Heat and mass transfer on magnetohydrodynamic peristaltic flow in a porous medium with partial slip

Alexandria Engineering Journal (2016) xxx, xxx–xxx

H O S T E D BY

Alexandria University

Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com

ORIGINAL ARTICLE

Heat and mass transfer on magnetohydrodynamic peristaltic flow in a porous medium with partial slip M. Gnaneswara Reddy Department of Mathematics, Acharya Nagarjuna University Campus, Ongole, A.P. 523 001, India Received 26 September 2014; revised 12 February 2016; accepted 4 April 2016

KEYWORDS Peristalsis; MHD; Porous medium; Slip flow; Compliant walls

Abstract The present study concerned with the impact of velocity slip on MHD peristaltic flow through a porous medium with heat and mass transfer is investigated. The relevant equations of flow with heat and mass transfer have been developed. Analytic solution is carried out under long-wavelength and small Reynolds number approximations. The expressions for the stream function, temperature and concentration and the heat transfer coefficient are obtained. Numerical results are graphically discussed for various values of physical parameters of interest. The velocity and temperature field increase with an increase in the velocity slip parameter and permeability parameter while it decreases with an increase in the Hartmann number. Ó 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction The transportation of many biological fluids is carried out with the help of naturally inherited mechanism inside the biological systems which is called peristalsis. It is nature’s way of moving the content within hollow muscular structures by successive contraction of their muscular fibers. This principle is responsible for transport of biological fluids such as urine in the ureter, chime in the gastrointestinal tract, semen in the vas deferens, ovum in the fallopian tube, lymph transport in the lymphatic vessels, blood pumps in the heart lung machine etc. In plant physiology, the peristalsis is involved in phloem translocation by driving a sucrose solution along tubules by peristaltic contractions. The corrosive and noxious fluids can also be transported by peristalsis. Such flows in presence of heat transfer also have great value. This process is useful for the analysis E-mail address: [email protected] Peer review under responsibility of Faculty of Engineering, Alexandria University.

of tissues, oxygenation and dialysis. Roller and finger pumps also work under the peristaltic mechanism. The seminal research on the peristaltic motion has been presented by Latham [1] and Jaffrin and Shapiro [2]. Since then the various experimental and theoretical studies have been presented in the viscous and non-Newtonian fluids [3–10]. In view of the importance of oxygenation and dialysis, the peristaltic flows with heat transfer have been also investigated [11–14]. Peristaltic transport of a Carreau fluid in a compliant rectangular duct was presented by Riaz et al. [15]. A mathematical study of non-newtonian micropolar fluid in arterial blood flow through composite stenosis was investigated by Ellahi et al. [16]. The influence of heat and mass transfer on MHD peristaltic flow through a porous space with compliant walls was investigated by Srinivas et al. [17]. Ellahi [18] have reported the effects of MHD and temperature dependent viscosity on the flow of non-Newtonian nanofluid in a pipe. Very recently, the influence of Joule heating on MHD peristaltic flow of a nanofluid with compliant walls was investigated by Gnaneswara Reddy and Venugopal Reddy [19].

http://dx.doi.org/10.1016/j.aej.2016.04.009 1110-0168 Ó 2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: M.G. Reddy, Heat and mass transfer on magnetohydrodynamic peristaltic flow in a porous medium with partial slip, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.04.009

