Influences of chemical reaction and wall properties on

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

Influences of chemical reaction and wall properties on MHD Peristaltic transport of a Dusty fluid with Heat and Mass transfer R. Muthuraj a,*, K. Nirmala b, S. Srinivas c a

Department of Mathematics, P.S.N.A. College of Engineering & Technology, Dindigul 624 622, India Department of Mathematics, Dhirajlal Gandhi College of Technology, Salem 636 309, India c Fluid Dynamics Division, VIT University, Vellore 632 014, India b

Received 24 July 2015; revised 10 January 2016; accepted 16 January 2016

KEYWORDS Peristaltic transport; Wall properties; Dusty fluid; MHD; Channel flow

Abstract The influence of elasticity of flexible walls on peristaltic transport of a dusty fluid with heat and mass transfer in a horizontal channel in the presence of chemical reaction has been investigated under long wavelength approximation. Expressions have been constructed for stream function, temperature and concentration by using perturbation technique. The effects of various parameters on heat and mass transfer characteristics of the flow are discussed through graphs. 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 physical mechanism of the flow induced by the traveling wave can be well understood and is known as the so-called peristaltic transport mechanism. Peristaltic pumping is also used in medical instruments such as the heart–lung machine. Investigation on peristaltic transport of non-Newtonian fluids is of utmost importance owing to its wide range of applications in engineering and biology. In particular, the study of such fluids has applications in a number of processes that occur in industry such as the extrusion of polymer fluids, solidification of liquid crystals, cooling of metallic plate in a bath, exotic lubricants, colloidal and suspension solutions. Further, such analysis may serve for the intrauterine fluid motion in a * Corresponding author. E-mail address: [email protected] (R. Muthuraj). Peer review under responsibility of Faculty of Engineering, Alexandria University.

sagittal cross section of the uterus under cancer therapy and drug analysis (see Refs. [1–14]). In view of these applications, many analytical, numerical and experimental attempts have been made to understand peristalsis in different situations for Newtonian and non-Newtonian fluids. Later, Hayat and Noreen [10] have investigated the influence of an induced magnetic field on the peristaltic flow of an incompressible fourth grade fluid in a symmetric channel with heat transfer using long wavelength, low Reynolds number and small Deborah number assumptions. The study of the peristaltic transport of viscoelastic non-Newtonian fluids with fractional Maxwell model in a channel was discussed by Tripathi [11]. Srinivas and Muthuraj [12] have investigated the effects of chemical reaction and space porosity on MHD mixed convective peristaltic flow in a vertical asymmetric channel. Alla et al. [13] have examined the effects of both rotation and magnetic field of a micro polar fluid through a porous medium induced by sinusoidal peristaltic waves traveling down the channel walls. The peristaltic transport of a Williamson nanofluid in a

http://dx.doi.org/10.1016/j.aej.2016.01.013 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: R. Muthuraj et al., Influences of chemical reaction and wall properties on MHD Peristaltic transport of a Dusty fluid with Heat and Mass transfer, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.01.013

2 tapered asymmetric channel under the action of a thermal radiation parameter was examined by Kothandapani and Prakash [14]. However, the wall properties are essential to be taken into consideration in various real situations. Therefore, some authors have developed theoretical models to describe peristaltic flow in channel/tube with wall properties [15–26]. Srinivas and Kothandapani [19] have examined the effects of heat and mass transfer on peristaltic transport in a porous space with compliant walls. Ali et al. [20] have analyzed the peristaltic motion of a non-Newtonian fluid in a channel having compliant boundaries. Hayat and Hina [21] have investigated the effects of heat and mass transfer on the MHD peristaltic flow of a Maxwell fluid in a planar channel with compliant walls. Mekheimer and Abdel-Wahab [22] have studied the effects of compliant wall properties on the flow of a Newtonian viscous compressible fluid when the wave propagating (surface acoustic wave) along the walls in a confined parallel-plane microchannel is conducted by considering the slip velocity. Muthuraj and Srinivas [23] have discussed the MHD peristaltic flow of a Newtonian fluid through porous space in a vertical channel with compliant walls under the assumptions of long wavelength and low Reynolds number. The influence of heat and mass transfer on the peristaltic transport of Johnson–Segalman fluid in a curved channel with flexible walls was investigated by Hina et al. [24]. Eldabe et al. [25] have analyzed the effect of wall properties on the peristaltic transport of a dusty fluid through porous channel with heat and mass transfer by using perturbation technique for small geometric parameter. They have also discussed the peristaltic motion of a power-law model, with heat and mass transfer through an asymmetric channel [26]. Hayat et al. [27] have examined the effect of elasticity of the flexible walls on the peristaltic flow of a power-law fluid with heat transfer. The flow of a MHD fluid through a channel in the presence of a transverse magnetic field is encountered in a variety of applications such as magnetohydrodynamic (MHD) generators, pumps, accelerators, and flow meters. In particular, the magnetohydrodynamic flows of non-Newtonian fluids are of great interest in magneto-therapy. Due to the importance of MHD flows, many studies have been carried out examining the effects of magnetic field on hydrodynamic flow in various configurations [28–32]. Moreover, two phase flows in which solid spherical particles are distributed in a clean fluid have attracted the interest of a number of researchers due to its practical applications such as petroleum industry, purification of crude oil, and physiological flows (see Refs. [33–45]). Saffman [33] pioneered the study of the fluid – particle system. They have discussed the convective stability of the particulate poiseuille flow with the assumption that the solid phase is distributed homogeneously. Later, the effects of dependence on temperature of the viscosity and electric conductivity, Reynolds number and particle concentration on the unsteady MHD flow and heat transfer of a dusty, electrically conducting fluid between parallel plates in the presence of an external uniform magnetic field have been investigated by Eguia et al. [41]. Hakan Erol [42] has studied the propagation of time harmonic waves in prestressed, anisotropic elastic tubes filled with viscous fluid containing dusty particles and the fluid is assumed to be incompressible and Newtonian. Ramesh et al. [43] have analyzed the steady two-dimensional MHD flow of a dusty fluid near the stagnation point over a permeable stretching

R. Muthuraj et al. sheet with the effect of non-uniform source/sink. Pavithra and Gireesha [44] have examined the boundary layer flow and heat transfer of a dusty fluid over an exponentially stretching surface in the presence of viscous dissipation and internal heat generation/absorption using Runge–Kutta method. Most recently, the problem of a steady two-dimensional magnetohydrodynamic (MHD) flow of a dusty fluid over a stretching hollow cylinder was analyzed by Rakesh et al. [45]. To the best of our knowledge, no investigation has been made on the heat and mass transfer effects on peristaltic transport of MHD dusty fluid in a horizontal channel with compliant walls in the presence of chemical reaction. Therefore, in this paper, we have extended the results of Eldabe et al. [25] for peristaltic transport of a MHD dusty fluid through a two-dimensional horizontal channel with combined effects of uniform magnetic field and chemical reaction. Analytical solutions of the momentum, heat and concentration equations are obtained by using perturbation technique for both fluid and solid particles. The features of flow, heat and mass transfer characteristics are presented and discussed graphically. The present paper is organized in the following fashion. The problem is formulated in Section 2. Section 3 deals with the solution of the problem under long wavelength assumption. Results and discussion are given in Section 4. The conclusions have been summarized in Section 5. 2. Formulation of the problem Consider the laminar flow of an incompressible fluid that contains small solid particles, whose number density N0 (assumed to be constant) is large enough to define average properties of the dust particles at a point through a symmetrical twodimensional channel. Choose the Cartesian coordinates (x, y), where x is along the walls and y is perpendicular to it (see Fig. 1). The geometry of the wall surface is described by 2p ðx ctÞ ð1Þ g ¼ d þ a sin k where d is the half width of the channel, ‘a’ is the amplitude of the wave, k is the wavelength, t is the time, and c is the wave velocity. The governing equation of motion of the flexible wall may be expressed as [21] LðgÞ ¼ p p0 L ¼ T

ð2Þ

@2 @2 @ @4 þ m 2 þ d þ B 4 þ K 2 @x @t @x @t

ð3Þ

where p0 is the pressure on the outside of the wall, T is the elastic tension in the membrane, m is the mass per unit area, d is the coefficient of viscous damping, B is the flexural rigidity of the plate, and K is the spring stiffness. The momentum, energy and concentration equations are 2 @uf @uf @uf @p @ uf @ 2 uf þl þ uf þ vf ¼ þ q @x @t @x @y @x2 @y2 þ KN0 ðus uf Þ rB20 uf

ð4Þ

2 @vf @vf @vf @p @ vf @ 2 vf q þ uf þ vf ¼ þl þ KN0 ðvs vf Þ þ @y @t @x @y @x2 @y2 ð5Þ

Please cite this article in press as: R. Muthuraj et al., Influences of chemical reaction and wall properties on MHD Peristaltic transport of a Dusty fluid with Heat and Mass transfer, Alexandria Eng. J. (2016), http://dx.doi.org/10.1016/j.aej.2016.01.013

MHD Peristaltic transport of a Dusty fluid

3

Figure 1

qcp

Flow geometry of the problem.

