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Research ArticleEffects of Heat and Mass Transfer on the PeristalticTransport of MHD Couple Stress Fluid through PorousMedium in a Vertical Asymmetric Channel

K. Ramesh and M. Devakar

Department of Mathematics, Visvesvaraya National Institute of Technology, Nagpur 440010, India

Correspondence should be addressed to M. Devakar; m [email protected]

Received 30 September 2014; Accepted 10 February 2015

Academic Editor: Jamshid M. Nouri

Copyright © 2015 K. Ramesh and M. Devakar. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

The intrauterine fluid flow due to myometrial contractions is peristaltic type motion and the myometrial contractions may occurin both symmetric and asymmetric directions. The channel asymmetry is produced by choosing the peristaltic wave train on thewalls to have different amplitude, and phase due to the variation of channel width, wave amplitudes and phase differences. Inthis paper, we study the effects of heat and mass transfer on the peristaltic transport of magnetohydrodynamic couple stress fluidthrough homogeneous porous medium in a vertical asymmetric channel. The flow is investigated in the wave frame of referencemoving with constant velocity with the wave. The governing equations of couple stress fluid have been simplified under the longwave length approximation. The exact solutions of the resultant governing equations have been derived for the stream function,temperature, concentration, pressure gradient, and heat transfer coefficients.Thepressure difference and frictional forces at both thewalls are calculated using numerical integration. The influence of diverse flow parameters on the fluid velocity, pressure gradient,temperature, concentration, pressure difference, frictional forces, heat transfer coefficients, and trapping has been discussed. Thegraphical results are also discussed for four different wave shapes. It is noticed that increasing of couple stresses and heat generationparameter increases the size of the trapped bolus.The heat generation parameter increases the peristaltic pumping and temperature.

1. Introduction

In recent years, the flow of non-Newtonian fluids has receivedmuch attention due to the increasing industrial, medical,and technological applications. Various researchers haveattempted diverse flow problems related to several non-Newtonian fluids and couple stress fluid is one of them. Thetheory of couple stress fluids originated by Stokes [1] hasmany biomedical, industrial, and scientific applications andwas used to model synthetic fluids, polymer thickened oils,liquid crystals, animal blood, and synovial fluid. Some earlierdevelopments in couple stress fluid theory with some basicflows can be found in the book by Stokes [2]. Recently, fewresearchers have studied some couple stress fluid flows fordifferent flow geometries [3–8].

Nowadays, peristaltic flows have gained much attentionbecause of their applications in physiology and industry.

Peristaltic transport is a form of fluid transport induced bya progressive wave of area contraction or expansion alongthe length of a distensible tube/channel and transportingthe fluid in the direction of the wave propagation. Thisphenomenon is known as peristalsis. In physiology this playsan important role in various situations such as the foodmovement in the digestive tract, urine transport from kidneyto bladder through ureter, movement of lymphatic fluids inlymphatic vessels, bile flow from the gall bladder into theduodenum, spermatozoa in the ductus efferentes of the malereproductive tract, ovum movement in the fallopian tube,blood circulation in the small blood vessels, the movementof the chyme in the gastrointestinal tract, intrauterine fluidmotion, swallowing food bolus through esophagus, andtransport of cilia. Many industrial and biological instrumentssuch as roller pumps, finger pumps, heart-lung machines,blood pumpmachines, and dialysis machines are engineered

Hindawi Publishing CorporationJournal of FluidsVolume 2015, Article ID 163832, 19 pageshttp://dx.doi.org/10.1155/2015/163832

2 Journal of Fluids

based on the peristaltic mechanism [9]. The intrauterinefluid flow due to myometrial contractions is peristaltic innature and these myometrial contractions occur in bothsymmetric and asymmetric directions and also when embryoenters the uterus for implantation there start the asymmetriccontractions.The contractions inside the nonpregnant uterusare very complicated because they are composed of variableamplitudes and different wave lengths [10]. In view of this,Pandey and Chaube [11] have investigated the peristaltictransport of a couple stress fluid in a symmetrical channelusing perturbationmethod in terms of small amplitude ratio.Ali and Hayat [12] have studied the peristaltic motion ofmicropolar fluid in an asymmetric channel. Naga Rani andSarojamma [13] have analyzed the peristaltic transport of aCasson fluid in an asymmetric channel. Hayat et al. [14] havediscussed the peristaltic flow of a Johnson-Segalman fluid inan asymmetric channel. Hayat and Javed [15] have studiedthe peristaltic transport of power-law fluid in asymmetricchannel.

The porous medium plays an important role in thestudy of transport process in biofluid mechanics, indus-trial mechanics, and engineering fields. The fluid transportthrough porous medium is widely applicable in the vascularbeds, lungs, kidneys, tumorous vessels, bile duct, gall bladderwith stones, and small blood vessels. In the pathological situ-ations, the distribution of fatty cholesterol, artery clogging,blood clots in the lumen of coronary artery, transport ofdrugs and nutrients to brain cells, and functions of organsare modeled as porous medium [16]. Recently, Tripathi [17]studied the peristaltic hemodynamic flow of couple stressfluids through a porous medium. Tripathi and Bég [18] haveinvestigated the peristaltic flow of generalized Maxwell fluidthrough a porous medium using homotopy perturbationmethod. Abd elmaboud and Mekheimer [19] have discussedperistaltic transport of a second-order fluid through a porousmedium using regular perturbation method. The magneto-hydrodynamic flows also gained much attention due to thewidespread applications in biofluid mechanics and industry.It is the fact that many fluids like blood are conductive innature and gave a new direction for research. The indispens-able role of biomagnetic fluid dynamics in medical sciencehas been very helpful with many problems of physiology.It has wide range of applications, such as magnetic woundor cancer tumour treatment, bleeding reduction duringsurgeries, provocation of occlusion of feeding vessels ofcancer tumor, cell separation, transport of drugs, blood pumpmachines, and magnetic resonance imaging to diagnose thedisease and the influence of magnetic field which may beutilized as a blood pump in carrying out cardiac operationsfor the blood flow in arteries with arterial disease like arterialstenosis or arteriosclerosis. Specifically, the magnetohydro-dynamic flows of non-Newtonian fluids are of great interestin magnetotherapy. The noninvasive radiological tests usethe magnetic field to evaluate organs in abdomen [20].Hayat et al. [21] have studied the peristaltic transport ofmagnetohydrodynamic Johnson-Segalman fluid for the caseof a planar channel. Wang et al. [22] have investigated theperistaltic motion of a magnetohydrodynamic generalizedsecond-order fluid in an asymmetric channel. Nadeem and

Akram [23, 24] have discussed the peristaltic transport of acouple stress fluid and Williamson fluid in an asymmetricchannel with the effect of the magnetic field.

Heat transfer plays a significant role in the cooling pro-cesses of industrial and medical applications. Such consider-ation is very important since heat transfer in the human bodyis currently considered as an important area of research. Inview of the thermotherapy and the human thermoregulationsystem, the model of bioheat transfer in tissues has beenattracted by the biomedical engineers. In fact the heat transferin human tissues involves complicated processes such asheat conduction in tissues, heat transfer due to perfusion ofthe arterial-venous blood through the pores of the tissue,metabolic heat generation, and external interactions suchas electromagnetic radiation emitted from cell phones [25].Heat transfer also involves many complicated processessuch as evaluating skin burns, destruction of undesirablecancer tissues, dilution technique in examining blood flow,paper making, food processing, vasodilation, and radiationbetween surface and its environment [26]. Mustafa et al.[27] have studied the peristaltic transport of nanofluid in achannel. The heat transfer characteristics of a couple stressfluid in an asymmetric channel have been analyzed by Abdelmaboud et al. [28]. Nadeem and Akbar [29] have discussedthe influence of heat transfer andmagnetic field on peristalticflow of a Johnson-Segalman fluid in a vertical symmetricchannel. Some more works regarding peristaltic flows withthe effect of heat transfer and magnetic field can be seenin [30–33]. Srinivas et al. [34] have studied the effects ofboth wall slip conditions and heat transfer on peristalticflow of MHD Newtonian fluid in a porous channel withelastic wall properties. Mass transfer is another importantphenomenon in physiology and industry. This phenomenonhas great applications such as nutrients’ diffusion out from theblood to neighboring tissues, membrane separation process,reverse osmosis, distillation process, combustion process, anddiffusion of chemical impurities [35]. Recently, Noreen [36]studied the problem of mixed convection peristaltic flow ofthird-order nanofluidwith an inducedmagnetic field. Saleemand Haider [37] have discussed the peristaltic transport ofMaxwell fluid with heat and mass transfer in an asymmetricchannel. Some more relevant works on the peristaltic trans-port with heat and mass transfer can be seen in [38–42].

