On the influence of heat transfer in peristalsis with variable viscosity

International Journal of Heat and Mass Transfer 52 (2009) 4722–4730

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On the influence of heat transfer in peristalsis with variable viscosity S. Nadeem *, T. Hayat, Noreen Sher Akbar, M.Y. Malik Department of Mathematics, Quaid-i-Azam University, Islamabad 44000, Pakistan

a r t i c l e

i n f o

Article history: Received 14 January 2009 Received in revised form 24 April 2009 Accepted 24 April 2009 Available online 22 July 2009

a b s t r a c t This study concentrates on the heat transfer characteristics and endoscope effects for peristaltic flow of a third order fluid. Two models of variable viscosity are chosen. Both perturbation and numerical solutions are obtained in each case. A comparative study is also made between the two solutions. The importance of pertinent flow parameters entering into the flow modeling is discussed. Ó 2009 Elsevier Ltd. All rights reserved.

Keywords: Peristaltic flow Heat transfer Variable viscosity Numerical solutions Shooting method Comparison

1. Introduction Peristaltic flows are relevant to a number of industrial and physiological applications including urine transport from kidney to the bladder, swallowing of food through the esophagus, chyme movement in the gastrointestinal tract, vasomotion of small blood vessels such as arteries, venules and capillaries, in roller and finger pumps and many others. Such flows are extensively studied in various geometries by using different assumptions of large wavelength, small amplitude ratio, small wave number, creeping flow etc. At present a wealth of literature on the topic dealing with the peristalsis in viscous and non-Newtonian fluids is available. However, few recent studies [1–16] and many refs there in may be mentioned in this direction. Existing literature indicates that little efforts are made to explain the heat transfer effects on the peristalsis. Radhakrishnamacharya and Srincvasulu [17] discussed the combined effects of wall properties and heat transfer on the peristaltic flow of a viscous fluid in a channel. Mekheimer and Elmaboud [18] analyzed the heat transfer and MHD effects on the peristaltic transport of a viscous fluid in a vertical annulus. Srinivas and Kothandapani [19] studied the peristaltic flow with heat transfer in an asymmetric channel when the wavelength is very large. In another attempts, Kothandapani and Srinivas [20] analyzed the influence of elasticity of the flexible walls on MHD peristaltic flow of a viscous fluid in a porous channel with heat transfer. Very recently, Hayat et al. [21]

* Corresponding author. Tel.: +92 5190642182; fax: +92 92512275341. E-mail address: [email protected] (S. Nadeem). 0017-9310/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2009.04.037

examined the MHD peristaltic flow of a third order fluid with an endoscope and variable viscosity. The purpose of the present investigation is to perform a study which can describe the heat transfer and endoscope effects on the peristaltic flow of a third order fluid when the viscosity is not constant. Two cases of variable viscosity are taken into account. The flow analysis is performed under the assumtion of long wavelength situation. Series solution is derived for small Deborah number. Numerical solution is obtained by employing shooting method. Comparison between the two presented solution is given and discussed. 2. Development of the problem We investigate the flow of a third order fluid through the gap between two concentric tubes. The inner tube is maintained at a temperature T0. A sinusoidal wave travels down on the wall of the outer tube having temperature T1. We choose cylindrical coordinate system in such a way that R is in the radial direction and the Z along the centre line of the inner and outer tubes. Shapes of the two walls are described by the following expressions

R1 ¼ a1 ;

ð1Þ

2p R2 ¼ a2 þ b sin ðZ  ctÞ: k

ð2Þ

In above expressions a1 is the radius of the inner tube, a2 is the radius of the outer tube at the inlet, b is the wave amplitude, k is the wavelength, c is the wave speed and t is the time. In the fixed frame ðR; ZÞ the continuity, momentum and energy equations give

S. Nadeem et al. / International Journal of Heat and Mass Transfer 52 (2009) 4722–4730

@U U @W þ þ ¼ 0; @R R @Z 

q

ð3Þ

  @  1 @  @ @ @ @p S  þ U W U ¼  RS þ þ ðS Þ  hh ; þ RR @t R @R @Z @R R @R @Z RZ ð4Þ



  @    1 @  @ @ @ @p q þ U þ W W¼ RSRZ þ S ; þ @t @R @Z @Z R @R @Z ZZ

qcp

ð5Þ



 @ @ @ @U @W @U U W T ¼ SRR þ þ SRZ þ SZR þ SZZ þ @t @R @Z @R @R @Z ! @W @ 2 T 1 @T @ 2 T ;  þ þk þ @Z @R2 R @R @Z 2

S¼

lþ

b3 trA21



2 2 A1

A1 þ a1 A2 þ a





þ b1 A3 þ b2 A1 A2 þ A2 A1 :

ð7Þ

t

A1 ¼ L þ L ; dAn þ An L þ Lt An ; dt

ð8Þ

 and w are the velocities in the wave frame. where u The boundary conditions in the wave frame are

w ¼ c;

 ¼ 0 at r ¼ r1 ¼ e; u

2p z; w ¼ c; at r ¼ r2 ¼ a2 þ b sin k T ¼ T 0 ; at r ¼ r1 ; T ¼ T1;

at r ¼ r2 :



If d, Pr, Re, Ec, e, / ¼ ab2

ð13Þ

@ 2 h 1 @h @2h þ þ d2 2 ; @r 2 r @r @z ð14Þ

w ¼ 1; at r ¼ r 1 ¼ e; w ¼ 1; at r ¼ r 2 ¼ 1 þ / sinð2pzÞ;

ð15aÞ ð15bÞ

h ¼ 1; at r ¼ r 1 ;