@Tf @Tf @Tf N0 cp þ uf þ vf ¼ ðTs Tf Þ @t @x @y sT i N0 h ðus uf Þ2 þ ðvs vf Þ2 þ sv 2 @ Tf @ 2 Tf þ þ kc @x2 @y2

Cf is the concentration of the fluid, kc is the thermal conductivity of the fluid, Df is the coefficient of mass diffusivity of the fluid particles, Ds is the coefficient of mass diffusivity of the solid particles, cm is the specific heat of solid particles, Ts is the temperature of solid particles, and sT is the thermal relaxation time of the solid particles. @wf ; @y

@wf ; @x

ð6Þ

uf ¼

2 @Cf @Cf @Cf @ Cf @ 2 Cf þ uf þ vf ¼ Df k1 C f þ @t @x @y @x2 @y2

ð7Þ

@us @us @us K þ us þ vs ¼ ðuf us Þ rB20 us @t @x @y m

ð8Þ

x y g ; y ¼ ; g ¼ ; k d d mws Tf Tf1 ; ws ¼ 2 ; hf ¼ Tf2 Tf1 kd

@vs @vs @vs K þ us þ vs ¼ ðvf vs Þ @t @x @y m

ð9Þ

N0 cm

@Ts @Ts @Ts þ us þ vs @t @x @y

N0 cp ¼ ðTs Tf Þ sT

2 @Cs @Cs @Cs @ Cs @ 2 Cs þ us þ vs ¼ Ds k1 C s þ @t @x @y @x2 @y2

ð10Þ

ð11Þ

with the boundary conditions uf ¼ us ¼ 0; Tf ¼ Tf1 ; Ts ¼ Ts1 ; Cf ¼ Cf1 and Cs ¼ Cs1 at y ¼ g uf ¼ us ¼ 0; Tf ¼ Tf2 ; Ts ¼ Ts2 ; Cf ¼ Cf2 and Cs ¼ Cs2 at y ¼ g

ð12Þ

ð13Þ

@p @ ¼ LðgÞ @x @x 2 @ uf @ 2 uf @uf @uf @uf þ u þ v q þ ¼l f f @x2 @y2 @t @x @y þ KN0 ðus uf Þ rB20 uf

ð14Þ

where r is the electrical conductivity of the fluid, B0 is the applied magnetic field, q is the density of the fluid and K is (resistance coefficient for the dust particles) a constant, Tf is the temperature of the fluid, cp is the specific heat of the fluid,

vf ¼

x ¼

/f ¼

Cf Cf1 ; Cf2 Cf1

/s ¼

us ¼

@ws ; @y

vs ¼

@ws @x

ð15Þ

wf mt ; wf ¼ ; kd m Ts Ts1 hs ¼ ; Ts2 Ts1

t ¼

Cs Cs1 Cs2 Cs1

ð16Þ

Introducing (15) and (16) into Eqs. (4)–(14), we obtain the following equations (after dropping primes and eliminating the pressure term) " ! ! 2 2 2 @wf @ @ 2 wf @ @ wf 2 @ wf 2 @ wf d þ þ d þ d @t @y2 @x2 @y @x @y2 @x2 !# 2 @wf @ @ 2 wf 2 @ wf þ d @x @y @y2 @x2 @ 2 wf 1 ð17Þ ¼ r41 wf þ P r21 ws r21 wf M2 R @y2 @hf @wf @hf @wf @hf þ d @t @y @x @x @y " 2 N0 l 1 E c N0 l 1 1 @ws @wf hs hf þ ¼ sT sv @y R @y 2 # 1 @ws @wf 1 2 @ 2 hf @ 2 hf þ d þ þ d2 @x @x2 @y2 R @x Pr

ð18Þ

" # 2 @/f @wf @/f @wf @/f @ 2 /f 1 2 @ /f d þ ¼ d þ 2 c/f þ K1 Sc1 @t @y @x @x @y @x2 @y ð19Þ


4

R. Muthuraj et al.

@ @w @ @w @ ðr21 ws Þ s ðr21 ws Þ d R ðr21 ws Þ þ s @t @y @x @x @y ¼ Rr21 wf r21 ws M2 A

@ 2 ws @y2

ð20Þ

@hs 1 @ws @hs 1 @ws @hs þ ¼ l2 ðhs hf Þ d @t R @y @x R @x @y @/s 1 @ws @/s 1 @ws @/s d þ @t R @y @x R @x @y 1 @ 2 /s @ 2 /s c/s þ K2 d2 þ ¼ Sc2 @x2 @y2

ð21Þ

ð22Þ

/f ¼ /f0 þ d/f1 þ d2 /f2 þ

ð31Þ

/s ¼ /s0 þ d/s1 þ d2 /s2 þ

ð32Þ

Substituting (27)–(32) in the system of Eqs. (17)–(22) subject to the boundary conditions (23)–(25), we get: Zeroth order equations: " # 2 @ 4 wf0 @ 2 wf0 1 @ 2 ws0 @ wf0 þ P ¼0 ð33Þ M2 4 2 2 @y @y @y2 R @y 2 1 @ 2 hf0 N0 l 1 N0 l1 Ec 1 @ws0 @wf0 þ ðh h Þ þ ¼0 s0 f0 sT sv @y Pr @y2 R @y ð34Þ

with the boundary conditions, @wf @ws ¼ 0; ¼ 0; hf ¼ 0; hs ¼ 0; /f ¼ 0; /s ¼ 0 at y ¼ g @y @y ð23Þ @wf @ws ¼ 0; ¼ 0; hf ¼ 1; hs ¼ 1; /f ¼ 1; /s ¼ 1 at y ¼ g @y @y ð24Þ @3 @2 @5 @3 @ g þ E þ E þ E þ E E1 2 3 4 5 @x @x@t2 @x@t @x5 @x3 @wf @ @wf @wf @ @wf @wf @ @wf ¼ r21 d þ @t @y @y @y @x @y @x @y @y @w 1 @ws @wf f þP M2 ð25Þ R @y @y @y @2 @2 d where r21 ¼ d2 @x is geometric parameter, 2 þ @y2 ; d ¼ k 2

2

c d2

ml mm ; l1 ¼ dl , l2 ¼ sTpcm m are non dimenP ¼ KNqm0 d ; R ¼ kd 2 ; A ¼ kd2 ffi qffiffiffiffiffiffiffiffi rB20 d2 sional parameters, M ¼ is the Hartmann number, l 2

c ¼ k1md

is

the

2

k1 cs1 d K2 ¼ mðc ; Ec ¼ d2 c s2 cs1 Þ

chemical m2 p ðTf2 Tf1 Þ

parameter,

k c d2

f1 K1 ¼ mðcf21 c ; f1 Þ

is the Eckert number, Pr ¼

lcp kc

is the Prandtl number, Sc1 ¼ Dmf is the Schmidt number of the fluid particles and Sc2 ¼ Dms is the Schmidt number of solid Particles, m is the kinematic coefficient of the viscosity, 3 4 4 2 d K d 4 ; E4 ¼ kBd E1 ¼ kT3 qmd 2 ; E2 ¼ mk3dq ; E3 ¼ kd2 qm 5 2 and E5 ¼ kqm2 qm are the non-dimensional elasticity parameters.

R

@ 2 wf0 @ 2 ws0 @ 2 ws0 2 M A ¼0 @y2 @y2 @y2

ð35Þ

ð36Þ

l2 ðhs0 hf0 Þ ¼ 0

ð37Þ

1 @ 2 /s0 c/s0 þ K2 ¼ 0 Sc2 @y2

ð38Þ

with the boundary conditions, @wf0 @ws0 ¼ 0; ¼ 0; hf0 ¼ 0; hs0 ¼ 0; @y @y /f0 ¼ 0; /s0 ¼ 0 at y ¼ g

ð39Þ

@wf0 @ws0 ¼ 0; ¼ 0; hf0 ¼ 1; hs0 ¼ 1; @y @y /f0 ¼ 1; /s0 ¼ 1 at y ¼ g

ð40Þ

@ 3 wf0 @wf0 1 @ws0 @wf0 M2 þ P R @y @y3 @y @y 3 2 5 @ @ @ @3 @ ¼ E1 þ E þ E þ E þ E g 2 3 4 5 @x@t2 @x@t @x5 @x3 @x at

y ¼ g

First order equations: " # 2 @ wf1 @ 2 wf1 1 @ 2 ws1 @ wf1 2 M þ P @y4 @y2 @y2 R @y2 " 3 # @ wf0 @wf0 @ 3 wf0 @wf0 @ 3 wf0 ¼ þ @t@y2 @y @x@y2 @x @y3

