Influence of heat and mass transfer on the peristaltic flow of a Johnson Segalman fluid in a vertical asymmetric channel with induced MHD

Influence of heat and mass transfer on the peristaltic flow of a Johnson Segalman fluid in a vertical asymmetric channel with induced MHD

Journal of the Taiwan Institute of Chemical Engineers 42 (2011) 58–66 Contents lists available at ScienceDirect Journal of the Taiwan Institute of C...

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Journal of the Taiwan Institute of Chemical Engineers 42 (2011) 58–66

Contents lists available at ScienceDirect

Journal of the Taiwan Institute of Chemical Engineers journal homepage: www.elsevier.com/locate/jtice

Influence of heat and mass transfer on the peristaltic flow of a Johnson Segalman fluid in a vertical asymmetric channel with induced MHD S. Nadeem *, Noreen Sher Akbar Department of Mathematics, Quaid-i-Azam University 45320, Near Third Avenue, Islamabad 44000, Pakistan

A R T I C L E I N F O

A B S T R A C T

Article history: Received 10 December 2009 Received in revised form 2 March 2010 Accepted 7 March 2010

This article deals with the influence of heat and mass transfer on the peristaltic flow of a Johnson Segalman fluid with induced magnetic field . The two dimensional equations of Johnson Segalman fluid are simplified by making the assumptions of long wave length and low Reynolds number. The arising equations are solved by using three types of techniques namely the perturbation, homotopy analysis method and numerical technique.Graphical results are sketched for various embedded parameters and interpreted. ß 2010 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

Keywords: Induced magnetic field Vertical channel Johnson Segalman fluid Heat transfer Mass transfer

1. Introduction The study of heat and mass transfer on the peristaltic flow problems is important in many engineering as well as physiological applications. Srinivas and Kothandapani (2009) have examined the influence of heat and mass transfer on MHD peristaltic flow through a porous space with compliant walls. The mixed convection heat and mass transfer in a non-Newtonian fluid at peristaltic surface with temperature dependent viscosity have been studied by Eldabe et al. (2007) and Ogulu (2006) have discussed the heat and mass transport of blood in a single lymphatic blood vessels with uniform magnetic field. Recently, peristaltic flows in asymmetric channel have encountered much success. This is because the physiologists observed that the intrauterine fluid flow caused by myometrial contractions may occur in both symmetric and asymmetric directions (Ealshahed and Haroun, 2005; Elshehawey et al., 2006; Haroun, 2007a,b). Some recent studies on peristaltic flows in symmetric and asymmetric channel are cited in the references Mekheimer (2008), Nadeem and Akram (2009a,b) and Nadeem and Akbar (2009a,b, 2010). More recently, Srinivas and Gayathri (2009) have discussed the peristaltic transport of Newtonian fluid in a vertical asymmetric channel with heat and mass transfer through porous medium. According to them the peristaltic flow of vertical

* Corresponding author. Tel.: +92 5190642182; fax: +92 512275341. E-mail address: [email protected] (S. Nadeem).

asymmetric channel in the presence of heat transfer is of great interest in medical research in understanding heat transfer process in human body. Motivated from the previous studies, the aim of the present paper is to discuss the peristaltic flow of Johnson Segalman fluid in a vertical asymmetric channel with the combined effects of heat and mass transfer. The properties of induced magnetic field is also carried out. The two dimensional equations of motion, energy, mass and induced magnetic field are simplified under the assumptions of longwave length and low Reynolds number. The reduced coupled nonlinear differential equations are solved analytically and numerically. The analytical solution have been calculated by using perturbation method and homotopy analysis method while the numerical solutions have been obtained by shooting method. The comparison of all the three methods have also been presented which show that for small values of parameters the three solutions are identical however, for large values the perturbation solution do not match with the HAM and numerical solutions. Lastly, the results for pressure rise, frictional force per wave length, the axial induced magnetic field, stream function, heat and mass transfer are discussed for various values of physical parameters and shown pictorially. 2. Mathematical formulation Let us consider the peristaltic flow of an incompressible, electrically conducting Johnson Segalman fluid in a two dimensional vertical asymmetric channel. Asymmetry in the flow is

1876-1070/$ – see front matter ß 2010 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.jtice.2010.03.006

S. Nadeem, N.S. Akbar / Journal of the Taiwan Institute of Chemical Engineers 42 (2011) 58–66

Nomenclature Br cp D Ec Gr hx k KT M Pr Q Sc Sr T¯ Tm u w We

$ ^ E ¼ me

local concentration Grashof specific heat coefficients of mass diffusivity Eckert number Grashof number axial induced magnetic field thermal conductivity thermal-diffusion ratio Hartmann number Prandtl number flow rate Schmidt number Soret number temperature temperature of the medium velocity component in r-direction velocity component in z-direction Weissenberg number