ð15cÞ

h ¼ 0; at r ¼ r 2

ð15dÞ

and

 2 @w Srr ¼ ð2k1 þ k2 Þ ; @r "    3 # @w @w þ 2C Srz ¼ lðrÞ ; @r @r "  # 2 @w Szz ¼ k2 ; @r

ð16aÞ ð16bÞ ð16cÞ

 2 ! 2  2  @u u @w u @u Shh ¼ lðrÞ  k1 þ 8C ; þ 5k1 þ 4k2 þ 4C @r r @r r @r

in which L ¼ gradV and t is the matrix transpose. The coordinates in the fixed ðR; ZÞ and wave ðr ; zÞ frames are related through the following transformations

 ¼ U; w ¼ W  c; u

ð12Þ



n ¼ 1; 2

r ¼ R; z ¼ Z  ct;

ð11Þ

    @ @ @u @w @u @w h ¼ Ec Pr d Srr þ Srz þ d2 Szr þ Szz d RedPr u þ w @r @z @r @r @z @z

ð6Þ

where l, ai(i = 1, 2) and bi(i = 1, 2, 3) are the material constants satisfying the thermodynamical conditions defined in [13]. The first three Rivlin–Ericksen tensors can be written as

Anþ1 ¼

  @ @ @p 1 @ @ w¼ þ Red u þ w ðrSrz Þ þ d ðSzz Þ; @r @z @z r @r @z

þ

 is the pressure and U; W are the respective velocity compowhere p nents in the radial and axial directions respectively, T is the temperature, q is the density, k denotes the thermal conductivity and cp is the specific heat at constant pressure. Expressions of an extra stress tensor S in a third order fluid is given by the following relation



@u u @w þ þ ¼ 0; @r r @z   @ @ @p d @ @ dShh u¼ þ ðrSrr Þ þ d2 ðSrz Þ  ; Red3 u þ w @r @z @r r @r @z r

4723

ð16dÞ where k1 ¼ a1 c=l0 a2 ; k2 ¼ a2 c=l0 a2 and the Deborah number C = bc2/l0a2. Invoking Eqs. (16a)–(16c) and using long wavelength and low Reynolds number assumptions, Eqs. (11)–(13) yield

@p ; @r    3 !! @p 1 @ @w @w ; r lðrÞ þ 2C 0¼ þ @z r @r @r @r ! !      3 @w @w @w @ 2 h 1 @h þ 2C 0 ¼ Br þ 2þ lðrÞ ; @r @r @r @r r @r

0¼ ð9aÞ ð9bÞ ð9cÞ

ð17Þ ð18Þ ð19Þ

ð9dÞ

 < 1 , l0, are respectively, the wave num-

where Br = Ec Pr is the Brinkman number and Eq. (17) shows that p – p(r).

ber, Prandtl number, Reynold number, Eckert number, radius ratio and reference viscosity then defining 3. Solution of the problem

z r R Z W w R¼ ; r¼ ; Z¼ ; z¼ ; W ¼ ; w¼ ; k a2 k c c a2  kU ku a22 P ðT  T 1 Þ l cp ; h¼ ; Pr ¼ 0 ; U¼ ; u¼ ; P¼ a2 c a2 c ckl0 k ðT 0  T 1 Þ

lðrÞ ¼

3.1. Case I In order to seek the solution of the problem we let

lðrÞ ¼ ear

l ðrÞ ct a2 qca2 a2 S b ; t ¼ ; d ¼ ; Re ¼ ; S¼ ; /¼ ; k c l0 a2 l0 k l0

which by Maclaurin series can be written as

r1 r2 c2 r1 ¼ ¼ e; r 2 ¼ ¼ 1 þ / sinð2pzÞ; Ec ¼ : a1 a2 cp ðT 0  T 1 Þ

lðrÞ ¼ 1  ar þ Oða2 Þ: ð10Þ

Eqs. (3)–(7) and (9) become

ð20Þ

ð21Þ

Substituting above equation into Eqs. (15), (18) and (19) and then solving the resulting problem by perturbation technique one obtains

S. Nadeem et al. / International Journal of Heat and Mass Transfer 52 (2009) 4722–4730

dp 1 w¼ 1þ ð3r2 þ 2ar3 þ 12L3 ðln r þ arÞ þ 12L4 Þ dz 12  þ C L56 r 8 þ L57 r 7 þ L58 r 6 þ L59 r 5 þ L60 r4 þ L61 r 3 þ L62 r 2  L49 L63 þL64 r þ L55 þ L65 ln r  þ 2 ; r r

a -0.99 Numerical solution

-1

ð22Þ

h ¼ L18 r 7 þ L19 r6 þ L20 r 5 þ L21 r 4 þ L22 r3 þ L23 r 2 þ L16 r þ L24 ðln rÞ2  þ L27 ln r þ L28 þ C L120 r 17 þ L121 r 16 þ L122 r 15 þ L123 r 14

  dp Q þ ðr 22  r 21 Þ L67 þC ¼ ; dz L66 L66

-1.04

ð23Þ -1.05 0.1

0.3

0.4

0.5

r

0.6

0.7

0.8

0.9

1

0.8

0.9

1

b -0.995 -1 Numerical solution

-1.005

Perturbation solution

-1.01

1

ð25Þ

-1.015

w(r,z)

dp dz; 0 dz  Z 1  dp ðoÞ dz; Fk ¼ r21  dz 0  Z 1  dp ðiÞ Fk ¼ dz; r 22  dz 0

Dpk ¼

0.2

ð24Þ

where C is used as the perturbation quantity. All the appearing quantities in above equations are included in the Appendix A. The dimensionless pressure rise Dpk and friction force on outer ðoÞ ðiÞ tube F k and inner tube F k , are defined by