The aim of the present study is to investigate the influenceof heat and mass transfer on the peristaltic flow of magne-tohydrodynamic couple stress fluid through homogeneousporous medium in a vertical asymmetric channel. This paperis arranged as follows. Section 2 presents the mathematicalformulation for the problem. The solution of the problemis obtained in Section 3. The four different wave forms arepresented in Section 4 while the computational results arediscussed in Section 5. The last section, Section 6, presentsthe conclusions of the present study.

2. Formulation of the Problem

Let us consider magnetohydrodynamic couple stress fluid ina vertical asymmetric channel through the porous medium

Journal of Fluids 3

B0

g

Y

X

T0

𝜆a1

H1C1 C0

c

H2

T1

a2

𝜙

d1

d2

o

Figure 1: Physical model of the problem.

with the width of 𝑑1+ 𝑑2. The surfaces 𝐻

1and 𝐻

2of the

asymmetric channel are maintained at constant temperatures𝑇0and 𝑇

1and the constant concentrations 𝐶

0and 𝐶

1,

respectively (see Figure 1).The porous medium is assumed tobe homogeneous. The motion is induced by sinusoidal wavetrains propagating with constant speed 𝑐 along the channelwalls as defined by the following:

𝐻1= 𝑑1+ 𝑎1cos(2𝜋

𝜆(𝑋 − 𝑐𝑡)) (right side wall) ,

𝐻2= −𝑑2− 𝑎2cos(2𝜋

𝜆(𝑋 − 𝑐𝑡) + 𝜙) (left side wall) ,

(1)

where 𝑎1and 𝑎2are the wave amplitudes, 𝜆 is the wave length,

𝑑1+ 𝑑2is the channel width, 𝑐 is the velocity of propagation,

𝑡 is the time, and𝑋 is the direction of wave propagation. Thephase difference 𝜙 varies in the range 0 ≤ 𝜙 ≤ 𝜋, in which 𝜙 =0 corresponds to symmetric channel with waves out of phaseand 𝜙 = 𝜋 corresponds to waves in phase, and further 𝑎

1, 𝑎2,

𝑑1,𝑑2, and𝜙meet the following relation 𝑎2

1+𝑎2

2+2𝑎1𝑎2cos𝜙 ≤

(𝑑1+ 𝑑2)2.

The continuity, momentum, energy, and concentrationequations for an MHD incompressible couple stress fluid, inthe absence of body couples, are [8, 16]

∇ ⋅ 𝑞 = 0,

𝜌𝐷𝑞

𝐷𝑡= −∇𝑝 − 𝜇∇

2𝑞 − 𝜂∇

4𝑞 + 𝐽 × 𝐵 + 𝑅

+ 𝜌𝑔𝛽𝑇(𝑇 − 𝑇

0) + 𝜌𝑔𝛽

𝐶(𝐶 − 𝐶

0) ,

𝜌𝑐𝑝

𝐷𝑇

𝐷𝑡= 𝑘∗∇2𝑇 + 𝑄

0,

𝐷𝐶

𝐷𝑡= 𝐷∇

2𝐶 +

𝐷𝐾𝑇

𝑇𝑚

∇2𝑇,

(2)

in which 𝐷/𝐷𝑡 represents the material derivative and 𝑅 isDarcy’s resistance in the porous medium which are given by

𝐷

𝐷𝑡=𝜕

𝜕𝑡+ 𝑢

𝜕

𝜕𝑥+ V

𝜕

𝜕𝑦, 𝑅 = −

𝜇

𝑘0

𝑞, (3)

where 𝑞 is the velocity vector, 𝜌 is the density, 𝑝 is thepressure, 𝜇 is the viscosity, 𝜂 is material constant associatedwith couple stress, 𝐽 is the electric current density, 𝐵 isthe total magnetic field, 𝑔 is the acceleration due to thegravity, 𝛽

𝑇is the coefficient of thermal expansion, 𝛽

𝐶is the

coefficient of expansion with concentration, 𝑐𝑝is the specific

heat at constant pressure, 𝑇 is the temperature, 𝐶 is themass concentration, 𝑘∗ is the thermal conductivity, 𝑄

0is the

constant heat generation parameter, 𝐷 is the coefficient ofmass diffusivity, 𝐾

𝑇is the thermal diffusion ratio, 𝑇

𝑚is the

mean temperature, and 𝑘0is the permeability parameter. The

viscous dissipation is neglected in the energy equation.In the fixed frame, governing equations for the peristaltic

motion of an incompressible magnetohydrodynamic couplestress fluid through homogeneous porous medium in thetwo-dimensional vertical channel are

𝜕𝑈

𝜕𝑋+𝜕𝑉

𝜕𝑌= 0,

𝜌 (𝜕𝑈

𝜕𝑡+ 𝑈

𝜕𝑈

𝜕𝑋+ 𝑉

𝜕𝑈

𝜕𝑌)

= −𝜕𝑃

𝜕𝑋+ 𝜇(

𝜕2𝑈

𝜕𝑋2+𝜕2𝑈

𝜕𝑌2)

4 Journal of Fluids

− 𝜂(𝜕4𝑈

𝜕𝑋4+ 2

𝜕4𝑈

𝜕𝑋2𝜕𝑌2+𝜕4𝑈

𝜕𝑌4)

− 𝜎𝐵2

0𝑈 −

𝜇

𝑘0

𝑈 + 𝜌𝑔𝛽𝑇(𝑇 − 𝑇

0) + 𝜌𝑔𝛽

𝐶(𝐶 − 𝐶

0) ,

𝜌 (𝜕𝑉

𝜕𝑡+ 𝑈

𝜕𝑉

𝜕𝑋+ 𝑉

𝜕𝑉

𝜕𝑌)

= −𝜕𝑃

𝜕𝑌+ 𝜇(

𝜕2𝑉

𝜕𝑋2+𝜕2𝑉

𝜕𝑌2)

− 𝜂(𝜕4𝑉

𝜕𝑋4+ 2

𝜕4𝑉

𝜕𝑋2𝜕𝑌2+𝜕4𝑉

𝜕𝑌4) −

𝜇

𝑘0

𝑉,

𝜌𝑐𝑝(𝜕𝑇

𝜕𝑡+ 𝑈

𝜕𝑇

𝜕𝑋+ 𝑉

𝜕𝑇

𝜕𝑌) = 𝑘

∗(𝜕2𝑇

𝜕𝑋2+𝜕2𝑇

𝜕𝑌2) + 𝑄

0,

(𝜕𝐶

𝜕𝑡+ 𝑈

𝜕𝐶

𝜕𝑋+ 𝑉

𝜕𝐶

𝜕𝑌)

= 𝐷(𝜕2𝐶

𝜕𝑋2+𝜕2𝐶

𝜕𝑌2) +

𝐷𝐾𝑇

𝑇𝑚

(𝜕2𝑇

𝜕𝑋2+𝜕2𝑇

𝜕𝑌2) ,

(4)

in which𝑈 and𝑉 are the respective velocity components,𝑃 isthe pressure, 𝑇 is the temperature and 𝐶 is the concentrationin the reference to fixed frame system, 𝜎 is the electricalconductivity of the fluid, and 𝐵

0is the applied magnetic field.

In the above, the inducedmagnetic field is neglected since themagnetic Reynolds number is assumed to be small.

The coordinates, velocities, pressure, temperature, andconcentration in the fixed frame (𝑋, 𝑌) and wave frame (𝑥, 𝑦)are related by the following expressions:

𝑥 = 𝑋 − 𝑐𝑡, 𝑦 = 𝑌, 𝑢 = 𝑈 − 𝑐, V = 𝑉,

𝑝 (𝑥, 𝑦) = 𝑃 (𝑋, 𝑌, 𝑡) , 𝑇 (𝑥, 𝑦) = 𝑇 (𝑋, 𝑌, 𝑡) ,

𝐶 (𝑥, 𝑦) = 𝐶 (𝑋, 𝑌, 𝑡) ,

(5)

in which 𝑢, V, 𝑝, 𝑇, and 𝐶 are velocity components, pressure,temperature, and concentration in the wave frame, respec-tively.