The Cauchy stress s for a Johnson Segalman fluid (Nadeem and Akbar, 2010) is related to the fluid motion such as

s ¼ PI þ T;

¯ ¯ þ S; T¯ ¼ 2mD

(8)

  dS¯ ¯ W ¯  aDÞ ¯ þ ðW ¯  aDÞ ¯ T S¯ ¼ 2hD; ¯ Sþm þ Sð dt

(9)

the energy equation in absence of dissipation term is defined as

rc p



(1)

In above equations a1 and b1 are the waves amplitudes, l is the wave length, d1 + d2 is the channel width, c is the wave speed, t is the time and X is the direction of wave propagation. The phase difference f varies in the range 0  f  p, when f = 0 then symmetric channel with waves out of phase can be described and for f = p, the waves are in phase. Moreover, a1, b1, d1, d2 and f meet the following relation 2

a21 þ b21 þ 2a1 b1 cos f  ðd1 þ d2 Þ : The equations which governs the MHD flow of a Johnson Segalman fluid (Nadeem and Akbar, 2010) is given as (i) Maxwell’s equations

$ ^ H ¼ J; J ¼ s 1 fE þ me ðV ^ HÞg;



!

@ ¯ @ @ ¯ @2 T¯ @2 T¯ þU þ V¯ þ 2 þ Q 0; T¼k 2 @t¯ @X¯ @Y¯ @X¯ @Y¯

(10)

In which PI is the spherical part of the stress due to the constraint of incompressibility, V¯ velocity, s Cauchy stress tensor, f¯ body force, m and h are the viscosities, m is called the relaxation time and ¯ and W ¯ are the symmetric and a is called slip parameter D skewsymmetric parts of the velocity gradient respectively and can be defined as follows ¯ ¼ D

1 T ðL¯ þ L¯ Þ; 2

  2p ðX  ctÞ ;

$  E ¼ 0;

(7)

where

l

$  H ¼ 0;

(5)

(iii) The equations of motion       @V 1  þ 2 þ ðV  $ ÞV ¼ divs  $ r me H me Hþ  $ Hþ 2 @t þ ragðT¯  T¯0 Þ þ rg aðC¯  C¯ 0 Þ; (6)

because of propagation of peristaltic waves of different amplitudes and phases on the channel walls. The heat transfer in the channel is taken into account by giving temperature to the left and right walls as T0 and T1 respectively. An external transverse uniform constant magnetic field H0, induced magnetic field H(hX(X, Y, t), H0 + hY(X, Y, t), 0) and the total magnetic field H+(hX(X, Y, t), H0 + hY(X, Y, t), 0) are taken into account. Finally the channel walls are considered to be non-conductive. The shapes of the channel walls are represented by the following expressions

l

(4)

$  V ¼ 0;

Greek symbols heat source parameter Kinmatic viscosity Kinmatic viscosity Density of the fluid phase difference

  2p ðX  ctÞ þ f : Y ¼ H2 ¼ d2  b1 cos

@H ; @t

(ii) The continuity equation

b m h r f

Y ¼ H1 ¼ d1 þ a1 cos

59

¯ ¼ W

(11)

1 T ðL¯  L¯ Þ: 2

(12)

¯ E is an induced electric field, J is the current density, me L¯ ¼ gradV, is the magnetic permeability, s1 is the electric conductivity, q =( K0 divT, K0 being the thermal conductivity) is the heat flux vector, r is the internal heat generation (radial heating) taken here to be zero, and e( = C0 )T, C0 being specific heat is the specific internal energy, C¯ is the concentration of fluid, r is the density, k denotes the thermal conductivity, Q0 constant heat absorption parameter, cp is the specific heat at constant pressure. Combining Eqs. (2)–(4), we obtain the induction equation (Nadeem and Akbar, 2010) and as follows

@Hþ 1 ¼ $ ^ ðV ^ Hþ Þ þ r2 Hþ ; @t j

(13)

(2)

1 where j ¼ sm is the magnetic diffusively. e Introducing a wave frame (x, y) moving with velocity c away from the fixed frame (X, Y) by the transformation

(3)

x ¼ X  ct;

y ¼ Y;

u ¼ U  c;

v ¼ V:

60

S. Nadeem, N.S. Akbar / Journal of the Taiwan Institute of Chemical Engineers 42 (2011) 58–66

Defining

v a ct H x¯ ¼ ; v¯ ¼ ; d ¼ 2 ; p¯ ¼ ; t¯ ¼ ; h ¼ ; a2 l cd l mc l l þ 2 ca2 C ¯ ¼ F ; p ¼ p¯ þ 1 Red me ðH Þ ; Rm ¼ sm a2 c; Re ¼ ; C¯ ¼ ; F e 2 2 n ca2 H 0 a2 rc rffiffiffiffiffiffi 0 2 T¯  T¯0 H0 me c rvC Sa2 ; P r ¼ 0 ; S¯ ¼ : ; Ec ¼ 0 ; u¼ S1 ¼ c r mc K C ðT 1  T 0 Þ T¯1  T¯0 Q 0 a22 g aa32 ðT¯1  T¯0 Þ mc b¼ ; We ¼ ; ; Gr ¼ a2 n2 kðT¯1  T¯0 Þ aga32 ðC¯ 1  C¯ 0 Þ rDK T ðT¯0  T¯1 Þ m ðC¯  C¯ 1 Þ Br ¼ ; Sr ¼ ; s¼ ; Sc ¼ ; Dr n2 mT m ðC¯ 0  C¯1 Þ ðC¯ 0  C¯ 1 Þ @C @C @F @F ;v ¼  ;h ¼ ; h ¼ d : u¼ @y @x x @y y @x x

y y¯ ¼ ; a2

u u¯ ¼ ; a2

(14) The governing equations of a Johnson Segalman fluid along with the effects of induced MHD in the presence of heat and mass transfer in dimensionless form or in terms of the stream function C(x, y) and magnetic-force function F(x, y) (dropping the bars and using u ¼ @C =@y; v ¼ @C =@x; hx ¼ @F=@y; hy ¼ d@F=@x) are 



(15)



@p m þ h @ 2 @ ðS Þ þ d Red ðC x C xy  C y C xx Þ ¼  þd @y m @x yx @y 2  ðSyy Þ þ Red S21 Fyy 3

3

 ReS21 d ðFy Fxx  Fx Fxy Þ;

C y  dðC y Fx  C x Fy Þ þ

RePrdðC y ux þ C x uy Þ ¼ d

2

(16)

1 2 ðFyy þ d Fxx Þ ¼ E; Rm

(17)

@2 u @2 u þ þ b; @x2 @y2

(18)

1 2 2 ðd s xx þ s yy Þ þ Sr ðd uxx þ u yy Þ; Sc

h @ c m @y2

Sxy ¼ 1þ

We2 ð1



a2 Þ

at y ¼ h1 and y ¼ h2 ;

(20c)

u ¼ 0;

at y ¼ h1 ;

u ¼ 1; at

y ¼ h2 ;

(20d)

s ¼ 0;

at y ¼ h1 ;

s ¼ 1; at

y ¼ h2 ;

(20e)

F ¼ 0;

at y ¼ h1 and y ¼ h2 :

Under the assumption of long wave length d  1 and low Reynolds number neglecting the terms of order d and higher, Eqs. (15)–(19) take the form 



m þ h @ p @Sxy @3 C ¼ þ þ ReS21 Fyy þ Gr u þ Br s ; @x @y m @x3

(21)

@p ¼ 0; @y

(22)





Fyy ¼ Rm E 

F 2



@C ; @y

(23)

2

(24) 2

1 @ s @ u þ Sr 2 ¼ 0: Sc @y2 @y

0

@2 B @ @y2

(25)





@2 c @2 c h 2 2 m þ 1 @y2 þ We ð1  a Þ @y2



2

2 1 þ We2 ð1  a2 Þ @@yc2

þ Br

3 1

@u C A þ ReS21 Fyyy þ Gr @y

@s ¼ 0; @y 

Fyy ¼ Rm E 

(26a) 

@C : @y

(26b)

With the help of Eq. (26a), Eq. (26) take the form 0

!3 1

@ @@2 c We2 ð1  a2 Þh @2 c A @C m  M2 þ þ Gu @y @y2 mþh @y m þ h r @y2 m þ B s ¼ M 2 E þ C 1 ; mþh r

(27)

where C1 is a constant and M12 ¼ Rm ReS21 , M 2 ¼ M12 mm þh.

3. Methods of solution ; The temperature and concentration Eqs. (24) and (25) with the boundary conditions (20d) and (20e) give

!2 :

@2 c @y2

u¼

The corresponding boundary conditions are

C¼ ;

(20f)

;

h @2 c m @y2

h @2 c m @y2

@C ¼ 1; @y

(19)

!2

Syy ¼ W e ð1  aÞ

(20b)

Elimination of pressure from Eqs. (21) and (22) yields

where b is heat source parameter, s is concentration in dimensionless sxx is second derivative of form concentration with respect to x. Sr is the Soret number, Sc Schmidt number, Br is the local concentration Grashof number and Gr is the local temperature Grashof number and the stresses are !2 2 Sxx ¼ W e ð1 þ aÞ

at y ¼ h2 ¼ d  bsin ðx þ fÞ;

@2 u þ b ¼ 0; @y2



@p m þ h @ @ ðS Þ þ d ðSxx Þ þ @x m @x @y xy  ReS21 Fyy  ReS21 dðFy Fxy  Fx Fyy Þ þ Gr u þ Br s ;