Z

-1.02

-1.03

þL124 r 13 þ L125 r 12 þ L126 r 11 þ L127 r 10 þ L128 r 9 þ L129 r8 þL130 r 7 þ L131 r 6 þ L132 r 5 þ L133 r 4 þ L134 r 3 þ L135 r2 L118 L137 þL116 r þ L136 ðln rÞ2 þ þ 2 þ L140 ln r þ L141 ; r r

Perturbation solution

-1.01

w(r,z)

4724

ð26Þ ð27Þ

-1.02 -1.025 -1.03 -1.035

in which dp/dz is defined through Eq. (24). -1.04

3.2. Case II 0.1

Here we are interested in the temperature-dependent viscosity. For that we choose Reynold’s model of viscosity i.e., bh

lðhÞ ¼ e ;

ð28Þ

where b is a parameter. Employing the same methodology as in Case I, the numerical solution of axial velocity is presented in Subsection 3.2. However the series solutions are

  dp r2 a11 þ ba4 ðrÞ  w ¼ 1 þ ðln r þ ba5 ðrÞ  a13 Þ dz 4 a12   a14 þC  ln r þ a7 ðrÞ  ba5 ðrÞ þ a15 ; a11 dp 2Q þ ðr 22  r 21 Þ ¼ ; dz 2a8 ðrÞ

0.2

0.3

0.4

0.5

r

0.6

0.7

Fig. 1. (a) Comparison of axial velocity for perturbation and numerical solutions ¼ 0:4, z ¼ 0:5. (b) Comparison of axial when C ¼ 0:005, a ¼ 0:2, / ¼ 0:6, e ¼ 0:1, dp dz velocity for perturbation and numerical solutions when C ¼ 0:005, b ¼ 0:2, / ¼ 0:6, dp e ¼ 0:1, dz ¼ 0:4, z ¼ 0:5, Br ¼ 0:1.

Table 1 Case I.

ð29Þ

ð30Þ

h ¼ Br ða14 ðrÞ þ a15 ðrÞ þ a20 ln r þ a21 Þ þ CðBr ða24 ðrÞ þ a25 ðrÞÞÞ: ð31Þ where all ai(r) and aij (i, j = 1, 2, 3 . . .) are given in Appendix A. (see Fig. 1)

4. Comparison between numerical and perturbation solutions Here the problem consisting of Eqs. (15a,b) and (18) is also solved numerically by employing shooting method. The numerical

r

Numerical solution

Perturbation solution

Error

0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00

1.000000 1.016465 1.029170 1.037464 1.042150 1.045808 1.047766 1.048315 1.048003 1.046604 1.044183 1.041439 1.037320 1.033230 1.027559 1.022216 1.015081 1.007172 1.000000

1.000000 1.010710 1.018960 1.025111 1.031621 1.035891 1.038451 1.039621 1.037822 1.038521 1.037642 1.029661 1.028631 1.025292 1.022292 1.018654 1.014380 1.006592 1.000000

0.000000 0.005755 0.010020 0.012050 0.010206 0.009573 0.008970 0.008362 0.009809 0.007783 0.006303 0.011438 0.008447 0.007421 0.005152 0.004967 0.000691 0.000576 0.000000

4725

S. Nadeem et al. / International Journal of Heat and Mass Transfer 52 (2009) 4722–4730 Table 2 Case II.

4

x 10

3

Γ = 0.010

r

Numerical solution

Perturbation solution

Error

0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00

1.000000 1.013079 1.020100 1.026625 1.030679 1.045808 1.036289 1.037758 1.038221 1.037908 1.036671 1.034967 1.032094 1.029015 1.024494 1.020031 1.013829 1.007942 1.000000

1.000000 1.014210 1.021160 1.030789 1.031621 1.038601 1.040463 1.041561 1.041632 1.040785 1.037642 1.036805 1.033389 1.029900 1.024958 1.020021 1.014380 1.007858 1.000000

0.000000 0.001115 0.000156 0.004039 0.000913 0.006939 0.008970 0.003652 0.002408 0.003952 0.006303 0.000935 0.001253 0.000859 0.000452 0.000001 0.000543 0.000083 0.000000

2

Γ = 0.015 Γ = 0.020

1

Γ = 0.025

F

(0)

0 -1 -2 -3 -4 -5 -1

-0.5

0

0.5

1

1.5

2

2.5

3

Q Fig. 4. Frictional force (on inner tube) versus flow rate when e = 0.3, / = 0.9, a = 0.8.

6

x 10

5

6

4

Γ = 0.010

5

Γ = 0.010

Γ = 0.015

4

Γ = 0.015

Γ = 0.020

4

Γ = 0.020

Γ = 0.025

2

3 2

Γ = 0.025

(i)

0

1

F

Δp

x 10

-2

0 -1

-4

-2 -6 -3 -4 -1

-0.5

0

0.5

1

1.5

2

2.5

-8 -1

3

-0.5

0

0.5

1

1.5

2

2.5

3

Q

Q

Fig. 5. Frictional force (on outer tube) versus flow rate when e = 0.3, / = 0.9, a = 0.8.

Fig. 2. Pressure rise versus flow rate when e = 0.3, / = 0.9, a = 0.8.