Using (5), the governing equations in the wave frame aregiven as follows:

𝜕𝑢

𝜕𝑥+𝜕V𝜕𝑦= 0,

𝜌 (𝑢𝜕𝑢

𝜕𝑥+ V𝜕𝑢

𝜕𝑦)

= −𝜕𝑝

𝜕𝑥+ 𝜇(

𝜕2𝑢

𝜕𝑥2+𝜕2𝑢

𝜕𝑦2)

− 𝜂(𝜕4𝑢

𝜕𝑥4+ 2

𝜕4𝑢

𝜕𝑥2𝜕𝑦2+𝜕4𝑢

𝜕𝑦4)

− 𝜎𝐵2

0(𝑢 + 𝑐) −

𝜇

𝑘0

(𝑢 + 𝑐) + 𝜌𝑔𝛽𝑇(𝑇 − 𝑇

0)

+ 𝜌𝑔𝛽𝐶(𝐶 − 𝐶

0) ,

𝜌 (𝑢𝜕V𝜕𝑥+ V𝜕V𝜕𝑦)

= −𝜕𝑝

𝜕𝑦+ 𝜇(

𝜕2V𝜕𝑥2

+𝜕2V𝜕𝑦2

)

− 𝜂(𝜕4V𝜕𝑥4

+ 2𝜕4V

𝜕𝑥2𝜕𝑦2+𝜕4V𝜕𝑦4

) −𝜇

𝑘0

V,

𝜌𝑐𝑝(𝑢𝜕𝑇

𝜕𝑥+ V𝜕𝑇

𝜕𝑦) = 𝑘

∗(𝜕2𝑇

𝜕𝑥2+𝜕2𝑇

𝜕𝑦2) + 𝑄

0,

(𝑢𝜕𝐶

𝜕𝑥+ V𝜕𝐶

𝜕𝑦)

= 𝐷(𝜕2𝐶

𝜕𝑥2+𝜕2𝐶

𝜕𝑦2) +

𝐷𝐾𝑇

𝑇𝑚

(𝜕2𝑇

𝜕𝑥2+𝜕2𝑇

𝜕𝑦2) .

(6)

We introduce the following dimensionless parameters:

𝑥 =𝑥

𝜆, 𝑦 =

𝑦

𝑑1

, 𝑢 =𝑢

𝑐, V =

V𝑐,

ℎ1=𝐻1

𝑎1

, ℎ2=𝐻2

𝑑1

, 𝑡 =𝑐𝑡

𝜆,

𝑝 =𝑑2

1

𝜆𝜇𝑐𝑝,

𝛿 =𝑑1

𝜆, 𝑑 =

𝑑2

𝑑1

, 𝑎 =𝑎1

𝑑1

, 𝑏 =𝑎2

𝑑1

,

Re =𝜌𝑐𝑑1

𝜇, 𝑀 = √

𝜎

𝜇𝐵0𝑑1, 𝐷

𝑎=𝑘0

𝑑21

,

𝛾 = √𝜇

𝜂𝑑1, Gr =

𝜌𝑔𝑑2

1𝛽𝑇(𝑇1− 𝑇0)

𝜇𝑐,

Gc =𝜌𝑔𝑑2

1𝛽𝐶(𝐶1− 𝐶0)

𝜇𝑐, 𝜓 =

𝜓

𝑐𝑑1

, Pr =𝜇𝑐𝑝

𝑘∗,

𝜃 =𝑇 − 𝑇

0

𝑇1− 𝑇0

, Φ =𝐶 − 𝐶

0

𝐶1− 𝐶0

, 𝛽 =𝑄0𝑑2

1

𝑘∗ (𝑇1− 𝑇0),

Sc =𝜇

𝜌𝐷, Sr =

𝜌𝐷𝐾𝑇(𝑇1− 𝑇0)

𝜇𝑇𝑚(𝐶1− 𝐶0),

(7)

where 𝛿 is the dimensionlesswave number, Re is the Reynoldsnumber, 𝑀 is the Hartmann number, 𝐷

𝑎is the Darcy

number, 𝛾 is the couple stress parameter, Gr is the localtemperature Grashof number, Gc is the local concentration

Journal of Fluids 5

Grashof number, Pr is the Prandtl number, 𝜃 is the dimen-sionless temperature, Φ is the dimensionless concentration,𝛽 is the heat generation parameter, Sc is the Schmidt number,and Sr is the Soret number.

In terms of these nondimensional variables, the govern-ing equations (6), after dropping the bars, become

𝛿𝜕𝑢

𝜕𝑥+𝜕V𝜕𝑦= 0, (8)

Re 𝛿(𝑢𝜕𝑢𝜕𝑥+1

𝛿V𝜕𝑢

𝜕𝑦)

= −𝜕𝑝

𝜕𝑥+ (𝛿2 𝜕2𝑢

𝜕𝑥2+𝜕2𝑢

𝜕𝑦2)

−1

𝛾2(𝛿4 𝜕4𝑢

𝜕𝑥4+ 2𝛿2 𝜕4𝑢

𝜕𝑥2𝜕𝑦2+𝜕4𝑢

𝜕𝑦4)

− (𝑀2+1

𝐷𝑎

) (𝑢 + 1) + Gr𝜃 + GcΦ,

(9)

Re 𝛿2 (𝑢 𝜕V𝜕𝑥+1

𝛿V𝜕V𝜕𝑦)

= −𝜕𝑝

𝜕𝑦+ (𝛿3 𝜕2V𝜕𝑥2

+ 𝛿𝜕2V𝜕𝑦2

)

−1

𝛾2(𝛿5 𝜕4V𝜕𝑥4

+ 2𝛿3 𝜕4V

𝜕𝑥2𝜕𝑦2+ 𝛿

𝜕4V𝜕𝑦4

) −1

𝐷𝑎

𝛿V,

(10)

RePr𝛿(𝑢𝜕𝜃𝜕𝑥+ V𝜕𝜃

𝜕𝑦) = (𝛿

2 𝜕2𝜃

𝜕𝑥2+𝜕2𝜃

𝜕𝑦2) + 𝛽, (11)

Re 𝛿(𝑢𝜕Φ𝜕𝑥

+ V𝜕Φ

𝜕𝑦)

=1

Sc(𝜕2Φ

𝜕𝑥2+𝜕2Φ

𝜕𝑦2) + Sr(𝛿2 𝜕

2𝜃

𝜕𝑥2+𝜕2𝜃

𝜕𝑦2) .

(12)

The dimensionless velocity components (𝑢, V) in terms ofstream function 𝜓 are related by the following relations:

𝑢 =𝜕𝜓

𝜕𝑦, V = −𝛿

𝜕𝜓

𝜕𝑥. (13)

Using (13), the governing equations (9)–(12) reduced to

Re 𝛿 [(𝜕𝜓

𝜕𝑦

𝜕

𝜕𝑥−𝜕𝜓

𝜕𝑥

𝜕

𝜕𝑦)𝜕𝜓

𝜕𝑦]

= −𝜕𝑝

𝜕𝑥+ (𝛿2 𝜕3𝜓

𝜕𝑥2𝜕𝑦+𝜕3𝜓

𝜕𝑦3)

−1

𝛾2(𝛿4 𝜕5𝜓

𝜕𝑥4𝜕𝑦+ 2𝛿2 𝜕5𝜓

𝜕𝑥2𝜕𝑦3+𝜕5𝜓

𝜕𝑦5)

− (𝑀2+1

𝐷𝑎

)(𝜕𝜓

𝜕𝑦+ 1) + Gr𝜃 + GcΦ,

− Re 𝛿3 [(𝜕𝜓

𝜕𝑦

𝜕

𝜕𝑥−𝜕𝜓

𝜕𝑥

𝜕

𝜕𝑦)𝜕𝜓

𝜕𝑥]

= −𝜕𝑝

𝜕𝑦− (𝛿4 𝜕3𝜓

𝜕𝑥3+ 𝛿2 𝜕3𝜓

𝜕𝑥𝜕𝑦2)

+1

𝛾2(𝛿6 𝜕5𝜓

𝜕𝑥5+ 2𝛿4 𝜕5𝜓

𝜕𝑥3𝜕𝑦2+ 𝛿2 𝜕5𝜓

𝜕𝑥𝜕𝑦4)

+1

𝐷𝑎

𝛿2 𝜕𝜓

𝜕𝑥,

RePr𝛿(𝜕𝜓

𝜕𝑦

𝜕𝜃

𝜕𝑥−𝜕𝜓

𝜕𝑥

𝜕𝜃

𝜕𝑦) = (𝛿

2 𝜕2𝜃

𝜕𝑥2+𝜕2𝜃

𝜕𝑦2) + 𝛽,

Re 𝛿(𝜕𝜓

𝜕𝑦

𝜕Φ

𝜕𝑥−𝜕𝜓

𝜕𝑥

𝜕Φ

𝜕𝑦)

=1

Sc(𝛿2 𝜕2Φ

𝜕𝑥2+𝜕2Φ

𝜕𝑦2) + Sr(𝛿2 𝜕

2𝜃

𝜕𝑥2+𝜕2𝜃

𝜕𝑦2) .