RedðC y C xy  C x C yy Þ ¼ 

dð C y s x þ C x s y Þ ¼

F 2

C ¼ ; a22 p

at y ¼ h1 ¼ 1 þ asin x;

(20a)



1 2ðh1  h2 Þ

 ðby2  ð2 þ h21 b  h22 bÞy þ 2h2 þ h21 h2 b  h1 h22 bÞ:

(28)

h2  y bSr Sc þ ðy  h1 Þðy  h2 Þ: ðh2  h1 Þ 2

(29)

S. Nadeem, N.S. Akbar / Journal of the Taiwan Institute of Chemical Engineers 42 (2011) 58–66

3.1. Perturbation solution To get the solution of Eq. (27) we have used regular perturbation taking We2 as a perturbation parameter. The solutions of stream function, magnetic force function and axial pressure gradient for small Weissenberg number can be written as 2 2



2

dp ¼ ðMð12a5 ðh1  h2 Þ þ ðh1  h2 Þð6a7 þ 3a6 ðh1 þ h2 Þ dx þ 2a5 ðh21 þ h1 h2 þ h22 ÞÞM 2   1 þ 6FM 4 Þcosh ðh1  h2 ÞM þ 6ð4a5 þ ð2a7 þ a6 ðh1 2 þ h2 Þ þ a5 ðh21 þ h22 ÞÞM 2   1  2M 4 Þsinh ðh1  h2 ÞM Þ=ð6M 2 ððh1 2     1 1 ðh1  h2 ÞM  2sinh ðh1  h2 ÞM ÞÞ  h2 ÞMcosh 2 2

3

12a5 y þ 3a6 ð2 þ M y Þ þ 2M ð3a9 þ 3ða7 þ a8 Þy þ a5 y Þ 6M4 þ a10 ðcosh ðMyÞ þ sin hðMyÞÞ þ a11 ðcosh ðMyÞ  sin hðMyÞÞ 1 þ We2 ð ða12 y þ a13 Þ  24ða6 þ 2a5 yÞÞ 24M10

þ We2 ða12 Þ

 ðM 2 ða26 þ 6a10 a11 M 8 Þ4a5 ða6 M 2 y þ a5 ð6 þ M2 y2 ÞÞÞa þ M2 ð6ð3a210 a11 M 10 ð1 þ 2MyÞa þ a10 ð6a5 a6 Mð1 þ 2Myð1 þ MyÞÞ  3M

2

ða26 ð1

61

(32)

The expression for axial induced magnetic field can be obtained with the help of hx ¼ @F @y , which are as follows

 2MyÞ

þ a211 M 8 ð1 þ 2MyÞÞ þ 2a25 ð3 þ 2Myð3 þ Myð3 þ 2MyÞÞÞÞa þ a11 ð3a26 M2 ð1 þ 2MyÞ þ 6a5 a6 Mð1 þ 2Myð1 þ MyÞÞ

hx ¼ Rm Ey þ Rm ðða11  a10 Þsin hðMyÞ  ða11 þ a10 Þcos hðMyÞÞ þ 12a5 y þ 3a6 ð2 þ M 2 y2 Þ þ 2M 2 ð3a9 þ 3ða7 þ a8 Þy