1000

4000 α α α α

3500 3000

= = = =

0.00 0 .05 0.10 0.15

500

α α α α

= = = =

0.00 0.05 0.10 0.15

0

2500 -500

(i)

F

Δp

2000 1500

-1000

1000 -1500

500 -2000

0

-500 -1000 -3

-2500 -3

-2

-1

0

1

2

-2

-1

0

Q Fig. 3. Pressure rise versus flow rate when e = 0.8, / = 0.85, C = 0.8.

1

2

3

Q

3

Fig. 6. Frictional force / = 0.85, C = 0.003.

(on outer

tube)

versus

flow

rate

when

e = 0.8,

4726

S. Nadeem et al. / International Journal of Heat and Mass Transfer 52 (2009) 4722–4730 18

40 Γ = 0.00

35

Q=1

16

Γ = 0.06 Γ = 0.12

30

Q=3

14

Γ = 0.18

Q=4

12

25

10 20

dp/dz

dp/dz

Q=2

15

8 6

10

4

5 0 0.5

2

0.6

0.7

0.8

0.9

1

0 0.4

1.1

z

0.7

0.8

z

0.9

1

1.1

-0.995

90

Γ = 0.000

α = 0.0

80

-1

α = 0.03

α = 0.09

-1.01

50

-1.015

w(r,z)

60

40

-1.025

20

-1.03

10

-1.035

0.7

0.8

0.9

z

1

Γ = 0.010 Γ = 0.015

-1.02

30

0.6

Γ = 0.005

-1.005

α = 0.06

70

dp/dz

0.6

Fig. 10. Axial pressure gradient dp/dz when e = 0.3, a = 0.4, C = 0.2, / = 0.3.

Fig. 7. Axial pressure gradient dp/dz when e = 0.3, / = 0.4, a = 0.2, Q = 3.

0 0.5

0.5

-1.04 0.4

1.1

0.5

0.6

0.7

0.8

r

0.9

Fig. 11. Variation of Deborah number on w when

Fig. 8. Axial pressure gradient dp/dz when e = 0.3, / = 0.4, C = 0.2, Q = 3.

1

1.1

1.2

1.3

e ¼ 0:1; a ¼ 0:2; dp ¼ 0:4; z ¼ 0:5: dz

-0.99 18 φ = 0.20

α = 0.10 α = 0.15

φ = 0.35

14

-1.005

12

-1.01

w(r,z)

dp/dz

α = 0.05

-1

φ = 0.30

10

-1.015 -1.02

8

-1.025

6

-1.03

4 2 0.5

α = 0.00

-0.995

φ = 0.25

16

0.4 0.6

0.7

z

0.8

0.9

Fig. 9. Axial pressure gradient dp/dz when e = 0.3, a = 0.4, C = 0.2, Q = 3.

1

0.5

0.6

0.7

0.8

r

0.9

1

Fig. 12. Variation of viscosity parameter on w when z ¼ 0:5:

1.1

1.2

1.3

e ¼ 0:1; C ¼ 0:01; dp ¼ 0:4; dz

4727

S. Nadeem et al. / International Journal of Heat and Mass Transfer 52 (2009) 4722–4730 6

1.6

Γ Γ Γ Γ

B = 0.2 r

B = 0.4

1.4

r

5

B = 0.6 r

B = 0.8 r

0.0 0.1 0.2 0.3

4

1

θ(r,z)

θ (r,z)

1.2

= = = =

0.8

3

2

0.6 1

0.4

0

0.2 0

0.4

0.6

0.8

1

r

Fig. 13. Temperature profile for e = 0.3, a = 0.8,

dp dz

-1 0.2

1.2

¼ 0:4; z ¼ 0:1; Q ¼ 1; C ¼ 0:1:

2.5

= = = =

0.0 0.1 0.2 0.3

2

θ(r,z)

0.6

0.8

r

1

Fig. 15. Temperature profile for e = 0.3, Br = 0.8,

1.2

dp dz

1.4

1.6

¼ 0:4; z ¼ 0:1; Q ¼ 1; a ¼ 0:1:

and ze[0.9, 1.1] and large when ze½0:61; 0:89: Moreover the pressure gradient increases by increasing Deborah number. Fig. 8 illustrates that the pressure gradient is small when ze½0:5; 0:6 and ze½0:9; 1:1 and large when ze½0:61; 0:89: It is also observed that pressure gradient decreases by increasing a. Figs. 9 and 10 show large pressure gradient as compared to the Figs. 7 and 8. It is also observed from the Figs. that pressure gradient increases by increasing / and Q. The velocity field w for different values of C and a are shown in the Figs. 11 and 12 respectively. It is noticed that axial velocity increases with an increase in C and decreases by increasing a (see Figs. 11 and 12). The temperature field for different values of Br, C and a against space variable r are displayed in Figs. 13–15. It is found that the temperature increases when Br, C and a are increased.

3 α α α α

0.4

1.5

1

0.5

Acknowledgement 0

0.4

0.6

0.8

1

r

Fig. 14. Temperature profile for e = 0.3, Br = 0.8,

dp dz

1.2

1.4

The corresponding author is thankful to the Quaid-i-Azam University for providing research funds.

¼ 0:4; z ¼ 0:1; Q ¼ 1; C ¼ 0:1:

Appendix A solution is also compared with the perturbation solution. The difference between the values of two solutions is given in Tables 1 and 2.

Here we present the values of all functions appearing in the solutions (22) to (24) i.e.