2

M.G. Reddy

In several applications the flow pattern corresponds to a slip flow, and the fluid presents a loss of adhesion at the wetted wall making the fluid slide along the wall. When the molecular mean free path length of the fluid is comparable to the distance between the plates as in nanochannels or microchannels, the fluid exhibits non-continuum effects such as slip-flow as demonstrated experimentally by Derek et al. [20]. Investigations of the effects of slip on the peristaltic motion have been recently reported in [21–23]. The aim of the present paper is to discuss the velocity slip effects on the MHD peristaltic transport of nonNewtonian fluid in a porous space with heat and mass transfer. Such an analysis is of great interest in bio-medical research. The momentum, temperature equations and concentration equations have been linearized under longwavelength and low-Reynolds number assumptions and exact solutions for the flow fluid dynamical variables have been derived. The contribution of several interesting parameters embedded in the flow system is examined by graphical representations. 2. Formulation of the problem The motion of heat and mass transfer peristaltic flow of a Newtonian viscous fluid through a two-dimensional channel of uniform thickness filled with a porous medium is considered. The motion in a channel is induced by imposing moderate amplitude sinusoidal waves on the compliant walls of the channel as shown in Fig. 1 and thus the walls are defined by
 2p y ¼ gðx; tÞ ¼  d þ a sin ðx  ctÞ : ð1Þ k where d is the mean half width of the channel, a is the amplitude, k is the wavelength, t is the time and c is the phase speed of the wave respectively. The magnetic Reynolds number and induced magnetic field are assumed to be small and neglected. Under these assumptions the governing equations of continuity, momentum, heat transfer and mass transfer are as follows: @u @v þ ¼ 0: @x @y

ð2Þ

 
2  @u @u @u @p @ u @2u q þu þv ¼ þl þ qgbðT  T0 Þ þ @t @x @y @x @x2 @y2 l ð3Þ þ qgb ðC  C0 Þ  rB20 u  u: k
2    @v @v @v @p @ v @2v l þ  u: q þu þv ¼ þl @t @x @y @y @x2 @y2 j
   @T @T @T j @2T @2T þ þu þv ¼ f @t @x @y q @x2 @y2 " 2  2  2 !# @u @v @u @v þ2 þ þ þm @y @x @x @y "    2 !  2 # 2 rB20 u2 l @u @v @u @v 2 þ þ : þ þ q @x @y @y @x
2 
 @C @C @C @ C @2C DKT @ 2 T @ 2 T þ : þ þ þu þv ¼D Tm @x2 @y2 @t @x @y @x2 @y2

Schematic diagram of the problem.

ð5Þ

ð6Þ

where u, v are the components of velocity along x -and y directions, p is the pressure, l is the coefficient of viscosity of the fluid, g is the gravitational acceleration, b is the coefficient of thermal expansion, b* is the coefficient of concentration expansion, r is the electrical conductivity of the fluid, k is the permeability parameter, B0 is the applied magnetic field, a is the thermal diffusivity, m is the kinematic viscosity, q is the density of the fluid, f is the specific heat at constant pressure, k1 is the chemical reaction of rate constant, T is the temperature, C is the concentration and D is the coefficient of mass diffusivity, KT is the thermal-diffusion ratio, and Tm is the mean temperature. The governing equation of motion of the flexible wall is expressed as L ðgÞ ¼ p  p0 :

ð7Þ



where L is an operator, which is used to represent the motion of stretching membrane with viscosity damping forces such that L ¼ s

@2 @2 @4 0 @ þ m þ C þ H: þ B @x2 @t2 @x4 @t

ð8Þ

Here s is the elastic tension in the membrane, m is the mass per unit area, C0 is the coefficient of viscous damping forces, B is the flexural rigidity of the plate, H is the spring stiffness and p0 is the pressure on the outside surface of the wall due to the tension in the muscles and assume that p0 = 0. The associated boundary conditions for the velocity slip, temperature and concentration at the wall interface are given by @u u ¼ h ; T ¼ T0 and C ¼ C0 at y ¼ g @y
 2p ¼  d þ a sin ðx  tÞ : k

Figure 1

ð4Þ

and the boundary conditions due to wall flexibility are

2  @  @p @ u @2u @u @u @u L ðgÞ ¼ ¼l þ u þ v þ  q @x @x @x2 @y2 @t @x @y l  rB20 u  uat y ¼ g: k

ð9Þ

ð10Þ

Please cite this article in press as: M.G. Reddy, Heat and mass transfer on magnetohydrodynamic peristaltic flow in a porous medium with partial slip, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.04.009

Heat and mass transfer on MHD peristaltic flow in a porous medium with partial slip Introducing the stream function w, the velocity fields are related by u¼