þ 2a25 ð3þ2Myð3þMyð3þ 2MyÞÞÞÞa  4M 6 ða14 þa15 ÞÞcosh ðMyÞ

1

þ 8M 5 ða210 ð3a6 M þ a5 ð8 þ 6MyÞÞ þ a211 ð3a6 M

þ a5 y3 Þ=6M 4 We2 ðða10 þ a11 Þða10  a11 Þ

þ a5 ð8 þ 6MyÞÞÞacosh ð2MyÞ  3ða310 þ a311 ÞaM10 cosh ð3MyÞ

 ðRm ð36a25 M 4 acos hðMyÞ  18a26 M 6 acos hðMyÞ

 6ð3a210 a11 M 10 ð1þ 2MyÞa þ a10 ð6a5 a6 Mð1 þ2Myð1þMyÞÞ þ 3M

2

ða26 ð1

þ 2MyÞ þ

a211 M8 ð1

24M 10

 18a10 a11 M 14 acos hðMyÞ  72a5 a6 M6 yacos hðMyÞ

þ 2MyÞÞ

þ 36a5 a6 M 5 asin hðMyÞ þ 72a25 M 5 yasin hðMyÞ

þ 2a25 ð3 þ 2Myð3 þ 2MyÞÞÞÞa þ a11 ð3a26 M2 ð1 þ 2MyÞ

þ 36a26 M 7 yasin hðMyÞ þ 36a10 a11 M 15 yasin hðMyÞ

þ 6a5 a6 Mð1þ 2Myð1 þ MyÞÞ2a25 ð3 þ2Myð3 þ Myð3 þ2MyÞÞÞÞa

1

þ 4M 6 ða15  a14 ÞÞsinh ðMyÞ þ 8M 5 ða210 ð3a6 M þ a5 ð8 þ 6MyÞÞ

þ 48a25 M 7 y3 asin hðMyÞÞÞÞ þ ða10  a11 Þ

 a211 ð3a6 M þ a5 ð8 þ 6MyÞÞÞasinh ð2MyÞ

 ðRm ð24ðM 8 ða13 þ a12 yÞ þ 24a25 a6 a þ ð48a35 y þ M 2 ða6

þ

3ða311



a310 ÞM 10

asinh ð3MyÞ:

(30)

þ 2a5 yÞð6a10 a11 M 8 þ ða6 þ 2a5 yÞ2 ÞÞaÞ  24ða14 þ a15 ÞM 10 cos hðMyÞ þ 36ða10  a11 Þa5 a6 M 5 acos hðMyÞ

R Ey2 þ Rm ðða11  a10 Þ12M 3 cos hðMyÞ F¼ m 2  ða11 þ a10 Þ12M 3 sin hðMyÞÞ þ 12a6 y þ yð12a5 y 2

2

þ M ð12a9 þ yð6ða7 þ a8 Þ þ 2a6 y þ a5 y ÞÞÞ=12M 1 þ We2 ð ðRm ð12M 8 yð2a13 þ a12 yÞÞÞÞ 24M10  24yða6 þ a5 yÞðM 2 ða26 þ 6a10 a11 M8 Þ

24M 10

þ 72ða10  a11 Þa25 M 5 yacos hðMyÞ þ 36ða10  a11 Þa26 M 7 yacos hðMyÞ þ 36a10 ða10 4

 a11 Þa11 M 15 yacos hðMyÞ  72a5 M 6 ða10 ða5  a6 MÞ þ a11 ða5 þ a6 MÞÞy2 acos hðMyÞ þ 48ða10  a11 Þa25 M 7 y3 acos hðMyÞ  8M 9 ða210 ð3a6 M þ a5 ð8

þ 2a5 ða6 M 2 y þ a5 ð12 þ M 2 y2 ÞÞÞa

þ 6MyÞÞ þ a211 ð3a6 M þ a5 ð8 þ 6MyÞÞacos hðMyÞ þ 3ða310

þ M 3 ð6ð4a14 M 5 þ 90ða10  a11 Þa25 a

þ a311 ÞM 14 acos hðMyÞ  24a14 sin hðMyÞ

 Mð4a15 M5 þ 42ða10 þ a11 Þa5 a6 a þ ð9ða10  a11 ÞMða26

þ 24a15 M 10 sin hðMyÞ  36ða10  a11 Þa25 M 4 asin hðMyÞ

8

þ a10 a11 M Þ þ 6ð14ða10 þ

a11 Þa25

 6ða10  a11 Þa5 a6 M þ ða10 þ

 18ða10  a11 ÞM 6 ða26 þ a10 a11 M 8 Þasin hðMyÞ  72ða10

a11 Þa26 M 2

 a11 Þa5 a6 M 6 ysin hðMyÞ þ 72a5 M 6 ða10 a5 þ a11 a5

10

þ a10 a11 ða10 þ a11 ÞM Þy þ 12a5 Mð3ða10  a11 Þa5 2

þ ða10 þ a11 Þa6 MÞy þ 8ða10 þ

þ a10 a6 M þ a11 a6 MÞy2 asin hðMyÞ  8M 4 ða210 ð3a6 M

a11 Þa25 y3 M2 Þ

aÞÞcos hðMyÞ

þ a5 ð8 þ 6MyÞÞ  a211 ð3a6 M þ a5 ð8

þ 4M 5 ða210 ð11a5 þ 3a6 M6a5 My

þ 6MyÞÞÞasin hð2MyÞ þ 3ða310  a311 ÞM 14 asin hð3MyÞÞÞÞ

 a211 ð11a5 þ 3a6 M þ 6a5 MyÞÞacos hð2MyÞ

þ a16 :

þ ða311  a310 ÞM 10 acos hð3MyÞ þ 6ð4ða14 þ a15 ÞM 6 þ 90ða10 þ a11 Þa25 a þ Mð42ða10  a11 Þa5 a6 þ 9ða10 þ

a11 Þa26 M

þ 9a10 a11 ða10 þ a11 ÞM

where all ai can be evaluated using Mathematica. In the fixed frame the flux at any axial station is written as