(14)

The dimensionless boundary conditions can be put in theforms

𝜓 =𝐹

2,

𝜕𝜓

𝜕𝑦= −1,

𝜕3𝜓

𝜕𝑦3= 0,

𝜃 = 0, Φ = 0

at 𝑦 = ℎ1= 1 + 𝑎 cos (2𝜋𝑥) ,

𝜓 = −𝐹

2,

𝜕𝜓

𝜕𝑦= −1,

𝜕3𝜓

𝜕𝑦3= 0,

𝜃 = 1, Φ = 1

at 𝑦 = ℎ2= −𝑑 − 𝑏 cos (2𝜋𝑥 + 𝜙) .

(15)

The dimensionless mean flow rate Θ in fixed frame is relatedto the nondimensional mean flow rate 𝐹 in wave frame by

Θ = 𝐹 + 1 + 𝑑, (16)

in which

𝐹 = ∫

ℎ1

ℎ2

𝜕𝜓

𝜕𝑦𝑑𝑦 = 𝜓 (ℎ

1(𝑥)) − 𝜓 (ℎ

2(𝑥)) . (17)

We note that ℎ1and ℎ

2represent the dimensionless forms of

the peristaltic walls

ℎ1(𝑥) = 1 + 𝑎 cos (2𝜋𝑥) ,

ℎ2(𝑥) = −𝑑 − 𝑏 cos (2𝜋𝑥 + 𝜙) ,

(18)

where 𝑎, 𝑏, 𝑑, and 𝜙 satisfy the relation 𝑎2 + 𝑏2 + 2𝑎𝑏 cos𝜙 ≤(1 + 𝑑)

2.

3. Solution of the Problem

Assuming that the wave length of the peristaltic wave is verylarge compared to the width of the channel, the wave number

6 Journal of Fluids

𝛿 becomes very small.This assumption is known as longwavelength approximation. Since 𝛿 is very small, all the higherpowers of 𝛿 are also very small. Therefore, neglecting termscontaining 𝛿 and its higher powers from (14), we get

𝜕𝑝

𝜕𝑥=𝜕3𝜓

𝜕𝑦3−1

𝛾2

𝜕5𝜓

𝜕𝑦5

− (𝑀2+1

𝐷𝑎

)(𝜕𝜓

𝜕𝑦+ 1) + Gr𝜃 + GcΦ,

(19)

𝜕𝑝

𝜕𝑦= 0, (20)

𝜕2𝜃

𝜕𝑦2+ 𝛽 = 0, (21)

1

Sc𝜕2Φ

𝜕𝑦2+ Sr𝜕

2𝜃

𝜕𝑦2= 0. (22)

Elimination of pressure form from (19) and (20) yields

𝜕4𝜓

𝜕𝑦4−1

𝛾2

𝜕6𝜓

𝜕𝑦6− (𝑀

2+1

𝐷𝑎

)𝜕2𝜓

𝜕𝑦2+ Gr𝜕𝜃

𝜕𝑦+ Gc𝜕Φ

𝜕𝑦= 0.

(23)

Solving (21) and (22) with the boundary conditions (15), thetemperature and concentration are obtained as

𝜃 = −𝛽𝑦2

2+ 𝐴1𝑦 + 𝐴

2,

Φ = SrSc𝛽𝑦2

2+ 𝐵1𝑦 + 𝐵2,

(24)

where

𝐴1=𝛽 (ℎ2

1− ℎ2

2) − 2

2 (ℎ1− ℎ2); 𝐴

2=𝛽ℎ2

1− 2𝐴1ℎ1

2;

𝐵1= −

ScSr𝛽 (ℎ21− ℎ2

2) + 2

2 (ℎ1− ℎ2)

; 𝐵2= −

ScSr𝛽ℎ21+ 2𝐵1ℎ1

2.

(25)

Inserting (24) in (23), with the help of boundary conditions(15), we obtain the stream function as

𝜓 = 𝐷1𝑦3+ 𝐷2𝑦2+ 𝐶3𝑦 + 𝐶

4+ 𝐶5cosh (𝑛

1𝑦)

+ 𝐶6sinh (𝑛

1𝑦) + 𝐶

7cosh (𝑛

2𝑦) + 𝐶

8sinh (𝑛

2𝑦) .

(26)

in which

𝑚1= √𝑀2 +

1

𝐷𝑎

; 𝑛1=√𝛾2+ 𝛾√𝛾2 − 4𝑚2

1

2;

𝑛2=√𝛾2− 𝛾√𝛾2 − 4𝑚2

1

2;

𝐶1= 𝛽𝛾2(GcScSr − Gr) ; 𝐶

2= 𝛾2(Gr𝐴

1+ Gc𝐵

1) ;

𝐶3= 𝐷19+ 𝐶8𝐷20;

𝐶4=𝐹

2− 𝐷1ℎ3

1− 𝐷2ℎ2

1− 𝐶3ℎ1− 𝐶5cosh (𝑛

1ℎ1)

− 𝐶6sinh (𝑛

1ℎ1) − 𝐶7cosh (𝑛

2ℎ1) − 𝐶8sinh (𝑛

2ℎ1) ;

𝐶5= 𝐷13+ 𝐶8𝐷14; 𝐶

6= 𝐷15+ 𝐶8𝐷16;

𝐶7= 𝐷17+ 𝐶8𝐷18; 𝐶

8=𝐷21

𝐷22

;

𝐷1=

𝐶1

6𝑚21𝛾2; 𝐷

2=

𝐶2

2𝑚21𝛾2;

𝐷3= sinh (𝑛

1ℎ1) − sinh (𝑛

1ℎ2) ;

𝐷4= cosh (𝑛

1ℎ1) − cosh (𝑛

1ℎ2) ;

𝐷5= sinh (𝑛

2ℎ1) − sinh (𝑛

2ℎ2) ;

𝐷6= cosh (𝑛

2ℎ1) − cosh (𝑛

2ℎ2) ;

𝐷7= 𝑛3

1(sinh (𝑛

1ℎ1) −

𝐷3sinh (𝑛

2ℎ1)

𝐷5

) ;

𝐷8= 𝑛3

1(− cosh (𝑛

1ℎ1) +

𝐷4sinh (𝑛

2ℎ1)

𝐷5

) ;

𝐷9= 𝑛3

2(cosh (𝑛

2ℎ1) +

𝐷6sinh (𝑛

2ℎ1)

𝐷5

) ;

𝐷10= 3𝐷1(ℎ2

2− ℎ2

1) + 2𝐷

2(ℎ2− ℎ1)

−6𝑛1𝐷1𝐷4

𝐷8

+6𝑛3

1𝐷1𝐷4

𝑛22𝐷8

;

𝐷11= 𝑛1𝐷3+𝑛1𝐷4𝐷7

𝐷8

−𝑛3

1𝐷3

𝑛22

−𝑛3

1𝐷4𝐷7

𝑛22𝐷8

;

𝐷12=𝑛1𝐷4𝐷9

𝐷8

−𝑛3

1𝐷4𝐷9

𝑛22𝐷8

;

𝐷13=𝐷10

𝐷11

; 𝐷14=𝐷12

𝐷11

;

𝐷15=𝐷7𝐷13+ 6𝐷1

𝐷8

; 𝐷16=𝐷7𝐷14− 𝐷9

𝐷8

;

𝐷17= −

𝑛3

1(𝐷3𝐷13+ 𝐷4𝐷15)

𝑛32𝐷5

;

𝐷18= −

𝑛3

1(𝐷3𝐷14+ 𝐷4𝐷16) + 𝑛3

2𝐷6

𝑛32𝐷5

;

𝐷19= − 1 − 3𝐷

1ℎ2

1− 2𝐷2ℎ1− 𝑛1𝐷13sinh (𝑛

1ℎ1)

− 𝑛1𝐷15cosh (𝑛

1ℎ1) − 𝑛2𝐷17sinh (𝑛

2ℎ1) ;

𝐷20= − 𝑛

1𝐷14sinh (𝑛

1ℎ1) − 𝑛1𝐷16cosh (𝑛

1ℎ1)

− 𝑛2𝐷18sinh (𝑛

2ℎ1) − 𝑛2cosh (𝑛

2ℎ1) ;

𝐷21= 𝐹 − 𝐷

1(ℎ3

1− ℎ3

2) − 𝐷2(ℎ2

1− ℎ2

2) − 𝐷19(ℎ1− ℎ2)

− 𝐷4𝐷13− 𝐷3𝐷15− 𝐷6𝐷17;

𝐷22= 𝐷20(ℎ1− ℎ2) + 𝐷4𝐷14+ 𝐷3𝐷16+ 𝐷6𝐷18+ 𝐷5.