L1 ¼

3ðr 21  r 22 Þ þ 2aðr 31  r 32 Þ L1 ; L2 ¼ ln r 1  ln r 2 ; L3 ¼  ; 12 L2

L4 ¼

ð3r 21  2ar 31 Þ  12L3 ðln r1 þ ar1 Þ ; L5 ¼ 8aL3 þ 4a2 L3 ; 12

5. Numerical results and discussion In this section the pressure rise, frictional forces, axial pressure gradient, axial velocity and temperature are analyzed carefully. For this object, the Figs. 2–15 are displayed. The pressure rise is calculated numerically by using Mathematica. Figs. 2 and 3 show the pressure rise Dp against volume flow rate Q for different values of Deborah number C and viscosity parameter a. These Figs. indicate that the relation between pressure rise and volume flow rate are inversely proportional to each other. As expected that pressure rise gives larger values for small volume flow rate and it gives smaller values for large Q. Moreover, the peristaltic pumping occurs in the region 1 6 Q < 1 for Fig. 2, 3 6 Q 6 1 for Fig. 3, other wise augmented pumping occurs. Moreover, the pressure rise increase with an increase in Deborah number and decreases when a is increased. Figs. 4–6 describe the variation of frictional forces. These Figs. show that the frictional forces have quite opposite features when compared with the pressure rise. The variations of pressure gradient are plotted in Figs. 7–10. Fig. 7 elucidates that pressure gradient is small when ze[0.5, 0.6]

 2 dp 1 ; L6 ¼ 4L3 þ 4a2 L23 ; L7 ¼ 8aL23 ; L8 ¼ 4L23 ; L9 ¼ Ec Pr dz 2 L10 ¼ L9 a3 ; L11 ¼ L9 a2 ; L12 ¼ aL9 ; L13 ¼ L9 ð1  aL5 Þ; L14 ¼ L9 ðL5  aL6 Þ; L15 ¼ L9 ðL6  aL7 Þ; L16 ¼ L9 ðL7  aL8 Þ; L17 ¼ L8 L9 ; L18 ¼ L22 ¼

L10 L11 L12 L13 ; L19 ¼ ; L20 ¼ ; L21 ¼ ; 49 36 25 16

L14 L15 L17 ; L23 ¼ ; L24 ¼ ; 9 4 2

L25 ¼ L18 r 71 þ L19 r61 þ L20 r 51 þ L21 r 41 þ L22 r 31 þ L23 r 21 þ L16 r1 þ L24 ðln r 1 Þ2 ;

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S. Nadeem et al. / International Journal of Heat and Mass Transfer 52 (2009) 4722–4730

L26 ¼ L18 r 72 þ L19 r 62 þ L20 r 52 þ L21 r 42 þ L22 r32 þ L23 r 22 þ L16 r 2

L27 ¼

L71 ¼ L32 þ aL33 þ 2L3 L30 þ 2aL3 L31 ;

þ L24 ðln r2 Þ ;

L72 ¼ L33 þ aL34 þ 2L3 L31 þ 2aL3 L32 ;

 3 1  L25 þ L26 @p 1 ; L ¼ 1  L  L ln r ; L ¼ ; 28 25 27 1 29 @z 2 ln rr12

L73 ¼ L34 þ aL35 þ 2L3 L32 þ 2aL3 L33 ;

2

L74 ¼ L35 þ aL36 þ 2L3 L33 þ 2aL3 L34 ; L75 ¼ L36 þ aL37 þ 2L3 L34 þ 2aL3 L35 ; L76 ¼ L37 þ aL38 þ 2L3 L35 þ 2aL3 L36 ;

L30 ¼ L29 a3 ; L31 ¼ L29 a2 ; L32 ¼ ð3a þ 2L3 a3 Þ L29 ; L33 ¼ L29 ð1 þ aL5 þ 6a2 L3 Þ; L34 ¼ L29 ðL5 þ aL6 þ 6aL3 Þ; L35 ¼ L29 ðL6 þ aL7 þ 2L3 þ 2L3 L5 aÞ; L36 ¼ L29 ðL7 þ aL8 þ 2L3 L5 þ 2L3 L6 aÞ;

L77 ¼ L38 þ aL39 þ 2L3 L36 þ 2aL3 L37 ; L78 ¼ L39 þ 2L3 L37 þ 2aL3 L38 ; L79 ¼ 2L3 L38 þ 2aL3 L39 ; L80 ¼ 2L3 L39 ; L81 ¼ 64ðL56 Þ2 ;

L37 ¼ L29 ðL8 þ 2L3 L6 þ 2L3 L7 aÞ; L38 ¼ ð2L3 L7 þ 2L3 L8 aÞ; L39 ¼ 2L3 L8 ; L40 ¼ 2aL30 ; L41 ¼ ð2L30  2aL31 Þ; L42 ¼ ð2L31  2aL32 Þ; L43 ¼ ð2L32  2aL33 Þ; L44 ¼ ð2L33  2aL34 Þ;

L82 ¼ 256L56 L57 ; L83 ¼ 248L56 L58 þ 49ðL57 Þ2 ; L84 ¼ 80L56 L59 þ 84L57 L58; L85 ¼ 64L60 L56 þ 70L57 L59 þ 36ðL58 Þ2 ; L86 ¼ 48L61 L56 þ 56L57 L60 þ 60L58 L59 þ