ð41Þ

4

3. Method of solution We seek perturbation solution in terms of small parameter d 1 as follows: f ¼ f0 þ df1 þ d2 f2 þ d3 f3 þ

2 1 @ /f0 c/f0 þ K1 ¼ 0 Sc1 @y2

ð26Þ

where ‘f ’ represents any flow variable. wf ¼ wf0 þ dwf1 þ d2 wf2 þ

ð27Þ

ws ¼ ws0 þ dws1 þ d2 ws2 þ

ð28Þ

hf ¼ hf0 þ dhf1 þ d2 hf2 þ

ð29Þ

hs ¼ hs0 þ dhs1 þ d2 hs2 þ

ð30Þ

ð42Þ

@hf0 @wf0 @hf0 @wf0 @hf0 þ @t @y @x @x @y 2N0 l1 Ec @ws0 @wf1 @ws1 @wf0 þ þ Rsv @y @y @y @y ¼

1 @ 2 hf1 N0 l1 þ ðhs1 hf1 Þ sT Pr @y2

2 @/f0 @wf0 @/f0 @wf0 @/f0 1 @ /f1 þ c/f1 ¼ @t @y @x @x @y Sc1 @y2

ð43Þ

ð44Þ


MHD Peristaltic transport of a Dusty fluid R

5 @ 3 wf1 1 @ws1 @wf1 þ P R @y @y3 @y " 2 # @ wf0 @wf0 @ 2 wf0 @wf0 @ 2 wf0 þ @t@y @y @x@y @x @y2

@ 2 wf1 @ 2 ws1 @ 2 ws1 2 M A @y2 @y2 @y2 ¼R

@ 3 ws0 @ws0 @ 3 ws0 @ws0 @ 3 ws0 þ @t@y2 @y @x@y2 @x @y3

ð45Þ

@hs0 1 @ws0 @hs0 1 @ws0 @hs0 þ l2 hs1 hf1 ¼ R @y @x R @x @y @t

ð46Þ

1 @ 2 /s1 @/s0 1 @ws0 @/s0 1 @ws0 @/s0 c/s1 ¼ þ 2 @t Sc2 @y R @y @x R @x @y

ð47Þ

M2

ð48Þ

@wf1 @ws1 ¼ 0; ¼ 0; hf1 ¼ 0; hs1 ¼ 0; /f1 ¼ 0; @y @y /s1 ¼ 0 at y ¼ g

y ¼ g

ð49Þ

wf0 ¼ A3 þ B3 y þ A4 cosh a1 y þ B4 sinh a1 y

ð51Þ

uf0 ¼ B3 þ A4 a1 sinh a1 y þ B4 a1 cosh a1 y

ð52Þ

ws0 ¼ T1 A4 cosh a1 y þ T1 B4 sinh a1 y þ Ey þ F

ð53Þ

us0 ¼ T1 A4 a1 sinh a1 y þ T1 B4 a1 cosh a1 y þ E

ð54Þ

1

1

1

0.5

0.5

0.5

y

0

a(i)

y0

a(ii)

y0 -0.5

-0.5

ð50Þ

Solving Eqs. (33)–(38) and (42)–(47) together with the boundary conditions (39)–(41) and (48)–(50), we get the expression for stream functions, velocity, temperature and concentration of fluid and solid particles

with the boundary conditions, @wf1 @ws1 ¼ 0; ¼ 0; hf1 ¼ 0; hs1 ¼ 0; /f1 ¼ 0; @y @y /s1 ¼ 0 at y ¼ g

@wf1 ¼ 0 at @y

b(i)

-0.5

M=1 -1 0

0.1

0.05

uf 0.15

0.2

0.25

-1 0

0.3

0.05

0.1

us 0.15

0.2

-1 0

0.3

0.25

1

1

1

0.5

0.5

0.5

y0

b(ii)

y

-0.5

0

y

c(i)

0.05

us

0.1

0.15

-1 0

0.2

0.05

uf

0.1

0.15

1

0.5

0.5

0.5

d(i)

y

-0.5

-1 0

0

y

d(ii)

1

uf

1.5

2

-1 0

0.25

0.3

c(ii)

0.01

us 0.03

0.04

0.05

0.15 u 0.2

0.25

0.3

0.02

(e)

0

-0.5

-0.5

0.5

0.2

M=2 -1 0

1

0

uf 0.15

M=2

1

y

0.1

-0.5

-0.5

M=1 -1 0

0

0.05

0.5

us

1

1.5

-1 0

0.05

0.1

f

Figure 2 Velocity distribution (c ¼ 0:5, R = 1, A = 0.5, Pr ¼ 0:71, K1 ¼ 1, K2 ¼ 1, Ec ¼ 0:2, Sc1 ¼ 0:5, E3 ¼ 0:5, E5 ¼ 0:2, Sc2 ¼ 0:5, N0 ¼ 10, sT ¼ 1, sv ¼ 1, l1 ¼ 0:1, l2 ¼ 2, e ¼ 0:001) (–) E1 ¼ 0:2, ( ) E1 ¼ 0:4, ( ) E1 ¼ 0:6, ( ) E1 ¼ 0:9, M = 1, E2 ¼ 0:2, E4 ¼ 0:1, (b) (–) E2 ¼ 0:2, ( ) E2 ¼ 0:5, ( ) E2 ¼ 0:7, ( ) E2 ¼ 1, E4 ¼ 0:1, E1 ¼ 0:01, (c) (–) E2 ¼ 0:2, ( ) E2 ¼ 0:5, ( ) E2 ¼ 0:7, ( ) E2 ¼ 1, E4 ¼ 0:1, E1 ¼ 0:01, (d) (–) E4 ¼ 0:1, ( ) E4 ¼ 0:3, ( ) E4 ¼ 0:5, ( ) E4 ¼ 0:7, M = 1, E2 ¼ 0:2, (e) (–) P = 0, ( ) P = 0.5, ( ) P = 1, ( ) P = 1.5, M = 1, E1 ¼ 0:2; E2 ¼ 0:5, E4 ¼ 0:01.


6

R. Muthuraj et al. 1

1

0.5

0.5

y

0

a(i)

y

-0.5

-1 0

a(ii) 0

-0.5

0.2

0.4

θ

0.6

0.8

-1 0

1

f

1

1

0.5

0.5

y

0.4

θ

0.4

θs

s

0.6

0.8

1

0.6

0.8

1

b(ii)

b(i)

y

0

-0.5

-1 0

0.2

0

-0.5

0.2

0.4

θf

0.6

0.8

-1 0

1

0.2

Figure 3 Temperature distribution, (c ¼ 0:5, R = 1, A = 1, K1 ¼ 1, K2 ¼ 1, Sc1 ¼ 0:5, Sc2 ¼ 0:5, M = 2, E1 ¼ 0:01, E2 ¼ 0:5; E3 ¼ 0:2; E4 ¼ 0:1; E5 ¼ 0:6, t = 1, N0 ¼ 10, l1 ¼ 0:1, l2 ¼ 2; sT ¼ 1; sv ¼ 1; e ¼ 0:001), (a) (–) Ec ¼ 1, ( ) Ec ¼ 3, ( ) Ec ¼ 5, ( ) Ec ¼ 7, Pr ¼ 1, (b) (–) Pr ¼ 1, ( )Pr ¼ 3, ( ) Pr ¼ 5, ( ) Pr ¼ 7, Ec ¼ 1.

1

1

0.5

y

0.5

a(i)

a(ii) y

0

-0.5

-1 0

0

-0.5

0.2

0.4

φf

0.6

0.8

-1 0

1

1

φs 0.6

0.8

1

0.5

b(i)

b(ii) y

0

-0.5

-1 0

0.4

1

0.5

y

0.2

0

-0.5

0.2

0.4

φf

0.6

0.8

1

-1 0

0.2

0.4

φ

s

0.6

0.8

1

Figure 4 Concentration distribution, (R = 1, A = 1, K1 ¼ 1, K2 ¼ 1, N0 ¼ 10, M = 2, E1 ¼ 0:01, E2 ¼ 0:5, E3 ¼ 0:2; E4 ¼ 0:1; E5 ¼ 0:6, t = 1, l1 ¼ 0:1, l2 ¼ 2, Ec ¼ 0:2, Pr ¼ 1, sT ¼ 1, sv ¼ 1, e ¼ 0:001) (a) (–) c ¼ 0:5, ( ) c ¼ 0:7, ( ) c ¼ 1, ( ) c ¼ 1:5, Sc1 ¼ 0:5, Sc2 ¼ 0:5, (b) (i) (–) Sc1 ¼ 0:1, ( ) Sc1 ¼ 0:5, ( ) Sc1 ¼ 0:7, ( ) Sc1 ¼ 1, Sc2 ¼ 0:5, c ¼ 0:5, (ii) (–) Sc2 ¼ 0:1, ( ) Sc2 ¼ 0:5, ( ) Sc2 ¼ 0:7, ( ) Sc2 ¼ 1, Sc1 ¼ 0:5, c ¼ 0:5.