(27)

Journal of Fluids 7

0 0.5 1 1.5

0

0.5

−1−1

−0.5

−0.5

u

y

M = 0.0

M = 1.0

M = 1.5

M = 2.0

(a)

0 0.5 1 1.5

00.20.40.6

−1 −0.5

u

y

−0.2

−0.4

−0.6

−0.8

−1

Sc = 0Sc = 1

Sc = 2Sc = 3

(b)

0 0.5 1 1.5

0

0.5

−1−1

−0.5

−0.5

u

y

𝛾 = 4

𝛾 = 5

𝛾 = 7𝛾 = 11

(c)

0 0.5 1 1.5

0

0.5

−1−1

−0.5

−0.5

u

y

Da = 0.2

Da = 0.3

Da = 0.4

Da = 0.6

(d)

0 0.5 1 1.5

00.20.40.6

−1 −0.5

u

y

−0.2

−0.4

−0.6

−0.8

−1

𝛽 = 0𝛽 = 3

𝛽 = 6

𝛽 = 9

(e)

0 0.5 1 1.5

0

0.5

−1−1

−0.5

−0.5

u

y

Gr = 0Gr = 1

Gr = 2Gr = 3

(f)

Figure 2: Velocity profile for (a) 𝑥 = 1, 𝑎 = 0.7, 𝑏 = 0.5, 𝑑 = 1, 𝐷𝑎= 0.5, 𝛾 = 5, 𝜙 = 𝜋/3, Gr = 0.5, Gc = 0.5, Sr = 0.4, Sc = 0.4, Θ = 2, and

𝛽 = 0.2; (b) 𝑥 = 1, 𝑎 = 0.7, 𝑏 = 0.5, 𝑑 = 1,𝑀 = 1,𝐷𝑎= 0.5, 𝛾 = 5, 𝜙 = 𝜋/3, Gr = 0.5, Gc = 0.5, Sr = 0.4, Θ = 2, and 𝛽 = 7; (c) 𝑥 = 1, 𝑎 = 0.7,

𝑏 = 0.5, 𝑑 = 1,𝑀 = 1,𝐷𝑎= 0.5, 𝜙 = 𝜋/3, Gr = 0.5, Gc = 0.5, Sr = 0.4, Sc = 0.4,Θ = 2, and 𝛽 = 0.2; (d) 𝑥 = 1, 𝑎 = 0.7, 𝑏 = 0.5, 𝑑 = 1,𝑀 = 1,

𝛾 = 5, 𝜙 = 𝜋/3, Gr = 0.5, Gc = 0.5, Sr = 0.4, Sc = 0.4, Θ = 2, and 𝛽 = 0.2; (e) 𝑥 = 1, 𝑎 = 0.7, 𝑏 = 0.5, 𝑑 = 1,𝑀 = 1,𝐷𝑎= 0.5, 𝛾 = 5, 𝜙 = 𝜋/3,

Gr = 0.5, Gc = 0.5, Sr = 0.8, Sc = 0.8, and Θ = 2; (f) 𝑥 = 1, 𝑎 = 0.7, 𝑏 = 0.5, 𝑑 = 1,𝑀 = 1, 𝐷𝑎= 0.5, 𝛾 = 5, 𝜙 = 𝜋/3, Gc = 0.5, Sr = 0.4,

Sc = 0.4, Θ = 2, and 𝛽 = 0.2.

Using (24) and (26) in (19), the pressure gradient is given by

𝜕𝑝

𝜕𝑥= 𝐸1𝑦2+ 𝐸2𝑦 + 𝐸3+ 𝐸4cosh (𝑛

1𝑦) + 𝐸

5sinh (𝑛

1𝑦)

+ 𝐸6cosh (𝑛

2𝑦) + 𝐸

7sinh (𝑛

2𝑦) ,

(28)

where

𝐸1=GcScSr𝛽 − Gr𝛽 − 6𝑚2

1𝐷1

2;

𝐸2= Gr𝐴

1+ Gc𝐵

1− 2𝑚2

1𝐷2;

𝐸3= Gr𝐴

2+ Gc𝐵

2+ 6𝐷1− 𝑚2

1(1 + 𝐶

3) ;

8 Journal of Fluids

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

x

dp/dx

M = 0.0

M = 1.0

M = 1.5

M = 2.0

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

25

30

x

dp/dx

𝛽 = 0𝛽 = 3

𝛽 = 6

𝛽 = 9

(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 105

10152025303540455055

x

dp/dx

Gr = 0Gr = 2

Gr = 4Gr = 6

(c)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

Da = 0.2

Da = 0.3

Da = 0.4

Da = 0.6

x

dp/dx

(d)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

25

x

dp/dx

Sc = 0Sc = 1

Sc = 2Sc = 3

(e)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

x

dp/dx

𝛾 = 4

𝛾 = 5

𝛾 = 7𝛾 = 11

(f)

Figure 3: Pressure gradient for (a) 𝑦 = 0, 𝑎 = 0.7, 𝑏 = 0.5, 𝑑 = 1, 𝐷𝑎= 0.5, 𝛾 = 5, 𝜙 = 𝜋/3, Gr = 0.5, Gc = 0.5, Sr = 0.4, Sc = 0.4, Θ = −1,

and 𝛽 = 0.2; (b) 𝑦 = 0, 𝑎 = 0.7, 𝑏 = 0.5, 𝑑 = 1,𝑀 = 1,𝐷𝑎= 0.5, 𝛾 = 5, 𝜙 = 𝜋/3, Gr = 0.5, Gc = 0.5, Sr = 0.4, Sc = 0.4, and Θ = −1; (c) 𝑦 = 0,

𝑎 = 0.7, 𝑏 = 0.5, 𝑑 = 1,𝑀 = 1, 𝐷𝑎= 0.5, 𝛾 = 5, 𝜙 = 𝜋/2, Gc = 0.5, Sr = 0.4, Sc = 0.4, Θ = −1, and 𝛽 = 0.2; (d) 𝑦 = 0, 𝑎 = 0.7, 𝑏 = 0.5, 𝑑 = 1,

𝑀 = 1, 𝛾 = 5, 𝜙 = 𝜋/3, Gr = 0.5, Gc = 0.5, Sr = 0.4, Sc = 0.4, Θ = −1, and 𝛽 = 0.2; (e) 𝑦 = 0, 𝑎 = 0.7, 𝑏 = 0.5, 𝑑 = 1,𝑀 = 1,𝐷𝑎= 0.5, 𝛾 = 5,

𝜙 = 𝜋/2, Gr = 0.5, Gc = 0.5, Sr = 0.4, Θ = −1, and 𝛽 = 0.2; (f) 𝑦 = 0, 𝑎 = 0.7, 𝑏 = 0.5, 𝑑 = 1,𝑀 = 1, 𝐷𝑎= 0.5, 𝜙 = 𝜋/3, Gr = 0.5, Gc = 0.5,

Sr = 0.4, Sc = 0.4, Θ = −1, and 𝛽 = 0.2.

𝐸4= 𝐶6(𝑛3

1− 𝑚2

1𝑛1−𝑛5

1

𝛾2) ;

𝐸5= 𝐶5(𝑛3

1− 𝑚2

1𝑛1−𝑛5

1

𝛾2) ;

𝐸6= 𝐶8(𝑛3

2− 𝑚2

1𝑛2−𝑛5

2

𝛾2) ;

𝐸7= 𝐶7(𝑛3

2− 𝑚2

1𝑛2−𝑛5

2

𝛾2) .

(29)

Journal of Fluids 9

0 0.5 1 1.51.5

22.5

33.5

44.5

55.5

6

x

dp/dx

d = 1.0d = 1.1

d = 1.2

d = 1.3

(a) Sinusoidal wave

0 0.5 1 1.52

2.5

3

3.5

4

4.5

x

dp/dx

d = 1.0d = 1.1

d = 1.2

d = 1.3

(b) Triangular wave

0 0.2 0.4 0.6 0.8 1 1.21.5

22.5

33.5

44.5

55.5

6

x

dp/dx

−0.2

d = 1.0d = 1.1

d = 1.2

d = 1.3

(c) Square wave

0 0.5 1 1.51.5

22.5

33.5

44.5

55.5

6

x

dp/dx

d = 1.0d = 1.1

d = 1.2

d = 1.3

(d) Trapezoidal wave

Figure 4: Pressure gradient for various wave forms for fixed values of 𝑦 = 0, 𝑎 = 0.7, 𝑏 = 0.5, 𝐷𝑎= 0.5,𝑀 = 1, 𝛾 = 5, 𝜙 = 0, Gr = 0.5,

Gc = 0.5, Sr = 0.4, Sc = 0.4, and 𝛽 = 0.2.

