low fluid in an asymmetric channel. Also Ali and Hayat [12] studied the peristaltic motion of a Carreau fluid in an asymmetric channel. Furthermore, Sobh[3] studied the slip flow in peristaltic transport of a Carreau fluid in an asymmetric channel and Wang et al.[18] have studied the magnetohydrodynamic peristaltic flow of a Sisko fluid in a symmetric or asymmetric channel. Recently, few studies have been done to understand the interaction between heat transfer and peristaltic flow for Newtonian fluids. Srinivas and Kothandapani [15] investigated the peristaltic transport of a Newtonian fluid with heat transfer in an asymmetric channel. Sobh [4] studied the slip flow of peristaltic transport of a magneto-Newtonian fluid through a porous medium with heat transfer. Also Mekhheimer and Abd elmaboud [7] studied the influence of heat transfer and magnetic field on peristaltic transport of a Newtonian fluid in a vertical annulus. Furthermore, Radhakrishnamacharya and Srinivasulu [6] investigated the influence of wall properties on peristaltic transport with heat transfer. Since it is well known that physiological fluids behave like non-Newtonian fluids, we'll study the heat transfer in peristaltic flow of Oldroyd fluid, as a viscoelastic fluid, in an asymmetric channel with peristaltic waves of different amplitudes and phase traveling on its walls. The problem is formulated and analyzed using perturbation series on the wave number as a parameter. Because of the complexity of the governing equation, long wavelength approximation is used to obtain analytic expressions for the axial velocity, the temperature and the heat transfer coefficient. Moreover, the influences of Reynolds number, phase difference, Weissenberg number, wave number, Prandtl number, Eckert number and channel

Heat transfer in peristaltic flow of viscoelastic fluid 1585 width on axial velocity, temperature and heat transfer coefficient have been discussed. 2 Formulation and Analysis Let us consider a two-dimensional flow of Oldroyd fluid in an asymmetric channel. We assume sinusoidal wave train moving with speed c along the channel walls. The upper wall is maintained at temperature 0T and the Lower wall at 1T . Let

21 dd + be the channel width. Taking X and Y as rectangular coordinates, the geometry of the wall surfaces are defined as

( ) ,)(2cos, 111 ⎥⎦⎤

⎢⎣⎡ −+= tcXadtXhλπ upper wall, (1)

( ) ( ) ,2cos, 222 ⎥⎦⎤

⎢⎣⎡ +−−−= φλπ tcXadtXh lower wall, (2)

Where 1a , 2a are the amplitudes of the waves, λ is the wavelength, t is the time and φ ( πφ ≤≤0 ) is the phase difference. Moreover, ,1a 2a , ,1d 2d and φ satisfy the following inequality, Mishra and Rao [10],

( ) .cos2 22121

22

21 ddaaaa +≤++ φ

The constitutive equation for Oldroyd fluid is

( ) ( )

( ) jijjj

kikii

kk

iii

kkjjj

kkij

jjiikkjjii

kkijij

vgx

gg

vgx

ggggx

vgggt

γμτ

τττ

τ

&−=⎥⎦

⎤∂∂

−

⎢⎢⎣

⎡

∂∂

−∂∂

∂

∂Γ+ )(2

1

(3)

where iig and jjg are respectively the diagonal components of covariant and contravariant metric tensor, ji, =1,2.

Taking moving coordinates ( )yx, , (wave frame), which travel in the X -direction with the same speed, as the wave, the unsteady flow in the laboratory frame ( )YX , can be treated as steady, Shapiro et al. [2]. The coordinates frame are related through

YytcXx =−= , , (4) VvcUu =−= , , (5)

where ( )VU , and ( )vu , are the velocity components in the corresponding coordinate system. Equations of motion in the moving coordinates are, Rathy [14]

1586 A. M. Sobh, S. S. Al Azab and H. H. Madi

0=∂∂

+∂∂

yv

xu (6)

yxxp

yuv

xuu xyxx

∂

∂−

∂

∂−

∂∂

−=⎥⎦

⎤⎢⎣

⎡∂∂

+∂∂ ττ

ρ (7)

yxyp

yvv

xvu yyyx

∂

∂−

∂

∂−

∂∂

−=⎥⎦

⎤⎢⎣

⎡∂∂

+∂∂ ττ

ρ (8)

The constitutive equations of Oldroyd fluid are, Rathy[14]

xxyxxxxxxx

xx yu

xu

yv

xu γμττ

τττ &−=⎥

⎦

⎤∂∂

−∂∂

−∂

∂⎢⎣

⎡+

∂

∂Γ+ 22 (9)

yxxxyyyxyx

yx xv

yu

yv

xu γμττ

τττ &−=⎥

⎦

⎤∂∂

−∂∂

−∂

∂⎢⎣

⎡+

∂

∂Γ+ (10)

yyyyyxyyyy

yy yv

xv

yv

xu γμττ

τττ &−=⎥

⎦

⎤∂∂

−∂∂

−∂

∂⎢⎣

⎡+

∂

∂Γ+ 22 , (11)