7 4

6 4

(a)

(b)

2

2

Z0

Z0

-2

-2 -4 -6 0

0.2

x

0.4

0.6

0.8

-4 0

1

0.4

x

0.6

0.8

1

x

0.6

0.8

1

1

2

(c)

1

Z

0

-1

-0.5

-2

-1

0.2

(d)

0.5

Z0

-3 0

0.2

x

0.4

0.6

0.8

-1.5 0

1

0.2

0.4

2

(e)

1

Z

0 -1 -2 -3 0

0.2

0.4

x

0.6

0.8

1

Figure 5 Coefficient of heat transfer rate, (c ¼ 5, R = 1, A = 0.2, K1 ¼ 1, K2 ¼ 1, Sc1 ¼ 2, Sc2 ¼ 2, M = 0.2, t = 1, l1 ¼ 0:1; l2 ¼ 0:1; Ec ¼ 1; sT ¼ 1; sv ¼ 1, E5 ¼ 1; N0 ¼ 1; e ¼ 0:001), (a) (–) E1 ¼ 0:5, ( ) E1 ¼ 0:7, ( ) E1 ¼ 0:9, ( ) E1 ¼ 1, Pr ¼ 1, E2 ¼ 1, E3 ¼ 0:7, E4 ¼ 0:01, (b) (–) E2 ¼ 1, ( ) E2 ¼ 1:2, ( ) E2 ¼ 1:4, ( ) E2 ¼ 1:6, Pr ¼ 1, E3 ¼ 0:7, E4 ¼ 0:01, E1 ¼ 0:5, (c) (–) E3 ¼ 0:1, ( ) E3 ¼ 0:5, ( ) E3 ¼ 1, ( ) E3 ¼ 2, Pr ¼ 1, E2 ¼ 1, E4 ¼ 0:01, E1 ¼ 0:5, (d) (–) E4 ¼ 0:01, ( ) E4 ¼ 0:015, ( ) E4 ¼ 0:2, ( ) E4 ¼ 0:025, Pr ¼ 1, E2 ¼ 1, E3 ¼ 0:7, E1 ¼ 0:5, (e) (–) Pr ¼ 0:71, ( )Pr ¼ 1, ( ) Pr ¼ 1:5, ( ) Pr ¼ 2, E2 ¼ 1, E3 ¼ 0:7, E4 ¼ 0:01, E1 ¼ 0:5.

hf0 ¼ A5 þ B5 y þ T11 cosh a1 y þ T12 sinh a1 y þ T13 cosh 2a1 y þ T14 sinh 2a1 y þ T15 y2

ð55Þ

þ T13 cosh 2a1 y þ T14 sinh 2a1 y þ T15 y /f0 ¼ A1 cosh ay þ B1 sinh ay þ

K1 Sc1 a2

K2 Sc2 /s0 ¼ A2 cosh by þ B2 sinh by þ 2 b

þ sinh a1 yðB9 þ T57 y þ T59 y2 Þ

ð59Þ

uf1 ¼ B8 þ sinh a1 yðA9 a1 þ T57 þ T62 y þ a1 T60 y2 Þ

hs0 ¼ A5 þ B5 y þ T11 cosh a1 y þ T12 sinh a1 y 2

wf1 ¼ A8 þ B8 y þ cosh a1 yðA9 þ T58 y þ T60 y2 Þ

þ cosh a1 yðB9 a1 þ T58 þ T61 y þ a1 T59 y2 Þ ð56Þ ð57Þ ð58Þ

ð60Þ

ws1 ¼ cosha1 yðA9 T100 þ T102 þ T103 y þ T106 y2 Þ þ sinha1 yðB9 T100 þ T104 þ T101 y þ T105 y2 Þ þ Gy þ H

ð61Þ

us1 ¼ sinh a1 yðA9 T93 þ T95 þ T96 y þ T99 y2 Þ þ cosh a1 yðB9 T93 þ T97 þ T94 y þ T98 y2 Þ þ G

ð62Þ


8

R. Muthuraj et al.

Figure 6 Effects of E1 , E4 , E5 and M on Stream line pattern, (c ¼ 5, R = 1, A = 0.2, K1 ¼ 1, K2 ¼ 1, Sc1 ¼ 2, Sc2 ¼ 2, M = 0.2, t = 1, l1 ¼ 0:1; l2 ¼ 0:1; Ec ¼ 1; sT ¼ 1; sv ¼ 1; E5 ¼ 1; N0 ¼ 1; e ¼ 0:001), (a) (i) E1 ¼ 0:05 (ii) E1 ¼ 0:1; (b) (i) E4 ¼ 0:05 (ii) E4 ¼ 0:1; (c) (i)E5 ¼ 0:05 (ii) E5 ¼ 0:1 (d) (i) M = 0.1 (ii) M = 0.5.

hf1 ¼ A10 þ B10 y þ cosh a1 yðB8 T161 þ B9 T162 þ GT163

hs1 ¼ A10 þ B10 y þ cosh a1 yðB8 T161 þ B9 T162 þ GT163

þ T164 þ T172 y þ T184 y Þ þ sinh a1 yðB8 T165

þ T192 þ T197 y þ T203 y2 Þ þ sinh a1 yðB8 T165

þ GT166 þ A9 T167 þ T168 þ T171 y þ T183 y2 Þ

þ GT166 þ A9 T167 þ T193 þ T196 y þ T202 y2 Þ

þ sinh 2a1 yðT170 þ B9 T173 þ A9 T174 þ T179 y

þ cosh 2a1 yðT194 þ A9 T175 þ B9 T176 þ T201 y

þ T181 y Þ þ cosh 2a1 yðA9 T175 þ B9 T176 þ T169

þ T182 y2 Þ þ sinh 2a1 yðT195 þ B9 T173 þ A9 T174

þ T180 y þ T182 y Þ þ T177 sinh 3a1 y þ T178 cosh 3a1 y

þ T200 y þ T181 y2 Þ þ T198 sinh 3a1 y þ T199 cosh 3a1 y

þ T185 y4 þ T186 y3 þ y2 ðGT187 þ B8 T188 þ T189

þ T185 y4 þ T186 y3 þ y2 ðGT187 þ B8 T188 þ T204

2

2

2

þ A9 T190 þ B9 T191 Þ

ð63Þ

þ A9 T190 þ B9 T191 Þ þ T205 y þ T206

ð64Þ



9

/f1 ¼ cosh ayðA6 þ T26 y T32 y T33 y2 Þ þ sinh ayðB6 þ T25 y þ T34 y þ T31 y2 Þ þ T27 sinhða1 þ aÞy þ T28 sinhða1 aÞy þ T29 coshða1 þ aÞy þ T30 coshða1 aÞy

ð65Þ

/s1 ¼ cosh byðA7 þ T46 y þ S3 y2 Þ þ sinh byðB7 þ T47 y þ S4 y2 Þ þ T42 sinhða1 þ bÞy þ T43 sinhðb a1 Þy þ T44 coshða1 þ bÞy þ T45 coshðb a1 Þy qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi P where a ¼ cSc1 ; b ¼ cSc2 ; a1 ¼ P þ M2 1þAM 2.

ð66Þ

The coefficient of heat transfer rate at the wall is given by Zf ¼ Zf0 þ dZf1

ð67Þ

where Zf0 ¼ gx hf0y ; Zf1 ¼ hf0x þ gx hf1y . 4. Results and discussion This section presents the graphical results in order to discuss the quantitative effects of the parameters involved in the analysis. In Figs. 2–7, we have shown the effects of Hartmann number (M), Schmidt number ðSc Þ, amplitude parameter ðdÞ and the elastic parameters E1 ; E2 ; E3 ; E4 and E5 . Fig. 2 shows the influences of E1 ; E2 ; E4 , P and M on velocity field. The effect of different values of E1 is graphed in Fig. 2a. It shows that the effect of increasing the above-mentioned elastic parameter leads to decrease the velocity throughout the channel. In Fig. 2b and c, fluid velocity is decreasing function with increasing the parameters E2 and M. This is because of the presence of the transverse magnetic field which creates a resistive force similar to the drag force that acts in the opposite direction of the fluid motion, thus causing the velocity of the fluid to decrease. An opposite result is true for increasing elastic parameter E4 , which is shown in Fig. 2d. Fig. 2e confirms that, increasing the material parameter P leads to decrease the fluid velocity which means that velocity of nonNewtonian fluid is significantly reduced as compared to Newtonian fluid (P = 0). To see the effects of Ec and Pr on temperature distribution, we have plotted Fig. 3. It is observed that the temperature profiles are linear for lower values of the parameters Ec and Pr while it becomes parabolic for higher values. Further, temperature is gradually enhanced with increasing the 1