The nondimensional expressions for the pressure differencefor one wave lengthΔ

𝑝𝜆

, the frictional forces at both walls𝐹𝜆1

at 𝑦 = ℎ1and 𝐹

𝜆2

at 𝑦 = ℎ2, and the heat transfer coefficients

𝑍ℎ1

and 𝑍ℎ2

at the right and left walls are defined as follows[32]:

Δ𝑝𝜆= ∫

1

0

(𝑑𝑝

𝑑𝑥)𝑑𝑥, 𝐹

𝜆1

= ∫

1

0

ℎ2

1(−𝑑𝑝

𝑑𝑥)𝑑𝑥,

𝐹𝜆2

= ∫

1

0

ℎ2

2(−𝑑𝑝

𝑑𝑥)𝑑𝑥,

𝑍ℎ1

=𝜕ℎ1

𝜕𝑥

𝜕𝜃

𝜕𝑦, 𝑍

ℎ2

=𝜕ℎ2

𝜕𝑥

𝜕𝜃

𝜕𝑦.

(30)

Using (30), the heat transfer coefficients at the right and leftwalls, respectively, are obtained as

𝑍ℎ1

= 2𝑎𝜋 (𝛽𝑦 − 𝐴1) sin (2𝜋𝑥) ,

𝑍ℎ2

= −2𝑏𝜋 (𝛽𝑦 − 𝐴1) sin (2𝜋𝑥 + 𝜙) .

(31)

The expressions for pressure rise Δ𝑝𝜆and frictional forces

at both walls 𝐹𝜆1

at 𝑦 = ℎ1and 𝐹

𝜆2

at 𝑦 = ℎ2involve the

integration of 𝑑𝑝/𝑑𝑥. Due to the complexity of 𝑑𝑝/𝑑𝑥, theanalytical integration of integrals of (30) is not possible. Inview of this, a numerical integration scheme is used for theevaluation of the integrals.

4. Expressions for Wave Shapes

The nondimensional expressions for the four consideredwave forms are given in the following.

(1) Sinusoidal wave:ℎ1(𝑥) = 1 + 𝑎 sin (2𝜋𝑥) ,

ℎ2(𝑥) = −𝑑 − 𝑏 sin (2𝜋𝑥 + 𝜙) .

(32)

(2) Triangular wave:

ℎ1(𝑥) = 1 + 𝑎(

8

𝜋3

∞

∑

𝑛=1

(−1)𝑛+1

(2𝑛 − 1)2sin (2𝜋 (2𝑛 − 1) 𝑥)) ,

ℎ2(𝑥) = −𝑑 − 𝑏(

8

𝜋3

∞

∑

𝑛=1

(−1)𝑛+1

(2𝑛 − 1)2sin (2𝜋 (2𝑛 − 1) 𝑥 + 𝜙)) .

(33)

(3) Square wave:

ℎ1(𝑥) = 1 + 𝑎(

4

𝜋

∞

∑

𝑛=1

(−1)𝑛+1

2𝑛 − 1cos (2𝜋 (2𝑛 − 1) 𝑥)) ,

ℎ2(𝑥) = −𝑑 − 𝑏(

4

𝜋

∞

∑

𝑛=1

(−1)𝑛+1

2𝑛 − 1cos (2𝜋 (2𝑛 − 1) 𝑥 + 𝜙)) .

(34)

10 Journal of Fluids

0 0.5 1 1.5 2

05

10152025

Θ

Δp𝜆

−2 −1.5 −1 −0.5

−5

−10

−15

M = 0.0

M = 1.0

M = 1.5

M = 2.0

(a)

0 0.5 1 1.5 2

05

10152025

Θ

Δp𝜆

−2 −1.5 −1 −0.5

−5

−10

−15

Da = 0.2

Da = 0.3

Da = 0.4

Da = 0.6

(b)

0 0.5 1 1.5 2

0

5

10

15

20

Θ

Δp𝜆

−2 −1.5 −1 −0.5

−5

−10

𝛾 = 4𝛾 = 5 𝛾 = 9

𝛾 = 6

(c)

0

0 0.5 1 1.5 2

5

10

15

20

Θ

Δp𝜆

−5

−2 −1.5 −1 −0.5−10

Sr = 0Sr = 1

Sr = 2Sr = 3

(d)

0

0 0.5 1 1.5 2

5

10

15

20

Θ

Δp𝜆

−5

−2 −1.5 −1 −0.5−10

𝛽 = 0𝛽 = 3

𝛽 = 6

𝛽 = 9

(e)

0

0 0.5 1 1.5 2

5

10

15

20

Θ

Δp𝜆

−5

−2 −1.5 −1 −0.5−10

Gc = 0Gc = 1

Gc = 2Gc = 3

(f)

Figure 5: Pressure difference for (a) 𝑦 = 0, 𝑎 = 0.7, 𝑏 = 0.5, 𝑑 = 1, 𝐷𝑎= 0.5, 𝛾 = 5, 𝜙 = 𝜋/3, Gr = 0.5, Gc = 0.5, Sr = 0.4, Sc = 0.4, Θ = 2,

and 𝛽 = 0.2; (b) 𝑦 = 0, 𝑎 = 0.7, 𝑏 = 0.5, 𝑑 = 1,𝑀 = 1, 𝛾 = 5, 𝜙 = 𝜋/3, Gr = 0.5, Gc = 0.5, Sr = 0.4, Sc = 0.4, Θ = 2, and 𝛽 = 0.2; (c) 𝑦 = 0,𝑎 = 0.7, 𝑏 = 0.5, 𝑑 = 1,𝑀 = 1,𝐷

𝑎= 0.5, 𝜙 = 𝜋/3, Gr = 0.5, Gc = 0.5, Sr = 0.4, Sc = 0.4,Θ = 2, and 𝛽 = 0.2; (d) 𝑦 = 0, 𝑎 = 0.7, 𝑏 = 0.5, 𝑑 = 1,

𝑀 = 1, 𝐷𝑎= 0.5, 𝛾 = 5, 𝜙 = 𝜋/2, Gr = 0.5, Gc = 0.5, Sc = 0.4, Θ = 2, and 𝛽 = 0.2; (e) 𝑦 = 0, 𝑎 = 0.7, 𝑏 = 0.5, 𝑑 = 1,𝑀 = 1, 𝐷

𝑎= 0.5, 𝛾 = 5,

𝜙 = 𝜋/3, Gr = 0.5, Gc = 0.5, Sr = 0.4, Sc = 0.4, and Θ = 2; (f) 𝑦 = 0, 𝑎 = 0.7, 𝑏 = 0.5, 𝑑 = 1,𝑀 = 1, 𝐷𝑎= 0.5, 𝛾 = 5, 𝜙 = 𝜋/3, Gr = 0.5,

Sr = 0.4, Sc = 0.4, Θ = 2, and 𝛽 = 0.2.

(4) Trapezoidal wave:

ℎ1(𝑥)

= 1 + 𝑎(32

𝜋2

∞

∑

𝑛=1

sin ((𝜋/8) (2𝑛 − 1))(2𝑛 − 1)

2sin (2𝜋 (2𝑛 − 1) 𝑥)) ,

ℎ2(𝑥)

= −𝑑

− 𝑏(32

𝜋2

∞

∑

𝑛=1

sin ((𝜋/8) (2𝑛 − 1))(2𝑛 − 1)

2sin (2𝜋 (2𝑛 − 1) 𝑥 + 𝜙)) .

(35)

Journal of Fluids 11

0 0.5 1 1.5 2

0

5

10

−5

−10

−2 −1.5 −1 −0.5

Θ

F𝜆1

M = 0.0

M = 1.0

M = 1.5

M = 2.0

(a)

0 0.5 1 1.5 2

0

5

10

−5

−10

−2 −1.5 −1 −0.5

Θ

F𝜆1

Da = 0.2

Da = 0.3

Da = 0.4

Da = 0.6

(b)

0 0.5 1 1.5 2

0246

−2

−4

−6

−8

−10−2 −1.5 −1 −0.5

Θ

F𝜆1

𝛾 = 4𝛾 = 5 𝛾 = 9

𝛾 = 6

(c)

0 0.5 1 1.5 2−2 −1.5 −1 −0.5Θ

02468

−2

−4

−6

−8

−10

F𝜆1

Sr = 0Sr = 1

Sr = 2Sr = 3

(d)

0 0.5 1 1.5 2−2 −1.5 −1 −0.5Θ

0246

−2

−4

−6

−8

−10

F𝜆1

𝛽 = 0𝛽 = 3

𝛽 = 6

𝛽 = 9

(e)

0 0.5 1 1.5 2−2 −1.5 −1 −0.5Θ

02468

−2−4−6−8−10−12

F𝜆1

Gc = 0Gc = 1

Gc = 2Gc = 3

(f)