@w @w ; v¼ : @y @x

ð11Þ

The following non-dimensional variables are defined: x y w ct g d2 p ; x0 ¼ ; y0 ¼ ; w0 ¼ ; t0 ¼ ; g0 ¼ ; p0 ¼ k d cd k d ckl T  T0 C  C0 a d qcd ; /¼ ; e¼ ; d¼ ; R¼ ; h¼ T0 C0 d k l rffiffiffi r k qmf c2 b ; b ¼ ; M¼ ; Ec ¼ B0 d; K ¼ 2 ; Pr ¼ fT0 l k d d 3 3 3 3 sd m1 cd cd Bd E1 ¼ 3 ; E2 ¼ 3 ; E3 ¼ 2 ; E4 ¼ 3 ; k lc kl kl k cl Hd3 l qDKT T0 ; Sc ¼ E5 ¼ ; Sr ¼ kcl lTm C0 qD

ð12Þ

Using the above non-dimensional quantities and Eq. (11) in Eqs. (2)–(6) with the boundary conditions (9) and (10) (dropping primes), we have
2  @ w @w @ 2 w @w @ 2 w Rd þ  @t@y @y @x@y @x @y2 ¼

@p @3w @3w @w 1 @w þ d2 2 þ 3  M2  : @x @x @y @y @y K @y

 @ 2 w @w @ 2 w @w @ 2 w Rd3 þ  @t@y @y @x@y @x @y2
 @p @3w @3w d2 @w :  ¼  þ d2 d2 3 þ 2 @y @x @x @y K @y

3

ability parameter, Prandtl number, Eckert number, Schmidt number, Velocity slip parameter and wall compliant parameters. With the assumptions of long wavelength approximation and neglecting the wave number (d  1) and small Reynolds number in Eqs. (13)–(16) along with the boundary conditions (17) and (18), we get 

@p @ 3 w @w þ 3  N2 ¼ 0: @x @y @y

ð19Þ



@p ¼ 0: @y

ð20Þ

 2  2 2 @2h @ w 2 @w þ 1 þ Br þ BrM ¼ 0: @y2 @y2 @y

ð21Þ

@2/ @2h  Sc/ þ ScSr ¼ 0: @y2 @y2

ð22Þ

Here N2 ¼ M2 þ K1 and Br( = Pr Ec) is the Brinkman number. The corresponding dimensionless boundary conditions are @w @2w ¼ b 2 ; h ¼ 0 and / ¼ 0 at y ¼ g @y @y

ð23Þ

ð13Þ 1




0.5



 @h @w @h @w @h 1 2 @2h @2h Rd þ  ¼ d þ @t @y @x @x @y Pr @x2 @y2 (  )   2 2 2 @2w @2w 2@ w þ Ec 4d2 þ  d @x@y @y2 @x2 "  2 # 2  2 !  2 @2w @2w @ w @2w 2 þ ð15Þ d þ  d 2 þM 2 @x@y @y@x @y2 @x

 @/ @w @/ @w @/ 1 2 @2/ @2/ þ  ¼ d Rd þ @t @y @x @x @y Sc @x2 @y2
 @2h @2h þ Sr d2 2 þ 2 : @x @y

y

ð14Þ

M M M M

0

= = = =

1.0 2.0 3.0 4.0

-0.5

-1

0

1

3

2

4

5

6

7

u

Figure 2a

ð16Þ

Variations of velocity u for various values of M.

1

The corresponding boundary conditions are 0.5

h ¼ 0 and / ¼ 0 at y ¼ g

¼ ½1 þ e sin 2pðx  tÞ 

ð17Þ

3

2

2

2

K = 0.1 K = 1.0 K = 2.0

0

K=∞



@ w 2 @ w @ w @w @ w @w @ w þ   Rd þd @y3 @x2 @y @t@y @y @x@y @x @y2 @w 1 @w  M2  @y K @y
 @3 @3 @2 @5 @ ðgÞat y ¼ g: ð18Þ ¼ E1 3 þ E2 þ E þ E þ E 3 4 5 @x @x @x@t2 @t@x @x5 3

y

@w @2w ¼ b 2 ; @y @y

where R, d, M, K, Pr, Ec, Sc, b and E1,E2, E3, E4 and E5 are the Reynolds number, wave number, Hartmann number, Perme-

-0.5

-1

0

2

4

6

8

10

12

u

Figure 2b

Variations of velocity u for various values of K.