The differential energy equation is

⎥⎦

⎤⎢⎣

⎡

∂∂

+∂∂

+∂∂

+∂∂

+∇=⎥⎦

⎤⎢⎣

⎡

∂∂

+∂∂

xv

yu

yv

xuTk

yTv

xTuC yxyyxxv τττρ 2 , (12)

where p is the pressure, xxτ , yxτ , yyτ are components of the extra stress tensor, Γ is relaxation time, μ is the coefficient of viscosity of the fluid,σ is the electrical conductivity, vC is the specific heat at constant volume, ν is kinematic viscosity, k is

thermal conductivity of the fluid, T is temperature and xxγ& , yxγ& , yyγ& are components of strain-rate tensor and given by

,2,2yv

xu

yyxx ∂∂

=∂∂

= γγ && and ,⎥⎦

⎤⎢⎣

⎡∂∂

+∂∂

=xv

yu

yxγ& (13)

Introducing the non-dimensional variables and parameters

jiij cd

dhh

dhhtct

cdvv

cuu

dyyxx τ

μτ

λλ

λ1

1

22

1

11

11

,,,,,,, ======== ,

jijiie cdp

cdp

dcWdcRd

γγλμμ

ρλ

δ && 12

1

1

11 ,,,, ==Γ

=== ,01

0

TTTT

−−

=θ ,

k

CP v

rμ

= ,)( 01

2

TTCcE

v −= , (14)

Heat transfer in peristaltic flow of viscoelastic fluid 1587 whereδ is the wave number, eR is the Reynolds number, iW is the Weissenberg number, θ is the dimensionless temperature, rP is the Prandtl number and E is the Eckert number, equations (6-12) are reduced to

0=∂∂

+∂∂

yv

xu (15)

yxx

pyuv

xuuR yxxx

e ∂

∂−

∂∂

−∂∂

−=⎥⎦

⎤⎢⎣

⎡∂∂

+∂∂ ττ

δδ , (16)

yxy

pyvv

xvuR yyxy

e ∂

∂−

∂

∂−

∂∂

−=⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂ τ

δτ

δδ 23 , (17)

xu

yu

xu

yv

xuWi xyxx

xxxxxx ∂

∂−=⎥

⎦

⎤⎢⎣

⎡∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

+∂∂

+ δττττ

δτ 222 , (18)

⎥⎦

⎤⎢⎣

⎡∂∂

+∂∂

−=⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂

∂+

∂

∂+

xv

yu

yu

xv

yv

xuWi yyxx

xyxyxy

2δτδτττ

δτ , (19)

xv

xv

yv

yv

xuWi xyyy

yyyyyy ∂

∂−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

−∂

∂+

∂

∂+ δδττ

ττδτ 222 . (20)

+⎥⎦

⎤⎢⎣

⎡∂∂

+∂∂

+⎥⎦

⎤⎢⎣

⎡∂∂

+∂∂

=⎥⎦

⎤⎢⎣

⎡∂∂

+∂∂

yv

xuE

xyPyv

xuR yyxx

re ττδθδθθθδ 2

22

2

21

⎥⎦

⎤⎢⎣

⎡∂∂

+∂∂

xv

yuE xy

2δτ (21)

Eliminating the pressure from equation (16), (17) we get

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

+∂∂

−∂∂

2

2

2

22

2

2

2

2

xuv

xvu

yvu

yuvRe δδ =

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂∂

−∂∂

∂+

∂

∂−

∂

∂

xyyxyxxxyyyxxy ττ

δττ

δ22

2

2

2

22 (22)

3 Rate of Volume Flow and Boundary Conditions The dimensional volume flow rate in laboratory frame is

( )( )

( )YdtYXUQ

tXh

tXh∫=

,1

2 ,

,, , (23)

1588 A. M. Sobh, S. S. Al Azab and H. H. Madi in which 1h and 2h are function of X , t The rate of volume flow in the wave frame is given by

( )( )

( )ydyxuq

xh

xh∫=1

2

, , (24)

where 1h and 2h are function of x only. On substituting Eqs. (4) and (5) into Eq. (23) and then integrating, we have )()( 21 xhcxhcqQ −+= (25) The time- mean flow over a period T is defined as

∫=T

tQdT

Q0

1 (26)

Substituting (25) into (26), and integrating, we get 21 dcdcqQ ++= . (27) Defining the dimensionless mean flows θ and F as follow

1dc

Q=θ and

1dcqF = , (28)

Equation (27) may be written as dF ++= 1θ , (29) where

( )

( )

dyuFx

h

xh∫=1

2

. (30)