Figure 6: Frictional force at the right wall for (a) 𝑦 = 0, 𝑎 = 0.7, 𝑏 = 0.5, 𝑑 = 1, 𝐷𝑎= 0.5, 𝛾 = 5, 𝜙 = 𝜋/3, Gr = 0.5, Gc = 0.5, Sr = 0.4,

Sc = 0.4, Θ = 2, and 𝛽 = 0.2; (b) 𝑦 = 0, 𝑎 = 0.7, 𝑏 = 0.5, 𝑑 = 1,𝑀 = 1, 𝛾 = 5, 𝜙 = 𝜋/3, Gr = 0.5, Gc = 0.5, Sr = 0.4, Sc = 0.4, Θ = 2, and𝛽 = 0.2; (c) 𝑦 = 0, 𝑎 = 0.7, 𝑏 = 0.5, 𝑑 = 1,𝑀 = 1, 𝐷

𝑎= 0.5, 𝜙 = 𝜋/3, Gr = 0.5, Gc = 0.5, Sr = 0.4, Sc = 0.4, Θ = 2, and 𝛽 = 0.2; (d) 𝑦 = 0,

𝑎 = 0.7, 𝑏 = 0.5, 𝑑 = 1,𝑀 = 1, 𝐷𝑎= 0.5, 𝛾 = 5, 𝜙 = 𝜋/2, Gr = 0.5, Gc = 0.5, Sc = 0.4, Θ = 2, and 𝛽 = 0.2; (e) 𝑦 = 0, 𝑎 = 0.7, 𝑏 = 0.5, 𝑑 = 1,

𝑀 = 1,𝐷𝑎= 0.5, 𝛾 = 5, 𝜙 = 𝜋/3, Gr = 0.5, Gc = 0.5, Sr = 0.4, Sc = 0.4, and Θ = 2; (f) 𝑦 = 0, 𝑎 = 0.7, 𝑏 = 0.5, 𝑑 = 1,𝑀 = 1,𝐷

𝑎= 0.5, 𝛾 = 5,

𝜙 = 𝜋/3, Gr = 0.5, Sr = 0.4, Sc = 0.4, Θ = 2, and 𝛽 = 0.2.

5. Results and Discussion

This section is dedicated to discussion and analysis of thevelocity distribution, pumping characteristics, heat and masscharacteristics, and trapping phenomena for different flowparameters.

5.1. Flow Characteristics. Figures 2(a)–2(f) illustrate theinfluence of Hartmann number 𝑀, Schmidt number Sc,couple stress parameter 𝛾, Darcy number𝐷

𝑎, heat generation

parameter 𝛽, and Grashof number Gr on axial velocity profileacross the channel. From these figures, it is observed thatthe maximum velocities are always located near the centre


0 0.5 1 1.5 2

01020304050607080

Trapezoidal waveSquare wave

Sinusoidal waveTriangular wave

−20

−10

−2 −1.5 −1 −0.5

Θ

Δp𝜆

(a)

0 0.5 1 1.5 2

02468

−2

−4

−6

−8

−10−2 −1.5 −1 −0.5

Θ



F𝜆1

(b)

0

5

10

0 0.5 1 1.5 2−2 −1.5 −1 −0.5Θ

−5

−10

−15

−20



F𝜆2

(c)

Figure 7: (a) Pressure difference, (b) frictional force at right wall, and (c) frictional force at left wall for different wave forms when 𝑦 = 0,𝑎 = 0.7, 𝑏 = 0.5, 𝑑 = 1,𝐷

𝑎= 0.5,𝑀 = 1, 𝛾 = 5, 𝜙 = 0, Gr = 0.5, Gc = 0.5, Sr = 0.4, Sc = 0.4, Θ = 2, and 𝛽 = 0.2.

of the channel and the velocity profiles are nearly parabolicin all cases. It is noted from Figure 2(a) that as Hartmannnumber 𝑀 increases, the velocity decreases near the centreof the channel and it is increased in the neighborhood of thewalls. This seems realistic because the magnetic field acts inthe transverse direction to the flow andmagnetic force resiststhe flow. The similar behavior is observed in [25]. The samebehavior can be seen with increasing of Schmidt number Scand couple stress parameter 𝛾 (see Figures 2(b) and 2(c)). It isobserved from Figure 2(d) that increasing of Darcy number𝐷𝑎increases the velocity near the centre of the channel

and decreases the velocity of the fluid near the peristalticwalls. The same trend is followed with the increasing of heatgeneration parameter 𝛽 (see Figure 2(e)). It is noticed fromFigure 2(f) that, with increasing of Grashof number Gr, thevelocity at the left wall increases while a reverse trend is seenat the right wall.

5.2. Pumping Characteristics. Figure 3 illustrates the varia-tion of pressure gradient over one wave length 𝑥 ∈ [0, 1].The effects of𝑀, 𝛽, and Gr on pressure gradient are displayedin Figures 3(a)–3(c). It can be seen from Figure 3(a) thatincreasing of Hartmann number 𝑀 increases the pressuregradient. It shows that when strong magnetic field is applied

to the flow field then higher pressure gradient is neededto pass the flow. This result suggests that the fluid pressurecan be controlled by the application of suitable magneticfield strength. This phenomenon is useful during surgeryand critical operation to control excessive bleeding. It is alsoobserved that increasing of 𝛽 and Gr increases the pressuregradient. From Figures 3(d)–3(f), it is noted that with theincreasing of 𝐷

𝑎, Sc, and 𝛾 the pressure gradient decreases.

It is noticed that, in the wider part of the channels 𝑥 ∈[0, 0.2] and 𝑥 ∈ [0.7, 1], the pressure gradient is small, sothe flow can be easily passed without the imposition of largepressure gradient. However, in the narrow part of the channel𝑥 ∈ [0.2, 0.7] the pressure gradient is large; that is, muchlarger pressure gradient is needed to maintain the same givenvolume flow rate. Figure 4 is prepared to see the behaviour ofpressure gradient for different four wave forms. It is observedfrom Figures 4(a)–4(d) that, in all the wave forms, increase in𝑑 decreases pressure gradient.

The dimensionless pressure difference per unit wavelength versus time mean flow rate Θ has been plotted inFigure 5. We split the whole region into four segments asfollows: peristaltic pumping region where Δ𝑝

𝜆> 0 and

Θ > 0 and augmented pumping region when Δ𝑝𝜆< 0 and

Θ > 0. There is retrograde pumping region when Δ𝑝𝜆> 0


0 0.5 1 1.50123456789

1011

𝜃

y

−0.5−1

𝛽 = 0𝛽 = 3

𝛽 = 6

𝛽 = 9

(a)

0 0.5 1 1.5

00.20.40.60.8

1

y

−0.5−1

−0.2−0.4−0.6−0.8−1

Φ

𝛽 = 0𝛽 = 3

𝛽 = 6

𝛽 = 9

(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

010203040

−10

−20

−30

−40

−50

−60

x

Zh1

𝛽 = 0𝛽 = 3

𝛽 = 6

𝛽 = 9

(c)

Figure 8: (a) Temperature profile, (b) concentration profile, and (c) heat transfer coefficient at the right wall for 𝑎 = 0.7, 𝑏 = 0.5, 𝑑 = 1,𝐷𝑎= 0.5,𝑀 = 1, 𝛾 = 5, 𝜙 = 𝜋/3, Gr = 0.5, Gc = 0.5, Sr = 0.4, Sc = 0.4, and Θ = 2.

and Θ < 0. Free pumping region corresponds to Δ𝑝𝜆=

0. The region where Δ𝑝𝜆> 0 and Θ > 0 is known as

peristaltic pumping region. In this region, the positive valueof Θ is entirely due to the peristalsis after overcoming thepressure difference. The region where Δ𝑝

𝜆< 0 and Θ > 0

is known as copumping or augmented pumping region. Inthis region, a negative pressure difference assists the flow dueto the peristalsis of the walls. The region where Δ𝑝

𝜆> 0

and Θ < 0 is called retrograde pumping region. In thisregion, the flow is opposite to the direction of the peristalticmotion. In the free pumping region, the flow is caused purelyby the peristalsis of the walls. It is evident from Figure 5 thatthere is an inversely linear relation betweenΔ𝑝

𝜆andΘ. From

Figure 5(a), it is clear that with the increasing of 𝑀, in theaugmented pumping and free pumping regions, the pumpingdecreases, in the peristaltic pumping region, the pumpingincreases up to a critical value of Θ and decreases afterthe critical value, and in the retrograde pumping region thepumping increases. It is observed from Figure 5(b) that, withthe increasing of𝐷

𝑎, the behaviour is quite opposite with𝑀.