0.5

y0 -0.5

-1 0

0.005

0.01

uf 0.015

0.02

0.025

Figure 7 Velocity distribution, (x = 1, t = 1, l1 ¼ 0:1, l2 ¼ 0:1, sT ¼ 1, sv ¼ 1, Ec ¼ 0:1, N0 ¼ 10, Pr ¼ 0:1, e ¼ 0:001, E3 ¼ 0:05, E1 ¼ 0:07, (–) E2 ¼ 0:01, (–) E2 ¼ 0:05, (–) E2 ¼ 0:1, (–) E2 ¼ 0:15, E4 ¼ 0, E5 ¼ 0), O__ Results of Eldabe et al. [25] (Non-porous case), +__ Results of the Present paper (when M; E4 ; E5 ! 0).

values of the parameters. The influence of the parameters of c and Sc on concentration distribution is displayed in Fig. 4. Fig. 4a is plotted for various values of chemical reaction parameter c. It shows that increasing c enhances the fluid concentration. The opposite trend is seen for the case when Sc is increased as noted in Fig. 4b. To see the influence of the parameters E1 ; E2 ; E3 ; E4 and Pr on coefficient of heat transfer (Z), we have displayed in Fig. 5. It shows the oscillatory behavior of heat transfer which may be due to peristalsis. It depicts that maximum amplitude of Z enhanced with increasing E1 ; E2 ; E3 , and Pr while the opposite is true for increasing E4 . The formation of an internally circulating bolus of fluid by closed streamlines is shown in Fig. 6. This trapped bolus pushed a head along peristaltic waves. The aim of Fig. 6 was to examine the influence of different parameters on trapping. Fig. 6a shows that the trapped bolus decreases with increasing elastic parameter E1 . The influence of E4 on trapping is displayed in Fig. 6b. It depicts that streamlines are increased with an increase of E4 . The quite opposite effect can be noticed when increasing the parameter E5 , which is shown in Fig. 6c. Fig. 6d shows the opposite effect of Fig. 6b, if E4 is replaced by magnetic parameter. Furthermore, our results are in good agreement with the results of Eldabe et al. [25] (for nonporous case) by choosing M ! 0; E4 ! 0; E5 ! 0 (see Fig. 7). 5. Conclusion In this section, numerical calculations have been performed in order to see the dependence of heat and mass transfer characteristics of MHD peristaltic transport of a dusty fluid through a horizontal channel for both fluid and solid particles. Analytical solutions for the problem are obtained by using perturbation technique for both fluid and solid particles. The features of flow, heat and mass transfer characteristics are presented and discussed graphically. The effects of pertinent parameters on flow, heat and mass transfer characteristics have been studied. Investigation of such analysis is of utmost importance owing to its wide range of applications in engineering and biological systems. In particular, it may serve for the intrauterine fluid motion in a sagittal cross section of the uterus under cancer therapy and drug analysis, the transport of lymph in the lymphatic vessels, and the vasomotion in small blood vessels such as arterioles, venues, and capillaries [1–14]. The main observations of the presented attempt may be summarized as follows: The effects of different values of E1 ; E2 and ‘M’ lead to suppressing the fluid velocity while the opposite trend can be seen with increasing E4 . Increasing values of Ec and P r , the fluid temperature profiles are parabolic in nature whereas they are linear for lower values of thermal parameters. Further, temperature is gradually enhanced with increasing these parameters. Increasing chemical reaction parameter enhanced the fluid concentration but increasing Schmidt number decreases the concentration of the fluid. The trapped bolus decreases and lines are also reduced with an increase in magnetic parameter while the elastic parameter E4 on trapping is opposite to magnetic parameter.


10

R. Muthuraj et al.

The results of hydrodynamics can be obtained as a limiting case of our analysis by taking M ! 0.

T1 A4 a1 A2x T1 B4x B2 b T1 B4 a1 B2x T1 A4x A2 b T36 ¼ Sc2 þ ; 2R 2R 2R 2R T1 A4 a1 A2x T1 B4x B2 b T1 B4 a1 B2x T1 A4x A2 b ; þ T37 ¼ Sc2 2R 2R 2R 2R

Appendix R ; 1 þ AM2 N0 l1 Ec Pr T1 A4 a21 2T1 B4 B3 a1 2EB4 a1 ; þ A4 a21 T2 ¼ 2 sv R R R T1 ¼

T3 ¼

N0 l1 Ec Pr T1 B4 a21 2T1 A4 B3 a1 2EA4 a1 2 ; þ B a 4 1 sv R R R2

T4 ¼

N0 l1 Ec Pr 2T1 A24 a21 ; sv R

T5 ¼

N0 l1 Ec Pr 4T1 A4 B4 a21 ; sv R

N0 l1 Ec Pr 2T1 B24 a21 N0 l1 Ec Pr 2EB3 ; T7 ¼ ; sv R sv R T4 T6 T5 ; T9 ¼ ; T8 ¼ 2 2 2 T6 ¼

T4 T6 T2 T7 ; T11 ¼ 2 ; T10 ¼ 2 2 a1 T3 T8 T12 ¼ 2 ; T13 ¼ 2 ; a1 4a1

T38 ¼ Sc2

T1 A2 bB4x T1 B2x A4 a1 T1 B4 a1 A2x T1 A4x B2 b þ ; T39 ¼ Sc2 2R 2R 2R 2R Fx E T40 ¼ Sc2 B2t A2 b þ B2x ; R R Fx E T41 ¼ Sc2 A2t B2 b þ A2x ; R R Ex Ex A2 b ; S2 ¼ Sc2 B2 b ; S1 ¼ Sc2 R R T42 ¼

T36 ; a21 þ 2a1 b

T43 ¼

T37 ; a21 2a1 b

T44 ¼

T38 ; a21 þ 2a1 b

T45 ¼

T39 ; a21 2a1 b

T46 ¼

T40 S2 ; 2b 4b2

T47 ¼

T41 S1 ; 2b 4b2

S3 ¼

T9 T10 ; ; T ¼ T14 ¼ 15 4a21 2 PT1 a1 Pa1 M2 a1 ; T16 ¼ a31 þ R

T1 A4 a1 B2x T1 B4x A2 b T1 B4 a1 A2x T1 A4x B2 b þ ; 2R 2R 2R 2R

S1 ; 4b

S4 ¼

S2 ; 4b

T49 ¼ ðRT1 A4t a21 þ ET1 A4x a21 Fx T1 B4 a31 Þ; T50 ¼ ðRT1 B4t a21 þ ET1 B4x a21 Fx T1 A4 a31 Þ;

T17 ¼ Sc1 ðA1t þ B3 A1x Þ;

T51 ¼ ðEx T1 B4 a31 Þ;

T52 ¼ ðEx T1 A4 a31 Þ;

T18 ¼ Sc1 ðB1t þ B3 B1x Þ; A4 a1 A1x B4x B1 a þ B1x B4 a1 A4x A1 a ; T19 ¼ Sc1 2

T53 ¼

A4 a1 A1x B4x B1 a B1x B4 a1 þ A4x A1 a T20 ¼ Sc1 ; 2

T54 ¼

A4 a1 B1x B4x A1 a þ A1x B4 a1 A4x B1 a ; T21 ¼ Sc1 2 B4 a1 A1x A4x B1 a B1x A4 a1 þ B4x A1 a ; T22 ¼ Sc1 2

P 3 T B B a ; 51 3x 4 1 Rð1 þ AM2 Þ P T52 B3x A4 a31 ; T56 ¼ Rð1 þ AM2 Þ

T23 ¼ Sc1 ðB3x B1 aÞ;

T24 ¼ Sc1 ðB3x A1 aÞ;

T17 T18 T19 ; T26 ¼ ; T27 ¼ 2 ; 2a 2a a1 þ 2aa1 T20 T21 ; T29 ¼ 2 ; T28 ¼ 2 a1 2aa1 a1 þ 2aa1 T25 ¼

T22 T23 ; ; T31 ¼ a21 2aa1 4a T24 T24 ; T34 ¼ 2 ; T33 ¼ 4a 4a

T30 ¼

T32 ¼

T23 ; 4a2

P 2 2 3 T ; þ A a þ B A a A B a 49 4t 3 4x 3x 4 1 1 1 Rð1 þ AM2 Þ

P 2 2 3 þ B a þ B B a A A a T 50 4t 1 3 4x 1 3x 4 1 ; Rð1 þ AM2 Þ

T55 ¼

T58 ¼

T54 5T55 4 ; 3 2a1 4a1

T59 ¼

T61 ¼ ða1 T57 þ 2T60 Þ;

T55 ; 4a31

T60 ¼

T56 ; 3 4a1

T62 ¼ ða1 T58 þ 2T59 Þ;