We note that ( )xh1 and ( )xh2 represent the dimensionless forms of the wall surfaces and defined by ( )xh1 =1+a cos 2π x and ( ) =xh2 -d-b cos ( )φπ +x2 , (31)

where 2

2

1

1 ,da

bda

a == , and 1

2

dd

d = ,

and a, b, d and φ satisfy the relation ( )222 1cos2 dabba +≤++ φ . (32) The boundary conditions for the dimensionless in the wave frame is ,1−=u at y= ( )xh1 , (33-a) .1−=u at y= ( )xh2 . (33-b)

dxdh

v 1−= , at )(1 xhy = , (34-a)

Heat transfer in peristaltic flow of viscoelastic fluid 1589

dxdh

v 2= . at )(2 xhy = . (34-b)

0=θ , at )(1 xhy = , (35-a) 1=θ . at )(2 xhy = . (35-b) 4 Method of Solution We expand the following in a power series of small parameter δ as follows ( )2

10 δδ Ouuu ++= , ),( 2

10 δδ Ovvv ++= ),( 2

10 δθδθθ O++= ( ) ( ) ( )210 δδτττ Oxxxxxx ++= , ( ) ( ) ( )210 δδτττ Oxyxyxy ++= ,

( ) ( ) ( )210 δδτττ Oyyyyyy ++= ,

( ),210 δδ OFFF ++=

).( 210 δδ Oxp

xp

xp

+∂∂

+∂∂

=∂∂ (36)

The use of expansions (36) with Eqs. (15), (16), (17), (18), (19), (20),(21),(22) and boundary conditions (33),(34),(35) we get system of order zero

000 =∂∂

+∂∂

yv

xu

, (37)

yx

p yx

∂

∂−=

∂∂ )0(

0 τ, (38)

( ) ( ) 02 000 =∂∂

−yu

Wi xyxx ττ , (39)

( ) ( )

yu

yu

Wi yyxy ∂∂

−=∂∂

− 0000 ττ , (40)

( ) 00 =yyτ , (41)

⎥⎦

⎤⎢⎣

⎡∂∂

+⎥⎦

⎤⎢⎣

⎡∂∂

=yu

EyP xy

r

0)0(20

210 τθ

, (42)


02

)0(2

=∂

∂

yxyτ

. (43)

With the dimensionless boundary conditions ,10 −=u at ( )xhy 1= , (44-a) .10 −=u at ( )xhy 2= . (44-b)

,10 dx

dhv −= at )(1 xhy = , (45-a)

dxdhv 2

0 = . at )(2 xhy = . (45-b)

00 =θ , at ),(1 xhy = (46-a) .10 =θ at ).(2 xhy = (46-b) The solution of (41),(40)and (38),subject the boundary conditions(44-a)and(44-b),is

[ ] 1)(21),( 2121

200 −++−⎥⎦

⎤⎢⎣⎡∂∂

= hhyhhyx

pyxu (47)

The instantaneous volume flow rate, 0F ,is given by

[ ] [ ]213

120

200 12

11

hhhhxp

dyuFh

h

−−−⎥⎦⎤

⎢⎣⎡∂∂

== ∫ (48)

The zero-order pressure gradient can be obtained by solving (48) for dxdp0 ,as

( )( )312

2100 12hh

hhFdxdp

−−+

= (49)

Substituting dx

dp0 into(47),we obtain the form of 0u as

322

10 ),( LyLyLyxu ++= (50)

where 312

2101 )(

)(6hh

hhFL

−−+

= , 312

212102 )(

))((6hh

hhhhFL

−+−+−

= ,

and 1)(

)(63

12

212103 −

−−+

=hh

hhhhFL . (51)

Substituting ),(0 yxu from (50) into(37),and then solving the equation , subject the boundary condition(45-a),we obtain the form of 0v as

)(21

31),( 1

321

231

10 xGyLyLyLyxv +−−−= (52)

where 113

21

12

31

11

11 2

131)( hLhLhLhxG +++−= . (53)

Heat transfer in peristaltic flow of viscoelastic fluid 1591 Substituting ),(0 yxu from (50)and(40) into (42),subject the boundary conditions(46-a)and(46-b),we obtain the form of 0θ as 54

23

32

410 ),( ByByByByByx ++++=θ (54)

where 211 3

1 ELPB r= , 212 32 LELPB r= , 2

23 21 ELPB r= ,

,)(21)(

32)(

311

)(1 2

122

22

31

3221

41

42

21

124 ⎥

⎦

⎤⎢⎣

⎡⎥⎦⎤

⎢⎣⎡ −+−+−−

−= hhLhhLLhhLEP

hhB r

and ⎥⎦

⎤⎢⎣

⎡⎥⎦⎤

⎢⎣⎡ −+−+−+−

−= )(

21)(

32)(

311 12

22

21

2221

31

32

212

12

15 hhLhhLLhhLEhP

hhhB r .(55)