L45 ¼ ð2L34  2aL35 Þ; L46 ¼ ð2L35  2aL36 Þ; L47 ¼ ð2L36  2aL37 Þ;

  dp1 1 ð24aL56 Þ; dz 2

L87 ¼ 32L62 L56 þ 42L57 L61 þ 48L58 L60 þ 25ðL59 Þ2 þ

L48 ¼ ð2L37  2aL38 Þ;



dp1 1 dz 2



 ð16L56 þ 21aL57 þ 16aL56 L3 Þ; L49 ¼ ð2L38  2aL39 Þ; L50 ¼ 2L39 ; L51 ¼

L56 r 81

L57 r 71

þ þ L49 L50   ; r 1 2r 21

L58 r 61

þ

L59 r 51

þ

L60 r 41

þ

L61 r 31

þ

L62 r 21

þ L47 r 1 þ L48 ln r

L56 ¼ L57

L53 ; L55 ¼ L54 ðln r1 þ ar 1 Þ  L51 ; L2

L40 ; 8

L41 L42 L43 L44 L45 L46 ¼ ; L58 ¼ ; L59 ¼ ; L60 ¼ ; L61 ¼ ; L62 ¼ ; 7 6 5 4 3 2 L50 ; L64 ¼ aL54 þ L47 ; L65 ¼ aL55 þ L48 ; 2   1 2a 5 ¼ 15ðr 42  r 41 Þ þ ðr  r51 Þ þ 20L3 6ðr 22 ln r 2  r21 ln r 1 Þ 120 5 2   3ðr 22  r 21 Þþþ4aðr 32  r 31 Þ þ 120L4 ðr 22  r 21 Þ ;

L63 ¼  L66

L67 ¼

  dp1 1 dz 2

 ð14L57 þ 18aL58 þ 16aL56 L3 Þ;

L52 ¼ L56 r 82 þ L57 r 72 þ L58 r 62 þ L59 r 52 þ L60 r 42 þ L61 r 32 þ L62 r 22 þ L47 r 2 þ L48 ln r 2 L49 L50   ; r 2 2r 22

L53 ¼ L51  L52 ; L54 ¼ 

L88 ¼ 16L64 L56 þ 28L57 L62 þ 36L58 L61 þ 40L59 L60 þ

 2  9  1   1  10 þ L57 r 2  r 91 þ L58 r 82  r81 L56 r 2  r 10 1 5 9 4  1   2   2  þ L59 r 72  r 71 þ L60 r62  r61 þ L61 r 52  r 51 7 3 5  2     1  þ L62 r 42  r 41 þ L64 r32  r31 þ L55 r 22  r 22 2 3   2   2 r 2 ln r 2  r 21 ln r1 r 2  r 21  2L49 ðr 2  r 1 Þ þ 2L65  2 4 þ 2L63 ðln r 2  ln r 1 Þ;

L68 ¼ aL30 ; L69 ¼ L30 þ aL31 ; L70 ¼ L31 þ aL32 þ 2aL3 L30 ;

L89 ¼ 8L65 L56 þ 14L57 L64 þ 24L58 L62 þ 30L59 L61   dp1 1 þ 16ðL60 Þ2 þ ð12L58 þ 15aL59 þ 32L56 L3 þ 28aL57 L3 Þ; dz 2 L90 ¼ 8L49 L56 þ 7L57 L65 þ 16L58 L64 þ 20L59 L62   dp1 1 þ 24L60 L61 þ ð10L59 þ 12aL60 þ 28L57 L3 þ 24aL58 L3 Þ; dz 2 L91 ¼ 16L63 L56 þ 7L57 L49 þ 6L58 L65 þ 10L59 L64 þ 16L60 L62   dp1 1 þ 9ðL61 Þ2 þ ð8L60 þ 9aL61 þ 24L58 L3 þ 20aL59 L3 Þ; dz 2 L92 ¼ 14L63 L57 þ 6L58 L49 þ 5L59 L65 þ 8L60 L64 þ 12L61 L62  2   dp1 1 dp1 1 þ 2a þ ð5L61 þ 6aL62 þ 20L59 L3 þ 16aL60 L3 Þ; dz 2 dz 2

L93 ¼ 12L63 L58 þ 5L59 L49 þ 4L60 L65 þ 3L61 L64 þ 3L61 L64 þ 4ðL62 Þ2  2   dp1 1 dp1 1 þ ð1 þ 4L3 a2 Þ þ dz 2 dz 2  ð4L62 þ 4aL64 þ 16L60 L3 þ 12aL61 L3 Þ; L94 ¼ 10L63 L59 þ 4L60 L49 þ 3L61 L65 þ 2L62 L63 þ 2L62 L64  2   dp1 1 dp1 1 þ ð8L3 aÞ þ dz 2 dz 2  ð2L64 þ 3aL65 þ 12L61 L3 þ 8aL62 L3 Þ;  2 dp1 1 L95 ¼ 8L63 L60 þ 3L62 L49 þ 3L62 L65 þ ðL64 Þ2 þ ð4L3 þ 4L23 a2 Þ dz 2   dp1 1 þ ð2L65 þ 3aL49 þ 8L62 L3 þ 4aL64 L3  2L63 aÞ; dz 2

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 2 dp1 1 L96 ¼ 2L62 L49 þ L65 L64  6L63 L61 þ ð2aL3 þ 4L23 aÞ dz 2   dp1 1 þ ð2L49 þ 4L64 L3 þ 4aL65 L3  4L63 aÞ; dz 2 L97 ¼ 2L62 L63 þ L49 L64 þ

a2 ðrÞ ¼

a6 ðrÞ ¼ w0 ðrÞ; a7 ðrÞ ¼ 2

  dp1 1 dz 2 a8 ðrÞ ¼

 ð4L63 þ 4L65 L3 þ 4aL49 L3  4L63 aÞ; L98 L99

r a1 ðrÞ a1 ðrÞ; a3 ðrÞ ¼ ; a4 ðrÞ ¼ 2 r

Z

r2



r1

  dp1 1 ¼ 2L64 L63 þ ð6L63 L3 a þ 4L49 L3 Þ; dz 2   dp1 1 ð8L63 L3 aÞ; ¼ dz 2

a9 ðrÞ ¼



a12 ðrÞ ¼ a14 ðrÞ ¼

L103 ¼ Ec PrðL83  aL84 Þ; L104 ¼ Ec PrðL84  aL85 Þ; L105 ¼ Ec PrðL85  aL86 Þ;