T63 ¼ ða1 T61 þ 4a1 T60 Þ;

T64 ¼ ða1 T57 þ T61 Þ;

T65 ¼ ða1 T62 þ 4a1 T59 Þ;

T66 ¼ ða1 T58 þ T62 Þ;

T67 ¼ a1 T63 þ 2a21 T60 ; T69 ¼ a1 T65 þ 2a21 T59 ;

T53 5T56 ; 2a31 4a41

T57 ¼

T68 ¼ ða1 T64 þ T63 Þ; T70 ¼ ða1 T66 þ T65 Þ;



11

1 T1 B4 a1 A5x þ ET11 T71 ¼ T11 R T1 B4 T13 a1 T1 A4 T14 a1 B5 T1 A4x Fx T12 a1 þ 2 T1 A4x T14 a1 þ T1 B4x T13 a1 ; T72 ¼

T73 ¼

T74 ¼

1 T1 A4 a1 A5x þ ET12 T12 R T1 B4 T14 a1 T1 A4 T13 a1 B5 T1 B4x Fx T11 a1 þ 2 T1 A4x T13 a1 T1 B4x T14 a1 ; 1 T1 A4 T12 a1 þ T1 B4 T11 a1 T13 ET13 þ 2 R T1 A4x T12 a1 þ T1 B4x T11 a1 ; 2Fx T14 a1 2 1 T1 A4 T11 a1 þ T1 B4 T12 a1 T14 ET14 þ 2 R T1 A4x T11 a1 þ T1 B4x T12 a1 ; 2Fx T13 a1 2

1 T75 ¼ ðT1 A4 a1 B5x 2T1 B4x T15 Ex T11 a1 Þ ; R 1 T76 ¼ ðT1 B4 a1 B5x 2T1 A4x T15 Ex T12 a1 Þ ; R 1 T77 ¼ ðT1 A4 a1 T15 Þ ; R

1 T78 ¼ ðT1 B4 a1 T15 Þ ; R

1 T1 A4 T13 a1 þ T1 B4 T14 a1 T1 A4x T13 a1 T1 B4x T14 a1 ; T79 ¼ R 2

T80 ¼

1 T1 A4 T14 a1 þ T1 B4 T13 a1 R 2 2Ex T13 a1 T1 A4x T14 a1 T1 B4x T13 a1 ; T81 ¼ ; R

2Ex T14 a1 ET15 2Ex T15 ; T83 ¼ T15 þ ; R R R EB5x Ex B5 þ 2Fx T15 þ ; T84 ¼ B5t R R

T82 ¼

1 1 T1 A4 a1 T12 T1 B4 a1 T11 T85 ¼ A5t ðEA5x Þ þ R R 2 1 T1 A4x a1 T12 T1 B4x a1 T11 þ Fx B5 ; þ R 2 Ra21 RT63 T52 ; T87 ¼ ; T86 ¼ 2 1 þ AM 1 þ AM2 RT64 T49 RT65 T51 T88 ¼ ; T89 ¼ ; 1 þ AM2 1 þ AM2

RT59 a21 RT60 a21 T86 ; T92 ¼ ; T93 ¼ ; 2 a1 1 þ AM 1 þ AM2 T87 2T92 T87 2T92 T88 ; 2 ; T95 ¼ þ þ T94 ¼ a1 a1 a21 a31 a1 T91 ¼

T89 2T91 T89 2T91 T90 ; 2 ; T97 ¼ þ 3 þ T96 ¼ a1 a1 a21 a1 a1 T91 T92 T93 ; T99 ¼ ; T100 ¼ ; T98 ¼ a1 a1 a1 T94 2T99 2 ; a1 a1 T96 2T98 ¼ 2 ; a1 a1

T94 T95 2T99 ; þ þ a2 a1 a31 1 T96 T97 2T98 ¼ þ þ 3 ; a21 a1 a1

T101 ¼

T102 ¼

T103

T104

T98 T99 2N0 l1 Ec ; T106 ¼ ; T107 ¼ ; a1 a1 Rsv ¼ PrT107 T1 B4 a1 ; T109 ¼ PrðT107 Ea1 þ T107 T93 B3 Þ;

T105 ¼ T108

T110 ¼ PrT107 B4 a1 ; T111 ¼ Pr ½T11 þ B3 T11 þ B4 a1 A5x ðA3x T12 a1 þ A4 B5 Þ Pr T71 N0 l1 þ T107 ET58 þ T107 T97 B3 ; sT l2 T112 ¼ PrT107 T1 A4 a1 ;

T113 ¼ PrT107 A4 a1 ;

T114 ¼ Pr ½T12 þ B3 T12 þ A4 a1 A5x ðA3x T11 a1 þ B4 B5 Þ Pr T72 N0 l1 ; þ T107 ET57 þ T107 T95 B3 sT l2 T115 T116

T73 N0 l1 ; ¼ Pr T13 þ B3 T13 2A3x T14 a1 sT l2 T74 N0 l1 ; ¼ Pr T14 þ B3 T14 2A3x T13 a1 sT l2

T117 ¼ Pr A4 a1 B5x ðB3x T11 a1 þ 2B4 T15 Þ T75 N0 l1 ; þ T107 ðET62 þ T96 B3 Þ s T l2 T118 ¼ Pr ½B4 a1 B5x ðB3x T12 a1 þ 2A4 T15 Þ T76 N0 l1 ; þ T107 ðET61 þ T94 B3 Þ s T l2 T119 ¼ Pr T107 ½T1 A4 a21 þ T93 A4 a1 ; T120 ¼ Pr T107 ½T1 B4 a21 þ T93 B4 a1 ; T121 ¼ Pr ½T107 ðT1 A4 a1 T58 þ T1 B4 a1 T57 þ B4 a1 T95 þ A4 a1 T97 Þ; T122 ¼ Pr ½A4 a1 T12 B4 a1 T11 þ T107 ðT1 A4 a1 T57 þ A4 a1 T95 Þ; T123 ¼ Pr ½B4 a1 T11 A4 a1 T12 þ T107 ðT1 B4 a1 T58 þ B4 a1 T97 Þ;

RT66 T50 T90 ¼ ; 1 þ AM2

T124 ¼ Pr ðA4 a1 T13 2B4 T14 a1 Þ; T125 ¼ Pr ðB4 a1 T13 2A4 T14 a1 Þ;


12

R. Muthuraj et al.

T126 ¼ Pr ðB4 a1 T14 2A4 T13 a1 Þ; T77 N0 l1 T127 ¼ Pr A4 a1 T15 þ T107 ðEa1 T60 þ T99 B3 Þ ; s T l2 T128 T129

T131 T132

T78 N0 l1 ¼ Pr B4 a1 T15 þ T107 ðEa1 T59 þ T98 B3 Þ ; sT l2 T79 N0 l1 T80 N0 l1 ; T130 ¼ Pr ; ¼ Pr sT l2 sT l2 T81 N0 l1 ; ¼ Pr 2B3x a1 T13 sT l2 T82 N0 l1 ; ¼ Pr 2B3x a1 T14 sT l2

T133 ¼ Pr ½T107 ðT1 A4 a1 T61 þ T1 B4 a1 T62 þ T94 A4 a1 þ T96 B4 a1 Þ;

T163 T165

T110 ¼ ; a2 1 T112 ; ¼ a21

T145 2T118 6T127 ; þ a2 a31 a41 1 T146 T154 3T157 ; ¼ 3 þ 4a2 4a1 8a41 1 T147 T155 3T156 ; ¼ 3 þ 4a21 4a1 8a41

T169 T170

T171 T173

T117 4T128 T118 4T127 ¼ 3 ; T172 ¼ 3 ; a2 a1 a21 a 1 1 T148 T149 T150 ; T174 ¼ ; T175 ¼ ; ¼ 4a21 4a21 4a21 T151 T152 T153 ¼ ¼ ; T ; T ; 177 178 4a2 9a21 9a21 1 T154 2T157 T155 T156 ; T180 ¼ ¼ 3 ; 4a21 4a31 4a21 2a1

T135 ¼ Pr ½T107 ðT1 A4 a21 T59 þ T1 B4 a21 T60 þ T98 A4 a1 þ T99 B4 a1 Þ;

T176 ¼

T136 ¼ Pr ðT94 B4 a1 Þ;

T179

T138 ¼ Pr ½T107 ðT1 B4 a21 T59 þ T98 B4 a1 Þ; T83 N0 l1 T139 ¼ Pr T15 þ B3 T15 2B3x T15 ; sT l2 T140

T84 N0 l1 ; ¼ Pr B5t ð2A3x T15 þ B3x B5 Þ þ B3 B5x sT l2

T141 ¼ Pr T107 B3 ;

T142 ¼ Pr ET107 ;