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4

M.G. Reddy 1

1

K = 0.1 K = 1.0

β = 0.0 β = 0.1 β = 0.2 β = 0.3

0

K=∞

0

-0.5

-0.5

-1

K = 2.0

0.5

y

y

0.5

0

1

2

3

4

5

6

7

-1

8

0

2

4

6

u

Variations of velocity u for various values of b.

1

1

0.5

0.5 E1 = 1.0, E2 = 0.5, E3 = 0.5, E4 = 0.2, E5 = 0.1 E1 = 2.0, E2 = 0.5, E3 = 0.5, E4 = 0.2, E5 = 0.1 E1 = 1.0, E2 =1.0, E3 = 0.5, E4 = 0.2, E5 = 0.1

0

E1 = 1.0, E2 = 0.5, E3 = 1.0, E4 = 0.2, E5 = 0.1 E1 = 1.0, E2 = 0.5, E3 = 0.5, E4 = 0.1, E5 = 0.1

-0.5

-1

14

12

β β β β

0

= = = =

0.0 0.02 0.06 0.1

E1 = 1.0, E2 = 0.5, E3 = 0.5, E4 = 0.2, E5 = 2.0

0

1

2

3

4

5

6

7

-1

8

Figure 2d Variations of the velocity u for various values of E1, E2, E3, E4 and E5.

0

0.5

1

1.5

2

θ

2.5

3

3.5

4

4.5

Figure 3c Variations of temperature distribution h for different values of b.

1

1 M M M M

= = = =

1.0 2.0 3.0 4.0

0

0.2 0.5 1.0 2.0

0

-0.5

-0.5

-1 0

Br = Br = Br = Br =

0.5

y

0.5

y

10

-0.5

u

-1

8

Figure 3b Variations of temperature distribution h for different values of K.

y

y

Figure 2c

θ

1

2

θ

3

4

5

Figure 3a Variations of temperature distribution h for different values of M.


@3w @3 @3 @2 @5 2 @w þ E  N þ E þ E ¼ E 1 2 3 4 @x3 @x@t2 @t@x @x5 @y3 @y  @ ðgÞ at y ¼ g: þE5 @x

0

10

20

θ

30

40

50

Figure 3d Variations of temperature distribution h for different values of Br.

3. Solution of the problem

ð24Þ

The set of Eqs. (19)–(22) subject to the subject boundary conditions (23) and (24) are solved exactly for expression w, h and / and are given by

Please cite this article in press as: M.G. Reddy, Heat and mass transfer on magnetohydrodynamic peristaltic flow in a porous medium with partial slip, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.04.009

Heat and mass transfer on MHD peristaltic flow in a porous medium with partial slip 1

1

y

y

E1 = 1.0, E2 = 0.5, E3 = 0.5,E4 = 0.2, E5 = 0.1

0

E1 = 2.0, E2 = 0.5, E3 = 0.5,E4 = 0.2, E5 = 0.1 E1 = 1.0, E2 = 1.0, E3 = 0.5,E4 = 0.2, E5 = 0.1 E1 = 1.0, E2 = 0.5, E3 = 1.0,E4 = 0.2, E5 = 0.1

-0.5

0

1

2

3

Figure 3e Variations of temperature distribution h for different values of E1, E2, E3, E4 and E5.

-3

-3.5

-2.5

φ

-2

-1.5

-1

-0.5

0

Figure 4c Variations of concentration distribution / for different values of Br.

1

1 M M M M

= = = =

1.0 2.0 3.0 4.0

0.5

y

0.5

0

y

-4

6

θ

Sc Sc Sc Sc

0

= = = =

0.1 0.6 1.0 2.0

-0.5

-0.5

-1

-0.8

-0.6

φ

-0.4

-0.2

1

-1 -1.5

0

Figure 4a Variations of concentration distribution / for different values of M.

-1

-0.5

0

φ

Figure 4d Variations of concentration distribution / for different values of Sc.