a16 ðrÞ ¼

L106 ¼ Ec PrðL86  aL87 Þ; L107 ¼ Ec PrðL87  aL88 þ 2L68 Þ; L108 ¼ Ec PrðL88  aL89 þ 2L69 Þ; L109 ¼ Ec PrðL89  aL90 þ 2L70 Þ; L110 ¼ Ec PrðL90  aL91 þ 2L71 Þ; L111 ¼ Ec PrðL91  aL92 þ 2L72 Þ; L112 ¼ Ec PrðL92  aL93 þ 2L73 Þ; L113 ¼ Ec PrðL93  aL94 þ 2L74 Þ;

L116 ¼ Ec PrðL96  aL97 þ 2L77 Þ; L117 ¼ Ec PrðL97  aL98 þ 2L78 Þ; L118 ¼ Ec PrðL98  aL99 þ 2L79 Þ; L119 ¼ Ec PrðL99 þ 2L80 Þ;

  3  dp r a11 1 þ ba2 ðrÞ  þ ba3 ðrÞ ; a11 ðrÞ dz 2 a12 r

Z Z

ra10 ðrÞdr; a13 ðrÞ ¼ a12 ðrÞ dr; a15 ðrÞ ¼ r

a24 ðrÞ ¼

Z

a22 ðrÞ dr; a25 ðrÞ ¼ r

a14 ¼ a7 ðr 1 Þ  a7 ðr 2 Þ;

L130 ¼

L110 L111 L112 L113 L114 ; L131 ¼ ; L132 ¼ ; L133 ¼ ; L134 ¼ ; 49 36 25 16 9

a16 ¼

L115 L117 ; L136 ¼ ; L137 ¼ 4L119 ; ¼ 4 2

a13 ¼

a13 ðrÞ dr; r

Z

Z

ra21 ðrÞdr;

a23 ðrÞ dr; r

a17

a14  a7 ðr 1 Þ: a11

 2  r1 a11 ðln r 1 þ ba5 ðr1 Þ  a13 Þ ; þ ba4 ðr1 Þ  4 a12

1 þ Br ða14 ðr 1 Þ þ a15 ðr 1 Þ  a14 ðr 2 Þ  a15 ðr2 ÞÞ ; ln r 1  ln r2 ¼ 1 þ Br ða14 ðr 1 Þ þ a15 ðr 1 ÞÞ  a16 ln r 1 ;

a18 ¼

16 15 14 13 12 L138 ¼ L120 r 17 1 þ L121 r 1 þ L122 r 1 þ L123 r 1 þ L124 r 1 þ L125 r 1

Br ða24 ðr 2 Þ  a25 ðr 1 Þ  a24 ðr 2 Þ þ a15 ðr 2 ÞÞ; ln r 1  ln r 2

a19 ¼ Br ða24 ðr 1 Þ þ a25 ðr 1 Þ  a18 ln r 1 Þ:

10 9 8 7 6 þ L126 r11 1 þ L127 r 1 þ L128 r 1 þ L129 r 1 þ L130 r 1 þ L131 r 1

þ þ þ þ L116 r 1 þ L136 ðln r1 Þ L118 L137 L138  L139 þ þ 2 ; L140 ¼ r1 ln r2  ln r 1 r1

a11 ðrÞdr;

a12 ¼ ln r 1  ln r 2 þ bða5 ðr 1 Þ  a5 ðr 2 ÞÞ; a15 ¼ 

L105 L106 L107 L108 L109 ; L126 ¼ ; L127 ¼ ; L128 ¼ ; L129 ¼ ; 144 121 100 81 64

þ

Z

a21 ðrÞ ¼ a17 ðrÞ þ a18 ðrÞ; a22 ðrÞ ¼ Z a23 ðrÞ ¼ Br a20 ðrÞdr;

L125 ¼

L135 r21

Z

  @w1 ; a17 ðrÞ ¼ ða16 ðrÞÞ2 ; a18 ðrÞ ¼ ða9 ðrÞÞ4 ; @r

L100 L101 L102 L103 L104 ; L121 ¼ ; L122 ¼ ; L123 ¼ ; L124 ¼ ; 289 256 225 196 169

L134 r 31

r21 r22  þ bða4 ðr 1 Þ  a4 ðr 2 ÞÞ; 4 4

 r2 a11 þ ba4 ðrÞ  ðln r þ ba5 ðrÞ  a13 Þ dr; a10 ðrÞ ¼ ða9 ðrÞÞ2 ; 4 a12

L120 ¼

L133 r 41

a2 ðrÞdr; a11 ¼

a3 ðrÞdr;

a20 ðrÞ ¼ bra1 ðrÞa17 ðrÞ;

L114 ¼ Ec PrðL94  aL95 þ 2L75 Þ; L115 ¼ Ec PrðL95  aL96 þ 2L76 Þ;

L132 r51

a2 ðrÞdr; a5 ðrÞ ¼

Z

¼ bra10 ðrÞa1 ðrÞ;

L100 ¼ Ec PraL81 ; L101 ¼ Ec PrðL81  aL82 Þ; L102 ¼ Ec PrðL82  aL83 Þ;