T85 N0 l1 T143 ¼ Pr A5t þ B3 A5x A3x B53 ; sT l2 T125 T124 T126 ; T145 ¼ T114 þ ; T144 ¼ T111 þ 2 2 2 T122 T123 T121 þ ; T147 ¼ T116 þ ; T115 þ 2 2 2 T119 T120 T119 ¼ ; T149 ¼ ; T150 ¼ ; 2 2 2

T146 ¼ T148

T120 T124 T126 ; T152 ¼ þ þ T129 ; 2 2 2 T125 T133 þ T130 ; T154 ¼ T131 þ ; ¼ 2 2

T151 ¼ T153

T155 T157

T159 T160 T161

T134 T136 T135 þ ; T156 ¼ ; ¼ T132 þ 2 2 2 T137 T138 T137 T138 þ ; T158 ¼ þ þ T139 ; ¼ 2 2 2 2 T134 T136 þ þ T140 ; ¼ 2 2 T122 T123 þ þ T143 ; ¼ 2 2 T108 T109 ; T162 ¼ ; ¼ a21 a21

T166

T168 ¼

T134 ¼ Pr ½T107 ðT1 A4 a1 T62 þ T96 A4 a1 Þ;

T137 ¼ Pr ½T107 ðT1 A4 a21 T60 þ T99 A4 a1 Þ;

T164

T144 2T117 6T128 ¼ þ ; a2 a31 a4 1 1 T113 T109 ; T167 ¼ ¼ a21 a21

T156 ; 4a2 1 T128 ; ¼ a21

T157 ; 4a2 1 T158 ¼ ; 12

T127 ; a2 1 T159 ¼ ; 6

T181 ¼

T182 ¼

T183 ¼

T184

T185

T186

T141 ; 2 T150 ; ¼ 2

T142 ; 2 T151 ; ¼ 2

T160 ; 2 T71 ; ¼ T164 þ l2

T187 ¼

T188 ¼

T189 ¼

T190

T191

T192

T72 ; T168 þ l2 T74 ; ¼ T170 þ l2

T73 ; l2 T75 ; ¼ T171 þ l2

T193 ¼

T194 ¼

T195

T196

T197 T199

T76 ; ¼ T172 þ l2 T80 ; ¼ T178 þ l2 T82 T180 þ ; l2 T78 ; ¼ T184 þ l2

T169 þ

T198 T200

T79 ; ¼ T177 þ l2 T81 ; ¼ T179 þ l2

T77 ; l2 T83 ; ¼ T189 þ l2

T201 ¼

T202 ¼

T203

T204

T183 þ

T205 ¼

T84 ; l2

T85 P ; T207 ¼ a31 þ T93 Pa1 M2 a1 ; R l2 P ¼ T67 B3x A4 a21 þ T94 PT61 M2 T61 ; R

T206 ¼ T208

T209 ¼ ðT68 þ B3 A4x a1 þ A4 a1 B3x þ

P T95 PT57 A3x B4 a21 R

A4t a1 M2 T57 Þ; T210 ¼ ðT69 B3x B4 a21 PT62 M2 T62 Þ;


MHD Peristaltic transport of a Dusty fluid T211

13

P ¼ T70 þ B3 B4x a1 þ B4 a1 B3x þ T97 PT58 A3x A4 a21 R ! B4t a1 M2 T58 ;

A6 ¼

P T59 a31 þ T98 Pa1 T59 M2 a1 T59 ; R P 3 ¼ T60 a1 þ T99 Pa1 T60 M2 a1 T60 ; R

B6 ¼

T217

T214 T215 þ ; ¼ 2 2

T218

þ sinhða1 þ aÞgT27 þ sinhða1 aÞgT28 ; T216 ¼ ðB3t þ B3 B3x Þ;

T214 T215 þ T216 ; ¼ 2 2

P T93 T207 a1 ðP þ M2 Þ cosh a1 g ; R P 2 ¼ T96 g þ T210 g þ ðP þ M ÞT62 g ; R

A7 ¼

1 ½coshða1 þ bÞgT44 þ coshðb a1 ÞgT45 cosh bg þ T47 g sinh bg þ S3 g2 cosh bg;

B7 ¼

T219 ¼ T220

1 ½sinh agðT25 g þ T34 g þ T31 g2 Þ þ coshða1 þ aÞgT29 sinh ag þ coshða1 aÞgT30 þ cosh agðA6 þ T26 g T32 g T33 g2 Þ

T214 ¼ ðA4 a21 A4x B4x B4 a21 Þ; T215 ¼ ðB4 a21 B4x A4x A4 a21 Þ;

1 ½sinh agðT25 g þ T34 gÞ þ coshða1 þ aÞgT29 cosh ag þ coshða1 aÞgT30 þ cosh agðT33 g2 Þ;

T212 ¼ T213

E ¼ T1 B4 a1 cosh a1 g; F ¼ T1 A4 ;

1 ½cosh bgðA7 þ T46 gÞ þ sinhða1 þ bÞgT42 þ sinhðb sinh bg a1 ÞgT43 þ coshða1 þ bÞgT44 þ coshðb a1 ÞgT45 þ T47 g sinh bg þ S3 g2 cosh bg þ S4 g2 sinh bg;

P T221 ¼ ðT97 þ T89 g2 Þ þ T211 þ T212 g2 þ ðP þ M2 ÞðT58 þ a1 T59 g2 Þ ; R

A8 ¼ A9 ;

a10 ¼ ½cosh a1 gðB8 T161 þ B9 T162 þ GT163 þ T184 g2 Þ;

B8 ¼

a11 ¼ ½cosh 2a1 gðA9 T175 þ B9 T176 þ T169 þ T182 g2 Þ; a12 ¼ ½g2 ðGT187 þ B8 T188 þ A9 T190 þ B9 T191 þ T189 Þ; b10 ¼ ½sinh a1 gðB8 T165 þ GT166 þ A9 T167 þ T202 g þ T193 Þ; 2

b11 ¼ ½sinh 2a1 gðB9 T173 þ A9 T174 þ T195 Þ; b12 ¼ ½T201 g cosh 2a1 g þ T181 g2 sinh 2a1 g þ T186 g3 þ T205 g; a2 2K1 Sc1 ; a2 ð2 cosh agÞ 1 K1 Sc1 1 þ A1 cosh ag ; B1 ¼ sinh ag a2

A1 ¼

A2 ¼

b2 2K2 Sc2 ; b2 ð2 cosh bgÞ

1 K2 Sc2 B2 ¼ þ A cosh bg ; 1 2 a2 sinh bg

1 P ½B9 T207 cosh a1 g þ G þ T210 g sinh a1 g R P þ M2 þ T211 cosh a1 g þ T212 g2 cosh a1 g þ T217 cosh 2a1 g þ T218 ;

A9 ¼

1 ½T61 g cosh a1 g þ T57 sinha1 g þ a1 T60 g2 sinha1 g; a1 sinha1 g

B9 ¼

1 ½T220 sinh a1 g þ T221 cosh a1 g þ T217 cosh 2a1 g þ T218 ; T219

A10 ¼ ½a10 þ T171 g sinh a1 g þ T179 g sinh 2a1 g þ a11 þ T178 cosh 3a1 g þ a12 ; 1 B10 ¼ ½b10 þ T197 g cosh a1 g þ b11 þ T198 sinh 3a1 g þ b12 ; g G ¼ ½B9 T93 cosh a1 g þ T96 g sinh a1 g þ T97 cosh a1 g þ T98 g2 cosh a1 g; H ¼ ðA9 T100 þ T102 Þ;

A3 ¼ A4 ; B3 ¼ B4 a1 cosh a1 g;

A4 ¼

ðE þ T1 B4 a1 cosh a1 gÞ ½ðE1 þ E2 Þ8p3 e cos2pðx tÞ 4p2 E3 e sin 2pðx tÞ 32p5 E4 ecos 2pðx tÞ 2pE5 e cos2pðx tÞ ; B4 ¼ ; T1 a1 sinha1 g cosha1 g RP T1 a1 T16 ðP þ M2 Þa1

1 A5 ¼ ½1 2ðT11 cosh a1 g þ T13 cosh 2a1 g þ T15 g2 Þ; 2 B5 ¼

1 ðA5 þ T11 cosh a1 g þ T12 sinh a1 g þ T13 cosh 2a1 g þ T14 sinh 2a1 g þ T15 g2 Þ ; g