1

K = 0.1 K = 1.0 K = 2.0

0.5

K=∞

0.5 E1 = 1.0, E2 = 0.5, E3 = 0.5, E4 = 0.2, E5 = 0.1

y

0

y

0

-1 -4.5

5

4

0.1 0.2 0.5 1.0

-0.5

E1 = 1.0, E2 = 0.5, E3 = 0.5,E4 = 0.1, E5 = 0.1 E1 = 1.0, E2 = 0.5, E3 = 0.5,E4 = 0.2, E5 = 2.0

-1

Br = Br = Br = Br =

0.5

0.5

-1

5

-0.5

E1 = 2.0, E2 = 0.5, E3 = 0.5, E4 = 0.2, E5 = 0.1

0

E1 = 1.0, E2 =1.0, E3 = 0.5, E4 = 0.2, E5 = 0.1 E1 = 1.0, E2 = 0.5, E3 =1.0, E4 = 0.2, E5 = 0.1 E1 = 1.0, E2 = 0.5, E3 = 0.5, E4 = 0.1, E5 = 0.1

-0.5

E1 = 1.0, E2 = 0.5, E3 = 0.5 ,E4 = 0.2, E5 = 2.0

-1

-3

-2.5

-2

-1.5

-1

-0.5

0

φ

Figure 4b Variations of concentration distribution / for different values of K.


w¼L

 sinh Ny y : Nðcosh Ng þ Nb sinh NgÞ

ð25Þ

-1 -35

-30

-25

-20

φ

-15

-10

-5

0

Figure 4e Variations of concentration distribution / for different values of E1, E2, E3, E4 and E5.

h ¼ l1 cosh 2Ny þ l2 y2 þ l3 cosh Ny þ l4 :

ð26Þ

Please cite this article in press as: M.G. Reddy, Heat and mass transfer on magnetohydrodynamic peristaltic flow in a porous medium with partial slip, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.04.009

6

M.G. Reddy 1.5

2.5 M = 1.0 M = 2.0 M = 3.0

2 1.5

β = 0.0 β = 0.1 β = 0.3

1

0.5

Z

Z

1 0.5

0

0 -0.5

-0.5 -1

0

0.2

0.4

x

0.6

0.8

-1

1

Figure 5a Variations of heat transfer coefficient Z for different values of M.

K = 0.5 K = 2.0

10

0.4

x

0.6

0.8

1

Figure 5d Variations of heat transfer coefficient Z for different values of b.

1.5

K=∞

1

5

Z

0.2

(a)

15

0.5

0

0

-5

-0.5

-10 -15

0

-1 0

0.2

0.4

x

0.6

0.8

1 -1.5

Figure 5b Variations of heat transfer coefficient Z for different values of K.

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

(b) 1.5

1.5

1

1 0.5

0.5

Z

E1 = 1.0, E2 = 0.5, E3 = 0.5,E4 = 0.2, E5 = 0.1 E1 = 2.0, E2 = 0.5, E3 = 0.5,E4 = 0.2, E5 = 0.1 E1 = 1.0, E2 =1.0, E3 = 0.5,E4 = 0.2, E5 = 0.1

0

0 -0.5

E1 = 1.0, E2 = 0.5, E3 =1.0,E4 = 0.2, E5 = 0.1 E1 = 1.0, E2 = 0.5, E3 = 0.5,E4 = 0.1, E5 = 0.1

-0.5

-1

E1 = 1.0, E2 = 0.5, E3 = 0.5,E4 = 0.2, E5 =2.0

-1

0

0.2

0.4

x

0.6

0.8

1

-1.5

Figure 5c Variations of heat transfer coefficient Z for different values of E1, E2, E3, E4 and E5.

/ ¼ l5 cosh 2Ny þ l6 þ l7 cosh Ny þ l8 cosh Scy:

ð27Þ

Figure 6

L¼

8ep3 N2

Streamlines for (a) M = 2.0 and (b) M = 4.0.