L135

Z

Z

2

16 15 14 13 12 L139 ¼ L120 r 17 2 þ L121 r 2 þ L122 r 2 þ L123 r 2 þ L124 r 2 þ L125 r 2 10 9 8 7 6 þ L126 r11 2 þ L127 r 2 þ L128 r 2 þ L129 r 2 þ L130 r 2 þ L131 r 2

þ L132 r52 þ L133 r 42 þ L134 r 32 þ L135 r22 þ L116 r 2 þ L136 ðln r2 Þ2 L118 L137 þ þ 2 ; L141 ¼ L138  L141 ln r 1 ; r2 r2 a1 ðrÞ ¼ L18 r7 þ L19 r 6 þ L20 r 5 þ L21 r 4 þ L22 r3 þ L23 r2 þ L16 r þ L24 ðln rÞ2  þ L27 ln r þ L28 þ C L120 r17 þ L121 r 16 þ L122 r15 þ L123 r 14 13 12 þL124 r þ L125 r þ L126 r11 þ L127 r10 þ L128 r 9 þ L129 r 8 þ L130 r7 þ L131 r6 þ L132 r 5 þ L133 r 4 þ L134 r 3 þ L135 r2 þ L116 r L118 L137 2 þ L136 ðln rÞ þ þ 2 þ L140 ln r þ L141 ; r r

References [1] S. Kh, Y. Mekheimer, Abd Elmaboud, Peristaltic flow of a couple stress fluid in an annulus: Application of an endoscope, Physica A 387 (2008) 2403– 2415. [2] Kh.S. Mekheimer, Effect of the induced magnetic field on peristaltic flow of a couple stress fluid, Phys. Lett. A. 372 (2008) 4271–4278. [3] M.H. Haroun, Effect of Deborah number and phase difference on peristaltic transport in an asymmetric channel, Commun. Nonlinear Sci. Numer. Simul. 12 (2007) 1464–1480. [4] Mohamed H. Haroun, Non-linear peristaltic flow of a fourth grade fluid in an inclined asymmetric channel, Comp. Mat. Sci. 39 (2007) 324–333. [5] K. Vajravelu, G. Radhakrishnamacharya, V. Radhakrishnamurty, Peristaltic flow and heat transfer in a vertical porous annulus with long wave approximation, Int. J. Non Linear. Mech. 42 (2007) 754–759. [6] K. Vajravelu, S. Sreenadh, V. Ramesh Babu, Peristaltic pumping of a Herschel– Bulkley fluid in a channel, Appl. Math. Comput. 169 (2005) 726–735. [7] N.T. M Eldabe, M.F. El-Sayed, A.Y. Ghaly, H.M. Sayed, Mixed convection heat and mass transfer in a non-Newtonian fluid at a peristaltic surface with temperature-dependent viscosity, Arch. Mech. 28 (2008) 599–624.

4730

S. Nadeem et al. / International Journal of Heat and Mass Transfer 52 (2009) 4722–4730

[8] A. Ebaid, A new numerical solution for the MHD peristaltic flow of a biofluid with variable viscosity in circular cylindrical tube via adomian decomposition method, Phys. Lett. A 372 (2008) 5321–5328. [9] T. Hayat, N. Ali, Z. Abbas, Peristaltic flow of a micropolar fluid in a channel with different wave forms, Phys. Lett. A. 370 (2007) 331–344. [10] T. Hayat, N. Ahmed, N. Ali, Effects of a an endoscope and the magnetic field on the peristalsis involving Jeffrey fluid, Commun. Nonlinear Sci. Numer. Simul. 13 (2008) 1581–1591. [11] T. Hayat, N. Ali, Peristaltic motion of a Jeffrey fluid under the effect of magnetic field in a tube, Commun. Nonlinear Sci. Numer. Simul. 13 (2008) 1343–1352. [12] T. Hayat, N. Ali, S. Asghar, A.M. Siddiqui, Exact peristaltic flow in tubes with an endoscope, Appl. Math. Comput. 182 (2006) 359–368. [13] T. Hayat, N. Ali, Peristaltically induced motion of a MHD third grade fluid in a deformable tube, Physica A 370 (2006) 225–239. [14] T. Hayat, N. Ali, Effects of an endoscope on a peristaltic flow of a micropolar fluid, Math. Comp. Mod. 48 (2008) 721–733. [15] T. Hayat, N. Ali, Q. Hussain, S. Asghar, Slip effects on the peristaltic transport of MHD fluid with variable viscosity, Phys. Lett. A 372 (2008) 1477– 1489.

[16] M. Kothandapani, S. Srinivas, Peristaltic transport of a Jeffrey fluid under the effect of magnetic field in an asymmetric channel, Nonlinear Mech. 43 (2008) 915–924. [17] G. Radhakrishnamacharya, C. Srinivasulu, Influence of wall properties on peristaltic transport with heat transfer, C. R. Mecanique 335 (2007) 369– 373. [18] Kh.S. Mekheimer, Y. Abd Elmaboud, The influence of heat transfer and magnetic field on peristaltic transport of a Newtonian fluid in a vertical annulus: an application of an endoscope, Phys. Lett. A 372 (2008) 1657–1665. [19] S. Srinivas, M. Kothandapani, Peristaltic transport in an asymmetric channel with heat transfer, a note, Int. Commun. Heat Mass Transfer 35 (2008) 514– 522. [20] M. Kothandapani, S. Srinivas, On the influence of wall properties in the MHD peristaltic transport with heat transfer and porous medium, Phys. Lett. A 372 (2008) 4586–4591. [21] T. Hayat, E. Momoniat, F.M. Mahmood, Peristaltic MHD flow of a third grade fluid with an endoscope and variable viscosity, J. Nonlinear Math. Phys. 15 (2008) 91–104.