9

 6ð14ða10  a11 Þa25  6ða10 þ a11 Þa5 a6 M Q¯ ¼

þ ða10  a11 Þa26 M 2 þ a10 ða10  a11 Þa11 M 10 Þy

Z

h1

ðu þ 1Þdy ¼

Z

h2

2

þ 12a5 Mð3ða10 þ a11 Þa5  ða10  a11 Þa6 MÞy

h1

udy þ h2

Z

h1

dy ¼ F þ h1  h2 :

(34)

h2

  The average volume flow rate over one period T ¼ lc of the peristaltic wave is given by the following expression

 8ða10  a11 Þa25 M 2 y3 ÞaÞsin hðMyÞ þ 4M 5 ða210 ð11a5 þ 3a6 M þ 6a5 MyÞ þ a211 ð11a5 þ 3a6 M þ 6a5 MyÞasinh ð2MyÞ  ða311 þ a310 ÞM 10 asinh ð3MyÞÞÞÞ þ a16 y þ a17 :

(33)

(31)



1 T

Z

T

¯ ¼ Qdt 0

1 T

Z

T

ðq þ h1  h2 Þ dt ¼ F þ 1 þ d: 0

(35)

S. Nadeem, N.S. Akbar / Journal of the Taiwan Institute of Chemical Engineers 42 (2011) 58–66

62

Table 1 Comparison of velocity profile for different solutions for Gr = 0.5, b = 0.3, M = 1, We = 0.2, E = 0.5, f ¼ p2 , b = 0.1, d = 0.9, a = 0.1, x = 0.5, Q = 0.2, Br = 1, Sc = 0.3, Sr = 0.5, m = 0.3, h = 0.2. y

Numerical solution

Perturbation solution

Error

HAM solution

Error

0.9 0.7 0.5 0.3 0.1 0.0 0.1 0.3 0.7 0.9

1.00000 0.99463 0.97014 0.94150 0.91776 0.90432 0.90435 0.91970 0.95138 1.00000

1.00000 1.00092 0.97634 0.94406 0.91590 0.89941 0.89895 0.91628 0.95089 1.00000

0.00000 0.00628 0.00635 0.00271 0.00203 0.00545 0.00600 0.00407 0.00051 0.00000

1.00000 1.00342 0.97893 0.94523 0.91525 0.89747 0.89673 0.91479 0.95057 1.00000

0.00000 0.00876 0.00897 0.00394 0.00274 0.00763 0.00849 0.00536 0.00085 0.00000

The non-dimensional expression for the pressure rise is defined as follows Z 1  dp dx: (36) Dp ¼ dx 0

Cm ¼

m ¯ 1 @ C ðy; qÞ m! @qm

(43)

:

q¼0

The problems at m th order are hC RC y ðyÞ; LC ½C m ðyÞ  xm C m1 ðyÞ ¼ 

(44)

3.2. Solution by homotopy analysis method (HAM) For HAM solution, the initial guess C0 and auxiliary linear operators LCy are chosen in the form

m1 X

0000

i¼0 m 1 X

1 mM2 mX G mby C 00m1  r ð ð m þ h Þ m þ hÞ i¼0 i¼0 Gr a1 Br mSr Sc by Br ma3 ð1  xm Þ þ þ ð1  xm Þ: þ (45) ðm þ hÞ ðm þ hÞ ðm þ hÞ

C 00m1 C 000m1i C 000mi 

þ 6We2 a

C0 ¼

ðF þ h1  h2 Þ

1 ð2y3  3h1 y2  3h2 y2 þ 6h1 h2 yÞ  y þ 3 3 ðh2  h1 Þ ðh2  h1 Þ      F F þ h1 ðh32  3h1 h22 Þ  h2  ðh31  3h21 h2 Þ ;  (37) 2 2

xm ¼ 0000

LC y ðC Þ ¼ C ;

(38)

LC y ðC 0 Þ ¼ 0;

(39)

The zeroth order deformation problems are ¯ ðy; qÞ  C 0 ðyÞ ¼ q ¯ ðy; qÞ; ð1  qÞLC y ½C hC NC y ½C F 2

C¯ ðy; qÞ ¼ ;

at y ¼ h1 ¼ 1 þ asin x;

F 2



0; 1;

m  1; m > 1:

at y ¼ h2 ¼ d  bsin ðx þ fÞ;

(40c)

@C¯ ðy; qÞ ¼ 1; @y

at y ¼ h1 and y ¼ h2 ;

(40d)

n¼1

"

(40a)

C¯ ðy; qÞ ¼  ;

(46)

By mathematica the solution can be computed as " !# 2Mþ1 2M 2mþ1n X X X C m ðyÞ ¼ lim akm;n y2nþ3 M!1

(40b)

00 00 C 0000 m1 C m1i C mi

RC y ¼ C m1 þ 3We2 a

þ

m¼n1

M X

a0m;0 lim M!1 m¼0

#

k¼0

;

where a0m;0 and akm;n are the constants.