 E1 þ E2  4p2 E4 

  E5 E3 sin 2pðx  tÞ ; cos 2pðx  tÞ  4p2 2p

where

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Heat and mass transfer on MHD peristaltic flow in a porous medium with partial slip

7

(a)

(a) 1.5

1.5

1

1

0.5

0.5

0

0

-0.5

-0.5

-1

-1

-1.5

-1.5 -0.1

0

0.1

0.3

0.2

0.4

0.5

0.7

0.6

(b)

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

(b)

1.5

1.5

1

1

0.5

0.5

0

0

-0.5

-0.5

-1

-1

-1.5

-1.5 -0.1

0

Figure 7

l1 ¼

l2 ¼

0.1

0.2

0.3

BrL2 ðN2 þ M2 Þ 8N ðcosh Ng þ Nb sinh NgÞ2 BrL2 N2 4ðcosh Ng þ Nb sinh NgÞ

l3 ¼ 

0.5

0.6

0.7

Streamlines for (a) K = 2.0 and (b) K = 1.

2



0.4

2



Figure 8

;

Streamlines for (a) b = 0.0 and (b) b = 0.2.

4. Results and discussion BrL2 M2

4ðcosh Ng þ Nb sinh NgÞ2

BrM2 ð1  LÞ2 ; 2 2BrM2 Lð1  LÞ ; N ðcosh Ng þ Nb sinh NgÞ 2

l4 ¼ ðl1 cosh 2Ng þ l2 g2 þ l3 cosh NgÞ; l5 ¼ 

4N2 l1 ScSr N2 ScSrl3 ; l6 ¼ 2Srl2 ; l7 ¼  2 ; 2 N  Sc 4N  Sc

l8 ¼ 

ðl5 cosh 2Ng þ l6 þ l7 cosh NgÞ cosh Scg

The heat transfer coefficient at the wall is given by Z ¼ gx h0 ðgÞ Z ¼ ð2ep cos 2pðx  tÞÞð2Nl1 sinh 2Ng þ 2l2 g þ Nl3 sinh NgÞ:

ð28Þ

The purpose of this section is to study the influences of various parameters on the velocity, temperature, concentration, heat transfer coefficient and streamlines (see Figs. 2–9). In the present study following default parameter values are adopted for computations: x ¼ 0:2, t ¼ 0:1, e ¼ 0:02, M ¼ 1:0, K ¼ 0:5, b ¼ 0:1, Br ¼ 0:2, Sc ¼ 1:0, E1 ¼ 1:0, E2 ¼ 0:5, E3 ¼ 0:5, E4 ¼ 0:2 and E5 ¼ 0:1. All graphs therefore correspond to these values unless specifically indicated on the appropriate graph. The influence of M, K, b and wall compliant parameters on the velocity distribution is shown graphically in Fig. 2. From Fig. 2a, it reveals that increasing M, leads to fall in the velocity. Because the effect of increasing magnetic field strength dampens the velocity. The effect of permeability parameter K on the velocity is displayed in Fig. 2b. It is clear that when the K increases, the velocity leads to enhance. An increasing K means reduce the drag force and hence cause the flow velocity to increase. Fig. 2c depicts to study the effects of b1 on the velocity field. From Fig. 2c, it can be seen that the velocity of the fluid increases with an increase in b1 : The influence of wall

Please cite this article in press as: M.G. Reddy, Heat and mass transfer on magnetohydrodynamic peristaltic flow in a porous medium with partial slip, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.04.009

8

M.G. Reddy

(a)

(b)

1.5

1.5

1

1

0.5

0.5

0

0

-0.5

-0.5

-1

-1

-1.5

-1.5

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

-0.1

0.7

(c)

(d)

1.5

1.5

1

1

0.5

0.5

0

0

-0.5

-0.5

-1

-1

-1.5

-1.5

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0

0

0.7

0.1

0.2

0.2

0.3

0.4

0.5

0.6

0.7

0.4

0.6

0.8

0.4

0.6

0.8

(f)

(e) 1.5

1.5

1

1

0.5

0.5

0

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Figure 9

0.2

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Streamlines for (a) E1 = 1.0, (b) E1 = 2.0, (c) E2 = 1.0, (d) E3 = 1.0, (e) E4 = 0.1 and (f) E5 = 1.0.