System of order one Equating the coefficients of δ on both sides in Eqs. (15), (16), (17), (18), (19), (20),(21),(22),(33),(34) and (35) we get

011 =∂∂

+∂∂

yv

xu

(56)

( ) ( )

yxxp

yu

vx

uuR yxxx

e ∂

∂−

∂∂

−∂∂

−=⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂ 10

100

00

ττ, (57)

( )( ) ( )

( ) ( ) ( )

xu

yu

yu

xu

yv

xuWi xyxyxx

xxxxxx ∂

∂−=⎥

⎦

⎤⎢⎣

⎡∂∂

−∂∂

−∂∂

−∂∂

+∂∂

+ 00110000

0

0

01 2222 τττ

τττ ,(58)

( )( ) ( )

( ) ( )

yu

yu

yu

yv

xuWi yyyy

xyxyxy ∂

∂−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂∂

−∂∂

−∂

∂+

∂

∂+ 10110

0

0

0

01 ττ

τττ , (59)

( )( ) ( )

( )

xv

yv

yv

xuWi yy

yyyyyy ∂

∂−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂∂

−∂

∂+

∂

∂+ 000

0

0

0

01 22τ

τττ , (60)

+⎥⎦

⎤⎢⎣

⎡∂∂

+∂∂

+⎥⎦

⎤⎢⎣

⎡∂∂

=⎥⎦

⎤⎢⎣

⎡∂∂

+∂∂

yv

xu

EyPy

vx

uR yyxxr

e0)0(0)0(

21

20

00

01 ττ

θθθ

⎥⎦

⎤⎢⎣

⎡∂∂

+∂∂

yu

yuE xyxy

0)1(1)0( ττ (61)

( ) ( ) ( )

xyyxyyv

uyu

vR xxyyyxe ∂∂

∂−

∂∂

∂+

∂

∂−=⎥

⎦

⎤⎢⎣

⎡∂∂

−∂∂ 0202

2

12

20

2

020

2

0τττ

, (62)

with the boundary conditions 01 =u at ( )xhy 1= (63-a)

1592 A. M. Sobh, S. S. Al Azab and H. H. Madi 01 =u at ( )xhy 2= (63-b) 01 =v at )(1 xhy = (64-a) 01 =v at )(2 xhy = (64-b) 01 =θ at )(1 xhy = (65-a) 01 =θ at )(2 xhy = (65-b) Substituting (41),(40),(39),(50),(52) into Eqs. (60), (59), (58), (57) and then solving the resulting system along the boundary conditions (63), we get velocity 1u in the form

+⎥⎦⎤

⎢⎣⎡∂∂

+++++= 2125

344352611 2

121

6122030),( y

xpyAyAyAyAyAyxu

+−+−+−+−+−⎜⎝⎛

−)(

2)(

6)(

12)(

20)(

30)(22

21

532

31

442

41

352

51

262

61

1

12

hhAhhAhhAhhAhhAhh

y

⎟⎟⎠

⎞−⎥⎦

⎤⎢⎣⎡∂∂ )(

21 2

22

11 hh

xp )(

6)(

12)(

20)(

30)(2

122

431

32

341

42

251

52

1

12

21 hhAhhAhhAhhAhh

hh−+−+−+−⎜

⎝⎛

−+

⎟⎟⎠

⎞−⎥⎦

⎤⎢⎣⎡∂∂

+−+ )(21)(

2 121

125 hh

xphhA (66)

where

,31 1

111 LLRA e= [ ])8()(32 11

112112 LLWLLRA ie += ,

)1226()21( 1

111112

11211

132

123

113 LLLLLLWLLLLLLRA ie −++−+= ,

( ) ( )121

112

1122

11311

1234 8428)(2 LLLLLLLLWixGLLLRA e −−+++= ,

( ) ( )1132

11

131

1223

112

1335 2)(4232)( LLxGLLLLLLLWixGLLLRA e +−−−++= . (67)

The instantaneous volume flow rate, 1F ,is given by

−+−+−+−⎥⎦⎤

⎢⎣⎡∂∂

== ∫ 512

21

62

72

712

61

1312

1

211 3(

120)7557(

420)(

1211

hhA

hhhhhhA

hhxp

dyuFh

h

)22(24

)5335(120

)322 321

42

412

31

4421

52

512

41

3521

62

61 hhhhhAhhhhhh

Ahhhh −+−+−+−+−+

.)(12

312

5 hhA

−+ (68)

The one-order pressure gradient can be obtained by solving (68) for dxdp1 ,as


( )

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−+−+

+−−+

+−−+

+−−+

−−++

−=

325

22152

215

315

1324

424

3124

414

4213

5232

413

513

5212

6222

512

612

2611

721

6211

7111

312

1

7021021070

70357035

3521352121142114

14101410840

701

hAhhAhhAhA

hhAhAhhAhA

hhAhAhhAhAhhAhAhhAhA

hhAhAhhAhAF

hhdxdp , (69)