[(Fig._1)TD$IG]

where 9C denotes the non-zero auxiliary parameter, q 2 [0, 1] is an embedding parameter and the nonlinear operator NCy is

¯ ðy; qÞ ¼ N C y ½C

@4 C @2 C þ 3We2 a @y4 @y2

!2

@4 C @2 C @3 C þ 6We2 a 2 @y4 @y @y3

!2

2

M2 m @ C m ðby þ a1 Þ þ Gr ðm þ hÞ @y2 ðm þ hÞ Br m ðSr Sc by þ a3 Þ: þ ðm þ hÞ 

(41)

¯ ðy; qÞ varies from initial Obviously when q varies from 0 to 1, then C ¯ ðy; qÞ in Taylor,s series guess C0 to the solution C(y). Expanding C with respect to an embedding parameter q, we have

C¯ ðy; qÞ ¼ C 0 ðyÞ þ

1 X

C m ðyÞqm ;

n¼1

(42) Fig. 1. Geometry of the problem.

(47)

[(Fig._2)TD$IG]

[(Fig._4)TD$IG]

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63

Fig. 2. h-Curve for velocity profile for Gr = 0.5, b = 0.1, M1 = 1, We = 0.1, E = 0.5, f ¼ p2 , b = 0.1, d = 0.9, a = 0.1, x = 0.5, Q = 0.2, Br = 1, Sc = 0.6, Sr = 0.1, m = 0.3, h = 0.2. Fig. 4. Pressure rise versus flow rate for Gr = 1, b = 0.1, M = 1, E = 0.3, f ¼ p2 , b = 0.1, d = 0.9, a = 0.1, Br = 1, Sc = 0.5, Sr = 0.3, m = 0.5, h = 0.4.

3.3. Numerical solution

[(Fig._5)TD$IG]

Here shooting method is used to solve Eqs. (26) and (20). Moreover, numerical solution is also compared with the perturbation and homotopy solutions. The values of different solutions are presented in Table 1.

4. Graphical results and discussion In this section, we have presented the graphical results of the solutions. Table 1. shows the numerical values of velocity profile for different solutions. Fig. 1 present the geometry of problem. Figs. 2 and 3 show the h-curve and comparison of velocity profiles. It is observed from these figures that both perturbation and homotopy solutions are almost identical for small values of parameters however, for large values the results are not identical which are not shown here. The expression for pressure rise is calculated numerically using a mathematics soft-ware. The graphical results of pressure rise, temperature, concentration, magnetic force function and stream functions are displayed in Figs. 4–16. Figs. 4 and 5 are prepared for pressure rise Dp against volume flow rate Q for different values of We, and M1. It is observed

[(Fig._3)TD$IG]

Fig. 5. Pressure rise versus flow rate for Gr = 1, b = 0.1, We = 0.1, E = 0.3, f = 0.6, b = 0.5, d = 0.9, a = 0.1, Br = 0.2, Sc = 0.5, Sr = 0.3, m = 0.5, h = 0.4.

[(Fig._6)TD$IG]

Fig. 3. Comparison of velocity profile for different solutions for Gr = 0.5, b = 0.3, M = 1, We = 0.2, E = 0.5, f ¼ p2 , b = 0.1, d = 0.9, a = 0.1, x = 0.5, Q = 0.2, Br = 1, Sc = 0.3, Sr = 0.5, m = 0.3, h = 0.2.

Fig. 6. Velocity profile for Gr = 1, b = 0.3, We = 0.1, E = 0.3, f = 0.6, b = 0.5, d = 0.9, a = 0.1, Br = 1, Sc = 0.5, Sr = 0.3, m = 0.3, h = 0.2, x = 0.5.

[(Fig._7)TD$IG]

64

[(Fig._10)TD$IG]

S. Nadeem, N.S. Akbar / Journal of the Taiwan Institute of Chemical Engineers 42 (2011) 58–66

Fig. 10. Axial induced magnetic field for m = 0.2, Gr = 1, b = 0.1, a = 0.1, h = 0.1, f = 0.2, M1 = 0.5, E = 0.5, Rm = 0.5, Sr = 0.3, Sc = 0.2, Br = 0.5.

Fig. 7. Velocity profile for Gr = 1, b = 0.3, M1 = 1, E = 0.3, f = 0.6, b = 0.5, d = 0.9, a = 0.1, Br = 1, Sc = 0.5, Sr = 0.3, m = 0.3, h = 0.2, x = 0.5.

[(Fig._1)TD$IG] [(Fig._8)TD$IG]

Fig. 11. Axial induced magnetic field for m = 0.2, Gr = 1, b = 0.1, a = 0.1, h = 0.1, f = 0.2, M1 = 0.5, E = 0.5, We = 0.1, Sr = 0.3, Sc = 0.2, Br = 0.5, b = 0.3, d = 0.9.