compliant parameters on the velocity field is shown in Fig. 2d. It is clear that the velocity decreases with increasing E1 and E2 whereas it increases with increasing E3, E4 and E5. This due to the fact that the elasticity of the walls has less resistance to the flow and so that the velocity field increases. Fig. 3 depicts the temperature profiles for various values of M, K, b, Br and wall compliant parameters. Figs. 3a and 3b show the effects of M and K respectively on the temperature distribution. It is observed that the temperature decreases with

increasing the Hartman number M while the opposite behavior for the permeability parameter K: The effect of velocity slip parameter on the temperature distribution is plotted in Fig. 3c. It reveals that the temperature profiles are almost curviness and when the slip parameter b1 increases the temperature field decreases. Fig. 3d depicts the influence of Br on the temperature. It can be found that an increase in Br results in the increase in the temperature field. Fig. 3e is presented to study the effects of E1, E2, E3, E4 and E5 on the concentration

Please cite this article in press as: M.G. Reddy, Heat and mass transfer on magnetohydrodynamic peristaltic flow in a porous medium with partial slip, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.04.009

Heat and mass transfer on MHD peristaltic flow in a porous medium with partial slip distribution. Fig. 3e reveals that the temperature increases with increasing E3, E4 and E5 and it decreases with increasing E1 and E2. Fig. 4 is plotted to the study the involved parameters on the concentration field. From Fig. 4a, it is clear that the concentration distribution increases with increasing the Hartmann number M. The influence of permeability parameter K on the concentration distribution is represented in Fig. 4b. From Figs. 4a and 4b, we can notice that an increase in Hartmann number affects the concentration profile in an opposite way to that of permeability parameter. The effect of the Brinkman number Br on the concentration field is illustrated in Fig. 4c. It can be noticed that the concentration distribution decreases with increasing Br. The concentration distribution for different values of Schmidt number Sc is plotted in Fig. 4d. The Schmidt number Sc embodies the ratio of the momentum diffusivity to the mass (species) diffusivity. It is evident that the concentration field decreases with an increase in Sc. Fig. 4e shows the variations of wall compliant parameters on the concentration field. It is observed that increase in E1 and E2 and decrease of E3, E4 and E5 result in the increase of the concentration. The variations of heat transfer coefficient at the wall for different values of the physical parameters of interest are presented in Fig. 5. From Figs. 5a and 5d, it is observed that the absolute value of heat transfer coefficient decreases with increasing M and b while from Fig. 5b, it increases when K is increased. The magnitude of heat transfer coefficient decreases with increasing wall compliant parameter which is observed in Fig. 5c. An interesting phenomenon of peristalsis is trapping in which stream lines split to trap a bolus in the wave frame. The effect of Hartman number M on trapping is analyzed through Fig. 6. It reveals that the volume of the trapped bolus decreases with increase of M. The effect of permeability parameter K on trapping is shown in Fig. 7 It is noticed that the streamlines closed loops creating a cellular flow pattern in the channel and the trapped bolus increases in size as K increases. The influence of velocity slip parameter on the trapping is presented in Fig. 8. It is observed that the size of the trapping bolus increases with increasing b: The influences of wall compliant parameters on trapping are presented through Fig. 9. It can be concluded that the size of trapped bolus increases by increasing E1, E2, E3 and E4 while it decreases with increasing E5. 5. Conclusions The influence of velocity slip on MHD peristaltic flow through a porous medium with heat and mass transfer is investigated. The exact solutions for stream function, temperature, concentration and heat transfer coefficient have been developed. The features of the flow characteristics are analyzed by plotting graphs and discussed in detail. The important findings of the present study are as follows: (1) The size of trapped bolus increases with increasing velocity slip parameter. (2) The velocity and temperature field increase with an increase in the velocity slip parameter and permeability parameter while it decreases with an increase in the Hartmann number.

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(3) An increase in the Brinkman number Br results in increase in the temperature distribution. (4) The concentration field decreases with an increase in the Schmidt number Sc. (5) The coefficient of heat transfer is oscillatory in nature.

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Please cite this article in press as: M.G. Reddy, Heat and mass transfer on magnetohydrodynamic peristaltic flow in a porous medium with partial slip, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.04.009