Substituting dxdp1 into(66),we obtain the form of 1u as

)()()(21

26122030),( 32

21

25344352611 xGyxGyxGy

Ay

Ay

Ay

Ay

Ayxu +++++++= (70)

where

,)( 11 dx

dpxG =

,2

)()()(60

)( 121

12

12

xGhhhh

xG+

−−

=η

where )(30)(10)(5)(3)(2 2

22

1532

314

42

413

52

512

62

6111 hhAhhAhhAhhAhhA −+−+−+−+−=η .

and ).(2)(60

)( 121

12

2213 xG

hhhh

hhxG +

−=

η

where )(30)(10)(5)(3)(2 125

21

224

31

323

41

422

51

5212 hhAhhAhhAhhAhhA −+−+−+−+−=η (71)

Substituting ),(1 yxu from (70) into(56),and then solving the equation , subject the boundary condition(64-a),we obtain the form of 1v as

2

)(6

)(62460120210

),(21

231

131

541

451

361

271

11

yxGyxGyAyAyAyAyAyxv −−−−−−−

=

).()(13 xNyxG +− (72)

where

113

21

12

31

11

31

15

41

14

51

13

61

12

71

11 )(

2)(

6)(

62460120210)( hxGhxGhxGhAhAhAhAhAxN +++++++= (73)

Substituting(39),(40),(41),(50),(52),(54),(59)and(60) into (61),subject the boundary conditions(65-a)and(65-b),we obtain the form of 1θ as

+++++++= 273645546372811 261220304256

),( yM

yM

yM

yM

yM

yM

yM

yxθ

)(12

)(20

)(30

)(42

)(56)(

1 42

41

552

51

462

61

372

71

282

81

1

12

hhMhhMhhMhhMhhMhh

−+−+−+−+−⎜⎝⎛

−


+−+−⎜⎝⎛

−+⎟

⎠⎞−+−+ )(

42)(

56)()(

2)(

661

62

271

72

1

12

2122

21

732

31

6 hhMhhMhh

hhyhhMhhM

)(30

51

52

3 hhM

− .)(2

)(6

)(12

)(20 12

721

22

631

32

541

42

4 ⎟⎠⎞−+−+−+−+ hhMhhMhhMhhM (74)

where

),54()

34( 11

111

1111 ALEPLBBLPRM rre +−=

),38()

52()2( 11

12112212

111

12

112

1212 LLEPWALALEPBLBLBLBLPRM rirre −++−−+=

)21

34()

32

234( 22313

112

121

13

113

122

1313 ALALEPBLBLBLBLBLBLPRM rre ++−−−++=

),3

16384( 2

111

11121

112

21 LLLLLLLEPW ri −+−

++−−−++= ))(4313( 14

113

122

13

123

132

1414 xGBBLBLBLBLBLBLPRM re

)322( 3241 ALALEPr + ),

3884

328( 1

12121

12

11221

111

22

113

21 LLLLLLLLLLLLEPW ri −−++−

++++−−++= 425124123

13

133

142

1515 4())(3

212( ALALEPxGBBLBLBLBLBLPRM rre

)),(812864())(4 121

112

22

13

21

11321

1221

113111 xGLLLLLLLLLLLLLLEPWxGL ri −+−+−−

)2)(2)(4())(2( 5212213413

143

1526 ALxGLxGLEPxGBBLBLBLPRM rre ++++−+=

),2102)(8)(42( 22

12

113

22

1321

1132

121

21

1231 LLLLLLLLLLxGLLxGLLLLEPW ri −+−+−+−

)(2)(2())(2())(( 12221

1232224

1537 xGLxGLLLLLEPWxGLEPxGBBLPRM rirre −+−++=

).2 2

213LL− (75)

Substituting ),(0 yxu from (50)and ),(1 yxu from(70)into (36) for ),( yxu we get the axial velocity u in the form

21

2534435261

322

1 )(21

26122030[),( yxG

yAy

Ay

Ay

Ay

ALyLyLyxu ++++++++= δ

)].()( 32 xGyxG ++ (76) Substituting ),(0 yxθ from (54)and ),(1 yxθ from(74)into (36) for ),( yxθ we have temperature θ in the form

5463728154

23

32

41 20304256

),( yMyMyMyMByByByByByx +++⎜⎝⎛+++++= δθ


+−+−⎩⎨⎧

−++++ )(

42)(

56)(1

261272

71

282

81

1

12

273645 hhMhhMhh

yMyMyM

yhhMhhMhhMhhMhhM

⎭⎬⎫−+−+−+−+− )(

2)(

6)(

12)(

20)(

3022

21

732

31

642

41

552

51

462

61

3

+−+−+−+−⎩⎨⎧

−+ )(

20)(

30)(

42)(

56)(4

142

451

52

361

62

271

72

1

12

21 hhMhhMhhMhhMhh

hh

⎟⎟⎠

⎞

⎭⎬⎫−+−+− )(

2)(

6)(

12 1272

122

631

32

5 hhMhhMhhM (77)