Fig. 8. Temperature profile for f = 0.1, b = 0.9, d = 0.9, a = 0.8, x = 0.1.

[(Fig._12)TD$IG] [(Fig._9)TD$IG]

Fig. 9. Concentration field for f = 0.1, b = 0.9, d = 0.9, a = 0.8, x = 0.1, Sr = 3, Sc = 0.2.

Fig. 12. Pressure gradient versus x for m = 0.5, Gr = 1, b = 0.1, a = 0.2, h = 0.4, M1 = 0.5, E = 0.5, We = 0.3, Sr = 0.3, Sc = 0.5, Br = 0.5, b = 0.5, d = 0.9, E = 0.3, Q =  0.3.

[(Fig._13)TD$IG]

[(Fig._14)TD$IG]

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Fig. 13. Pressure gradient versus x for m = 0.5, Gr = 1, b = 0.1, a = 0.2, h = 0.4, f = 0.3, M1 = 0.5, E = 0.5, Sr = 0.3, Sc = 0.5, Br = 0.5, b = 0.5, d = 0.9, E = 0.3, Q =  0.2.

Fig. 14. Pressure gradient versus x for m = 0.5, Gr = 1, b = 0.1, a = 0.2, h = 0.4, f = 0.3, We = 0.3, E = 0.5, Sr = 0.3, Sc = 0.5, Br = 0.5, b = 0.5, d = 0.9, E = 0.3, Q =  0.2.

from the figures that the relation between pressure rise and volume flow rate are inversely proportional to each other. It means that pressure rise give larger values for small volume flow rate and it gives smaller values for large Q. Moreover, the peristaltic pumping occurs in the region 1  Q < 1 for We, and 1  Q  2.3 for M1 other wise augmented pumping occurs. Moreover with the increase in We pressure rise decreases, while pressure rise increases with an increase in M1. Velocity profile could be analyzed through Figs. 6 and 7. It is observed that with the increase in M1 velocity profile increase for y 2[ 0.8 to 0], while decreases with increase in M1 and with an increase in We velocity profile decrease for y 2[ 0.8 to 0], otherwise increases

with increase in We. Effects of temperature and concentration field have been shown through Figs. 8 and 9. It is seen that with the increase in b temperature profile increases while concentration field decreases with an increase in b. The expression for axial induced magnetic field hx against space variable y for different values of magnetic Reynolds number Rm and Weissenberg number We are shown in Figs. 10 and 11. It is observed that with the increases in Rm and We, hx increases in the upper half of the channel while in the lower half the behavior is opposite. Figs. 12– 14 are prepared to see the behavior of pressure gradient for different parameters of interest. It is observed from the figures that with an increase in f pressure gradient increases in

[(Fig._15)TD$IG]

Fig. 15. Streamlines for different values of M1: (a) for M1 = 0.5, (b) for M1 = 1 (c) for M1 = 1.5 other parameters are m = 0.5, Gr = 1, b = 0.1, a = 0.2, h = 0.4, f = 0.3, We = 0.3, E = 0.5, Sr = 0.3, Sc = 0.5, Br = 0.5, b = 0.5, d = 0.9, E = 0.3, Q =  0.2.

[(Fig._16)TD$IG]

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Fig. 16. Streamlines for different values of We. (a) for We = 0.3, (b) for We = 0.5, (c) for We = 0.7, other parameters are m = 0.5, Gr = 1, b = 0.1, a = 0.2, h = 0.4, f = 0.3, E = 0.5, Sr = 0.3, Sc = 0.5, Br = 0.5, b = 0.5, d = 0.9, E = 0.3, Q =  0.2, M1 = 0.5.

x 2 [0–0.5], otherwise decreases with an increase in f, while pressure gradient increases with an increase in We and pressure gradient decreases with an increase in M1. The trapping phenomena for different values of We, and M1 are shown in Figs. 15 and 16. It is observed from Fig. 15. that with an increase in M1 the size of the trapping bolus decreases. It is depicted from Fig. 16 that with the increase in We the size of the trapped bolus increases. 5. Conclusion This study examines the influence of heat and mass transfer on the peristaltic flow of a Johnson Segalman fluid in a vertical asymmetric channel with induced MHD. Three types of solution have been evaluated. The main points of the performed analysis are as follows.

1. The perturbation, homotopy analysis and numerical solutions are identical up to three digits. 2. The effects of We and M1 on the pressure rise are opposite. 3. The influence of We and M1 on the velocity is qualitatively opposite. 4. Temperature profile increases with an increase in b. 5. Concentration profile decreases with an increase in b. 6. Axial induced magnetic field increases with an increase in We and Rm. 7. Pressure gradient increases with an increase in We, while decreases with an increase in M1. 8. The size of trapped bolus decreases with an increase in M1, while the size of the trapped bolus increases with an increase in We.

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