14

R. Muthuraj et al.

References [1] Y.C. Fung, C.S. Yih, Peristaltic transport, Trans. ASME J. Appl. Mech. 35 (1968) 669–675. [2] C. Pozrikidis, A study of peristaltic flow, J. Fluid Mech. 180 (1987) 515–527. [3] L.M. Srivastava, V.P. Srivastava, Interaction of peristaltic flow with pulsatile flow in a circular cylindrical tube, J. Biomech. 18 (1985) 247–253. [4] E.F. El Shehawey, Kh.S. Mekheimer, Couple-stresses in peristaltic transport of fluids, J. Phys. D: Appl. Phys. 27 (1994) 1163–1170. [5] Kh.S. Mekheimer, Non-linear peristaltic transport through a porous medium in an inclined planner channel, J. Porous Media 6 (2003) 189–201. [6] M. Elshahed, M.H. Haroun, Peristaltic transport of Johnson– Segalman fluid under effect of a magnetic field, Math. Probl. Eng. 6 (2005) 663–677. [7] M.H. Haroun, Effect of Deborah number and phase difference on peristaltic transport of a third-order fluid in an asymmetric channel, Commun. Non-linear Sci. Numer. Simul. 12 (2007) 1464–1480. [8] T. Hayat, N. Ali, S. Asghar, Hall effects on peristaltic flow of a Maxwell fluid in a porous medium, Phys. Lett. A 363 (2007) 397. [9] M. Kothandapani, S. Srinivas, Nonlinear peristaltic transport of Newtonian fluid in an inclined asymmetric channel through a porous medium, Phys. Lett. A 372 (2008) 1265–1276. [10] T. Hayat, S. Noreen, Peristaltic transport of fourth grade fluid with heat transfer and induced magnetic field, CR Mec. 338 (2010) 518–528. [11] D. Tripathi, Peristaltic transport of a viscoelastic fluid in a channel, Acta Astronaut. 68 (2011) 1379–1385. [12] S. Srinivas, R. Muthuraj, Effects of chemical reaction and space porosity on MHD mixed convective flow in a vertical asymmetric channel with peristalsis, Math. Comput. Model. 54 (2011) 1213–1227. [13] A.M. Abd-Alla, S.M. Abo-Dahab, R.D. Al-Simery, Effect of rotation on peristaltic flow of a micropolar fluid through a porous medium with an external magnetic field, J. Magn. Magn. Mater. 348 (2013) 33–43. [14] M. Kothandapani, J. Prakash, Effect of radiation and magnetic field on peristaltic transport of nanofluids through a porous space in a tapered asymmetric channel, J. Magn. Magn. Mater. 378 (2015) 152–163. [15] T.K. Mittra, S.N. Prasad, On the influence of wall properties and Poiseuille flow in peristalsis, J. Boimech. 6 (1973) 681–693. [16] P. Muthu, B.V. Rathishkumar, P. Chandra, On the influence of wall properties in the peristaltic motion of micro polar fluid, ANZIAM J. 45 (2003) 245–260. [17] G. Radhakrishnamacharya, Ch. Srinivasulu, Influence of wall properties on peristaltic transport with heat transfer, CR Mec. 335 (2007) 369–373. [18] T. Hayat, Maryiam Javed, S. Asghar, MHD peristaltic motion of Johnson–Segalman fluid in a channel with compliant walls, Phys. Lett. A. 372 (2008) 5026–5036. [19] S. Srinivas, M. Kothandapani, The influence of heat and mass transfer on MHD peristaltic flow through a porous space with compliant walls, Appl. Math Comput. 213 (2009) 197–208. [20] N. Ali, T. Hayat, S. Asghar, Peristaltic flow of a Maxwell fluid in a channel with compliant walls, Chaos, Solitons Fract. 39 (2009) 407–416. [21] T. Hayat, S. Hina, The influence of wall properties on the MHD peristaltic flow of a Maxwell fluid with heat and mass transfer, Nonlinear Anal.: Real World Appl. 11 (2010) 3155–3169. [22] Kh.S. Mekheimer, A.N. Abdel-Wahab, Effect of wall compliance on compressible fluid transport induced by a

[23]

[24]

[25]

[26]

[27]

[28] [29]

[30]

[31]

[32]

[33] [34]

[35]

[36]

[37]

[38]

[39]

[40]

[41]

surface acoustic wave in a micro channel, Numer. Methods Partial Diffl. Eqs. 27 (2011) 621–636. R. Muthuraj, S. Srinivas, Effects of compliant wall properties and space porosity in the MHD peristaltic transport with heat and mass transfer in a vertical channel, Int. J. Appl. Math. Mech. 7 (2011) 97–112. S. Hina, T. Hayat, A. Alsaedi, Heat and mass transfer effects on the peristaltic flow of Johnson–Segalman fluid in a curved channel with compliant walls, Int. J. Heat Mass Transfer 55 (2012) 3511–3521. N.T. Eldabe, K.A. Kamel, G.M. Abd- Allah, S.F. Ramadan, Peristaltic motion with heat and mass transfer of a dusty fluid through a Horizontal porous channel under the effect of wall properties, Int. J. Res. Rev. Appl. Sci. 15 (2013) 300–311. N.T. Eldabe, A.S. Zaghrout, H.M. Shawky, A.S. Awad, effects of chemical reaction with heat and mass transfer on peristaltic motion of power-law fluid in an asymmetric channel with wall’s properties, Int. J. Res. Rev. Appl. Sci. 15 (2013) 280–292. T. Hayat, M. Javed, S. Asghar, A.A. Hendi, Wall properties and heat transfer analysis of the peristaltic motion in a power-law fluid, Int. J. Numer. Meth. Fluids 71 (2013) 65–79. T.G. Cowling, Magneto Hydrodynamics, Inter Science Publishers, New York, 1957. M.F. El-Amin, Magneto hydrodynamic free convection and mass transfer flow in micro polar fluid with constant suction, J. Magn. Magn. Mater. 234 (2001) 567–574. V.M. Soundalgekar, H.S. Takhar, MHD forced and free convective flow past a semi-infinite plate, AIAA J. 15 (1977) 457–458. J. Zueco, S. Ahmed, Combined heat and mass transfer by mixed convection MHD flow along a porous plate with chemical reaction in presence of heat source, Appl. Math. Mech. – Engl. Ed. 31 (2010) 1217–1230. J. Prathap Kumar, J.C. Umavathi, B.M. Biradar, Mixed convection of magneto hydrodynamic and viscous fluid in a vertical channel, Int. J. Non-Linear Mech. 46 (2011) 278–285. P.G. Saffman, On the stability of laminar flow of a dusty gas, J. Fluid Mech. 13 (1962) 120–128. K.K. Singh, Unsteady flow of a conducting dusty fluid through a rectangular channel with time dependent pressure gradient, Indian J. Pure Appl. Math. 8 (1976) 1124. P. Mitra, P. Bhattacharyya, Unsteady hydro magnetic laminar flow of a conducting dusty fluid between two parallel plates started impulsively from rest, Acta Mech. 39 (1981) 171–182. K. Vajravelu, J. Nayfeh, Hydro magnetic flow of a dusty fluid over a stretching sheet, Int. J. Nonlinear Mech. 27 (1992) 937– 945. N. Datta, D.C. Dalal, S.K. Mishra, Unsteady heat transfer to pulsatile flow of a dusty viscous incompressible fluid in a channel, Int. J. Heat Mass Transfer 36 (1993) 1783–1788. I.J. Uwanta, Oscillatory free convection flow of a conducting fluid with a suspension of spherical particles, in: G.O.S. Ekhaguere, O. Ugbebor (Eds.), Direction in Mathematics, Associated Book Makers Nig. Ltd., 1999, pp. 159–167. H.A. Attia, Unsteady MHD Couette flow and heat transfer of dusty fluid with variable physical properties, Appl. Math. Comput. 177 (2006) 308–318. H.A. Attia, Unsteady hydro magnetic Couette flow of dusty fluid with temperature dependent viscosity and thermal conductivity, Int. J. Non-Linear Mech. 43 (2008) 707–715. P. Eguı´ a, J. Zueco, E. Granada, D. Patinõ, NSM solution for unsteady MHD Couette flow of a dusty conducting fluid with variable viscosity and electric conductivity, Appl. Math. Model. 3 (2011) 303–316.


MHD Peristaltic transport of a Dusty fluid [42] H. Erol, Analyzing of the effect of the dusty fluid, anisotropy and layered of the wall on the wave propagation, Appl. Math. Model. 36 (2012) 2486–2497. [43] G.K. Ramesh, B.J. Gireesha, C.S. Bagewadi, MHD flow of a dusty fluid near the stagnation point over a permeable stretching sheet with non-uniform source/sink, Int. J. Heat Mass Transfer 55 (2012) 4900–4907. [44] G.M. Pavithra, B.J. Gireesha, Effect of internal heat generation/ absorption on dusty fluid flow over an exponentially stretching

15 sheet with viscous dissipation, J. Math. 2013 (2013) 583615– 583624. [45] A. Rasekh, D.D. Ganji, S. Tavakoli, H. Ehsani, S. Naeejee, MHD flow and heat transfer of a dusty fluid over a stretching hollow cylinder with a convective boundary condition, Heat Transfer Asian Res. 43 (2014) 221–232.