The heat transfer coefficient ( Z ) at the(upper) wall is given by .1 yxhZ θ= (78) Substituting Eq. (77) in Eq. (78), we get

⎜⎜⎝

⎛++++

⎩⎨⎧++++= 3544536271

432

23

11 34567234 yMyMyMyMyMByByByBhZ x δ

)(20

)(30

)(42

)(56

12

52

51

462

61

372

71

282

81

1

127

26 hhMhhMhhMhhMhh

yMyM−+−+−+−⎢⎣

⎡−

+++

⎟⎟⎠

⎞

⎭⎬⎫⎥⎦⎤−+−+−+ )(

2)(

6)(

1222

21

732

31

642

41

5 hhMhhMhhM . (79)

Results and Discussion It is seen form equations (76), (77) and (79) that we have obtained the axial velocity, the temperature and the heat transfer coefficient in explicit form. The effect of the physical parameters of the problem on the axial velocity and the temperature is seen through figures (1-10). Fig (1) shows the effect of the Weissenberg number iW on the axial velocity u

at x=0.2, δ =0.02, E=1, rP =1, eR =10, a=0.5, b=0.7, d=1, θ =1, φ =4π and ( iW =0,

0.04, 0.08). We observe that there is no effect for Weissenberg number iW on the axial velocity as the curves coincide. Fig (2) represents the graph of the axial velocity u versus y at x=0.2, E=1,

rP =1, iW =0.01, eR =10, a=0.5, b=0.7, d=1, θ =1, φ =4π and (δ =0, 0.04, 0.08). It can

be seen that an increase in the wave number δ increase the magnitude of the axial velocity u.

1596 A. M. Sobh, S. S. Al Azab and H. H. Madi The effects of Reynolds number eR on the axial velocity u is seen through

Fig.(3) at x=0.2, δ =0.02, E=1, rP =1, iW =0.04, a=0.5, b=0.7, d=1, θ =1, φ =4π and

( eR =0, 10, 20). It is noted that an increase in the Reynolds number eR increase the magnitude of the axial velocity. Fig.(4) gives the effects of phase difference φ on the axial velocity u at x=0.2,

δ =0.02, E=1, rP =1, iW =0.04, a=0.3, b=0.5, d=0.7, θ =1, eR =10 and (φ =0, 6π ,

4π ).

We have observed that the magnitude of the axial velocity increase with increasing phase difference φ . To see the effects of channel width d on the axial velocity we have prepared Fig.(5). Obviously, the magnitude of the axial velocity increase as d increases. Fig.(6) displays the influence of the Reynolds number eR on the temperature

distribution for x=0, δ =0.02, E=1, rP =1, iW =0.04, a=0.7, b=1.2, d=2, θ =1, φ =4π

and ( eR =90, 80, 70). We note that the temperature θ increases as Reynolds number

eR increases. Fig.(7) shows the effect of the Prandtl number rP on the temperature for x=0,

δ =0.02, E=1, eR =50, iW =0.04, a=0.5, b=0.7, d=2, θ =1, φ =4π and ( rP =1.5, 1.4,

1.3). It is noticed that the temperatureθ increases with increasing the Prandtl number rP .

Fig.(8) depicts the variation of the fluid temperature θ with y, for different value of the Eckert number and at x=0, δ =0.02, rP =1.5, iW =0.04, a=0.5, b=0.7, d=2,

θ =1, eR =50, φ =4π and (E=1, 0.8, 0.6). It is evident that the temperatureθ increases

with the increase in Eckert number E. In Fig. (9), the temperature is graphed versus y at x=0, δ =0.02, E=1, rP =2,

iW =0.04, a=0.5, b=0.7, θ =1, eR =50, φ =4π and (d=2, 1.9, 1.8). We note that the

temperature distribution θ increase as the channel width d increases. Fig.(10) is the graph of temperature distribution versus y for different values of phase difference φ and at x=0, δ =0.02, rP =1, E=1, iW =0.03, a=0.3, b=0.4, d=0.9, θ =1, and eR =10. It can be noticed that an increase in the phase difference φ result increase in the magnitude of the temperature distribution. Variations of the heat transfer coefficient (Z) at wall have been presented in Table(1), (a)-(d). The results reveal that the heat transfer coefficient (Z) increases

Heat transfer in peristaltic flow of viscoelastic fluid 1597 with increasing upper wave amplitude a, Eckert number E, Reynolds number eR and Weissenberg number iW . Concluding Remarks In this paper we presented a theoretical approach to study the effect of heat transfer on peristaltic flow of viscoelastic fluid in an asymmetry channel. The governing equations of motion and energy are solved analytically using perturbation expansion on wave as a parameter. Furthermore, The effect of various values of parameters of interest on axial velocity, temperature and heat transfer coefficient are discussed numerically and explained graphically through figures (1-10). Moreover, the heat transfer coefficient is discussed through table(1). The main results can be summarized as follows : •. There is no appreciable effect of Weissenberg number iW on the axial velocity u. •. The magnitude of the axial velocity increases with increasing wave numberδ , Reynolds number eR , phase difference φ and channel width d. •. The temperatureθ increases with increasing Reynolds number eR , Prandtl number rP , Eckert number E and channel width d . •. The magnitude of the temperature increases with an increase in phase difference φ . •. The heat transfer coefficient (Z) increases with increasing upper wave amplitude a, Eckert number E, Reynolds number eR and Weissenberg number iW . References [1] A. Ebaid, Effect of magnetic field and wall slip condition on the peristaltic

transport of a Newtonian fluid in an asymmetric channel, Phys.Lett.A372(2008),4493-4499.

[2] A.H. Shapiro, M.Y. Jaffrin , S.L. Weinberg, Peristaltic pumping with long

wavelengths at low Reynlod number ,J.Fluid Mech,37 (1969), 799-825. [3] A. M. Sobh, Slip flow in peristaltic transport of a Carreau fluid in an

asymmetric channel, Can.J.Phys.87,(2009),1-9. [4] A. M. Sobh, Heat transfer in a slip flow of peristaltic transport of a magneto-

Newtonian fluid through a porous medium, Inter.J.Biomath., vol.2, No.3, (2009), 299-309.

1598 A. M. Sobh, S. S. Al Azab and H. H. Madi [5] A.R. Rao, and M. Mishra , Nonlinear and curvature effects on peristaltic

flow of a viscous fluid in an asymmetric channel ,Acta Mech.168(2004),35-59.

[6] G.Radhakrishnamacharya and Ch.Srinivasulu, Influence of wall properties on

peristaltic transport with heat transfer, Compt.Rendus Mec.335(2007),369-373.

[7] Kh.S.Mekheimer and Y. Abd elmaboud, The influence of heat transfer and

magnetic field on peristaltic transport of a Newtonian fluid in a vertical annulus: Application of an endoscope, Phys.Lett.A372(2008),1567-1665.

[8] M.H.Haroun ,Effect of wall compliance on peristaltic transport of a Newtonian

fluid in an asymmetric channel, Mathematical Problems in Engineering, Vol. 2006, Article ID 61475,pp.1-19,doi:10.1155/MPE/61475(2006).

[9] M. Kothandapani and S.Srinivas, Non-linear peristaltic transport of a Newtonian

fluid in an inclined asymmetric channel through a porous medium, Phys. Lett. A 372, (2008),1265-1276.

[10] M. Mishra, A.R. Rao, Peristaltic Transport of a Newtonian fluid in an

asymmetric channel, ZAMP54(2003),532-550. [11] M.V. Subba Reddy, A. Ramachandra Rao and S. Sreenadh, Peristaltic motion

of a power-law fluid in an asymmetric channel, Int. J. of Non-linear Mech. 42(2007),1153-1161.

[12] N. Ali and T. Hayat, Peristaltic motion of a Carreau fluid in an asymmetric

channel, Appl. Math. Comput.193 (2007),535-552. [13] O. Eytan and D.Elad, Analysis of Intra-Uterine fluid motion induced by uterine

contraction,Bull.Math.Biology 61 (1999), 221-238. [14] R.K.Rathy,An Introduction to Fluid Dynamics, Oxford and IBH publishing

Co., New Delli, Bombay ,Calcutta.,1976. [15] S.Srinivas and M.Kothandapani, Peristaltic transport in an asymmetric channel

with heat transfer, Inter.Commun.in heat and Mass Trans.35,(2008),514-522. [16] T.Hayat, N. alvi and N.Ali, Peristaltic mechanism of a Maxwell fluid in an

asymmetric channel, Nonlinear Anal. Real World Appl.9,(2008),1474-1490.

Heat transfer in peristaltic flow of viscoelastic fluid 1599 [17] Y.C. Fung and C.S.Yih, Peristaltic transport Transport, J.Appl.Mech.,35

(1968),669-675. [18] Y. Wang, T. Hayat, N. Aliand, M.Oberlack, Magnetohydrodynamic peristaltic

motion of a Sisko fluid in an asymmetric or asymmetric channel, Phy.A387(2008),347-362.

1.6 1.4 1.2 1.0 0.8 0.6 0.4

1.0

0.5

0.0

0.5

1.0

1.5

Fig.1. Axial velocity versus y at x=0.2,δ =0.02,E=1, rP =1, eR =10,a=0.5,b=0.7,d=1,θ =1,4πφ = .

Wi= 0 Wi=0.04 Wi=0.08

u

y


1.2 1.0 0.8 0.6 0.4

0.5

0.0

0.5

1.0

Fig.2.Axial velocity versus y at x =0.2 ,E=1, rP =1,W=0.01, eR =10,a=0.5,b=0.7,d=1,θ =1,4πφ = .

0.38 0.37 0.36 0.35 0.34 0.33

0.0

0.1

0.2

0.3

0.4

Fig.3.Axial velocity versus y at x =0.2 ,E=1,δ =0.02, rP =1,W=0.04,a=0.5,b=0.7,d=1,θ =1,4πφ = .

u

y

δ=0 δ=0.04 δ=0.08

u

y

Re=0 Re=10 Re=20


0.30 0.25 0.20 0.15 0.10 0.050.4

0.2

0.0

0.2

0.4

0.6

Fig.4.Axial velocity versus y at x =0.2 ,E=1,δ =0.02, rP =1,W=0.04, eR =10,a=0.3,b=0.5,d=0.7,θ =1.

0.26 0.24 0.22 0.20 0.18 0.16 0.14 0.12

0.0

0.2

0.4

0.6

Fig.5.Axial velocity versus y at x =0.,E=1,δ =0.02, rP =1,W=0.04,a=0.5,b=0.5, eR =10,θ =1,4πφ = .

u

y

u

y

φ=0 φ=π/6 φ=π/4

d=0.7 d=0.8 d=0.9


2.0 2.5 3.0 3.5 4.0

2

1

0

1

Fig.6.Temperature distribution y at x =0 ,E=1,δ =0.02, rP =1,W=0.04,a=0.7,b=1.2,d=2, ,θ =1,4πφ =

1.60 1.65 1.70 1.75 1.80 1.85 1.90

0.0

0.5

1.0

Fig.7.Temperature distribution y at x =0 ,E=1,δ =0.02, eR =50,W=0.04,a=0.5,b=0.7,d=2, ,θ =1,4πφ =

θ

y

θ

y

Re=90 Re=80 Re=70

Pr=1.5 Pr=1.4 Pr=1.3


1.60 1.65 1.70 1.75 1.80 1.85 1.90

0.0

0.5

1.0

Fig .8.Temperature distribution y at x =0 ,δ =0.02, rP =1.5,W=0.04,a=0.5,b=0.7,d=2, eR =50,

θ =1,4πφ = .

2.0 2.2 2.4 2.6

0.5

0.0

0.5

1.0

Fig .9.Temperature distribution y at x =0 ,E=1,δ =0.02, rP =2,W=0.04,a=0.5,b=0.7, eR =50,

θ =1,4πφ =

θ

θ

y

y

E=1 E=0.8 E=0.6

d=2 d=1.9 d=1.8


0.8 0.7 0.6 0.5 0.4 0.3 0.2

1.5

1.0

0.5

Fig.10.Temperature distribution y at x =0 ,E=1,δ =0.02, rP =1,W=0.03,a=0.3,b=0.4,d=0.9,

eR =10,θ =1.

θ

y

φ=0 φ=π/6 φ=π/4

Heat transfer in peristaltic flow of viscoelastic fluid 1605 Table 1

Variation of heat transfer coefficient

(a):δ =0.02,E=1, rP =1, iW =0.04, eR =10,b=1.2,d=1.5,θ =1,φ =4π

x a

0.5 0.7 0.9 1.1

0.1 1.36228 1.38401 1.42048 1.47375

0.2 1.64078 1.67317 1.6991 1.71983

0.3 2.15405 2.72392 3.81008 5.80024

(b):δ =0.02, rP =1,W=0.04, eR =10,a=0.5,b=1.2,d=2,θ =1,4πφ =

x E

1 2 3 4

0.1 1.21571 1.235 1.25429 1.27358

0.2 1.36437 1.41494 1.46551 1.51609

0.3 1.52971 1.57427 1.61882 1.66338

(c):δ =0.02, rP =1,E=1,W=0.04 ,a=0.5,b=1.2,d=1.5,θ =1,4πφ =

x eR

10 2 0 3 0 40

0.1 1.36228 1.40381 1.44535 1.48689

0.2 1.64078 1.75637 1.87196 1.98755

0.3 2.15405 2.61615 3.07824 3.54034


(d):δ =0.02, rP =1,E=1, eR =10 ,a=0.7,b=1.2,d=1.5,θ =1,4πφ =

x W

0.02 0.04 0.08 0.16

0.1 1.38303 1.38401 1.38599 1.38993

0.2 1.67104 1.67317 1.67741 1.6859

0.3 2.71924 2.72392 2.73297 2.75106

Received: October, 2009

Heat Transfer in Peristaltic Flow of Viscoelastic Fluid in ... · Heat transfer in peristaltic flow of viscoelastic fluid 1587 whereδis the wave number,Re is the Reynolds number,

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