It is noticed from Figure 5(c) that in the augmented pumpingregion the pumping increases and in the peristaltic pumpingand retrograde pumping regions the pumping decreases.Figure 5(d) depicts that, in all the pumping regions, the

pumping decreases by increasing Sr. It is noted from Figures5(e)-5(f) that the behaviour is quite opposite with Sr whileincreasing 𝛽 and Gc.

Figure 6 describes the variation of frictional forces againstflow rate Θ for different values of 𝑀, 𝐷

𝑎, 𝛾, Sr, 𝛽, and Gc.

It is observed that there is a direct linear relation betweenfrictional forces and Θ. The frictional forces have exactlyopposite behaviour when compared with that of pressuredifference. Figure 7(a) indicates the effects of four differentwave forms on pressure difference. It is noticed that thetrapezoidal wave has best peristaltic pumping characteristics,while the triangular wave has poor peristaltic pumping ascompared to the other waves. Figures 7(b)-7(c) show thatthe frictional forces at the walls have opposite behaviour ascompared to the pressure difference.

5.3. Heat and Mass Characteristics. Figure 8 depicts theeffects of heat transfer, concentration, and heat transfercoefficient on the peristaltic transport for various values of𝛽. We can observe that the temperature and concentrationprofiles are almost parabolic except when𝛽 = 0. It is observedfrom Figure 8(a) that the temperature increases with 𝛽. Itis clear from Figure 8(b) that the concentration profile hasquite opposite behaviour of temperature profile. It is noticed


0 0.5 1 1.5

00.10.20.30.40.5

x

y

−0.1−0.2−0.3−0.4−0.5

−1.5 −1 −0.5

(a) 𝐷𝑎 = 0.3, 𝜙 = 0

0 0.5 1 1.5

00.10.20.30.40.5

x

y

−0.1−0.2−0.3−0.4−0.5

−1.5 −1 −0.5

(b) 𝐷𝑎 = 0.3, 𝜙 = 𝜋/3

0 0.5 1 1.5

00.10.20.30.40.5

x

y

−0.1−0.2−0.3−0.4−0.5

−1.5 −1 −0.5

(c) 𝐷𝑎 = 0.35, 𝜙 = 0

0 0.5 1 1.5

00.10.20.30.40.5

x

y

−0.1−0.2−0.3−0.4−0.5

−1.5 −1 −0.5

(d) 𝐷𝑎 = 0.35, 𝜙 = 𝜋/3

0 0.5 1 1.5

00.10.20.30.40.5

x

y

−0.1−0.2−0.3−0.4−0.5

−1.5 −1 −0.5

(e) 𝐷𝑎 = 0.4, 𝜙 = 0

0 0.5 1 1.5

00.10.20.30.40.5

x

y

−0.1−0.2−0.3−0.4−0.5

−1.5 −1 −0.5

(f) 𝐷𝑎 = 0.4, 𝜙 = 𝜋/3

Figure 9: Streamlines for 𝑎 = 0.5, 𝑏 = 0.5, 𝑑 = 1,𝑀 = 1, 𝛾 = 5, Gr = 0.5, Gc = 0.5, Sr = 0.4, Sc = 0.4, Θ = 2, and 𝛽 = 0.2.

from Figure 8(c) that, due to the peristalsis, the heat transfercoefficient is in oscillatory behaviour. Moreover, the absolutevalue of heat transfer coefficient increases with increase of 𝛽.

5.4. Trapping Phenomenon. In the wave frame, the stream-lines, in general, have a shape similar to the walls as thewalls are stationary. However under certain conditions somestreamlines can split to enclose a bolus of fluid particles inclosed streamlines. Hence some circulating regions occur. Inthe fixed frame of reference the fluid bolus is trapped with thewave and itmoves as awholewith thewave speed. To examinethe effects of 𝐷

𝑎, 𝛾, and 𝜙 in the symmetric and asymmetric

channels we have plotted Figures 9–11. In Figures 9 and 10,the left panels (a), (c), and (e) related to symmetric channeland the right panels (b), (d), and (f) are corresponding toasymmetric channel. It is observed from Figure 9 that when𝐷𝑎increases, the size of the trapped bolus decreases near

the left wall and increases near the right wall in both panels.However, the increase of Darcy number 𝐷

𝑎increases the

trapped bolus near both walls when Gc = 0 and Gr = 0.The effects of 𝛾 on the trapping phenomena are displayedin Figure 10. It is shown that with the increasing 𝛾 decreasethe trapping phenomena for the symmetric and asymmetricchannels. Since 𝛾 = √𝜇/𝜂𝑑

1, 𝛾 decreases as 𝜂 increases

and hence increasing of couple stresses increases the sizeof trapped bolus. Figure 11 gives the trapping behaviour forvarious values of 𝜙. It is evident that as 𝜙 increases, the trap-ping bolus decreases and when it reaches to 𝜋 the trappingdisappears. Moreover, with the increase of 𝜙 the bolus movesupward with decreasing effect. Figures 12 and 13 providethe variations of 𝛽 on trapping for different wave shapes:(a) sinusoidal, (b) triangular, (c) square, and (d) trapezoidal.From these figures we observe that the size of the trappedbolus increases with increasing 𝛽 in all the wave forms.


0 0.5 1 1.5

00.10.20.30.40.5

x

y

−0.1−0.2−0.3−0.4−0.5

−1.5 −1 −0.5

(a) 𝛾 = 5, 𝜙 = 0

0 0.5 1 1.5

00.10.20.30.40.5

x

y

−0.1−0.2−0.3−0.4−0.5

−1.5 −1 −0.5

(b) 𝛾 = 5, 𝜙 = 𝜋/3

0 0.5 1 1.5

00.10.20.30.40.5

x

y

−0.1−0.2−0.3−0.4−0.5

−1.5 −1 −0.5

(c) 𝛾 = 5.05, 𝜙 = 0

0 0.5 1 1.5

00.10.20.30.40.5

x

y

−0.1

−0.2

−0.3

−0.4

−0.5−1.5 −1 −0.5

(d) 𝛾 = 5.05, 𝜙 = 𝜋/3

0 0.5 1 1.5

00.10.20.30.40.5

x

y

−0.1

−0.2

−0.3

−0.4

−0.5−1.5 −1 −0.5

(e) 𝛾 = 5.1, 𝜙 = 0

0 0.5 1 1.5

00.10.20.30.40.5

x

y

−0.1

−0.2

−0.3

−0.4

−0.5−1.5 −1 −0.5

(f) 𝛾 = 5.1, 𝜙 = 𝜋/3

Figure 10: Streamlines for 𝑎 = 0.5, 𝑏 = 0.5, 𝑑 = 1,𝑀 = 1,𝐷𝑎= 0.5, Gr = 0.5, Gc = 0.5, Sr = 0.4, Sc = 0.4, Θ = 2, and 𝛽 = 0.2.

6. Conclusions

The effects of heat and mass transfer on the peristalticflow of magnetohydrodynamic couple stress fluid throughporous medium in a vertical asymmetric channel havebeen analyzed. The governing equations are modeled underthe assumption of long wave length approximation. Theexact solutions for the stream function, pressure gradient,temperature, heat transfer coefficients, and concentrationare obtained. The effects of involved parameters on thevelocity characteristics, pumping characteristics, heat andmass characteristics, and the trapping due to the peristalsis ofthe walls are discussed in detail. From the analysis the mainfindings can be summarized as follows:

(i) Increasing of heat generation increases the peristalticpumping, size of the trapped bolus, and the magni-tude of heat transfer coefficient at the peristaltic walls.

(ii) Increasing of couple stresses increases the size oftrapped bolus.

(iii) Increasing of heat generation increases the tempera-ture and decreases the concentration.

(iv) The trapezoidal wave has best peristaltic pumping ascompared to the other wave shapes.

(v) The frictional forces have an opposite behaviour ascompared to the pressure difference.

Symbols

𝑎1, 𝑎2: Wave amplitudes

𝜆: Wave length𝑐: Propagation velocity𝑡: Time


0 0.5 1 1.5

00.10.20.30.40.5

x

y

−0.1

−0.2

−0.3

−0.4

−0.5−1.5 −1 −0.5

(a) 𝜙 = 0

0 0.5 1 1.5

00.10.20.30.40.5

x

y

−0.1

−0.2

−0.3

−0.4

−0.5−1.5 −1 −0.5

(b) 𝜙 = 𝜋/4

0 0.5 1 1.5

00.10.20.30.40.5

x

y

−0.1

−0.2

−0.3

−0.4

−0.5−1.5 −1 −0.5

(c) 𝜙 = 𝜋/2

0 0.5 1 1.5

00.10.20.30.40.5

x

y

−0.1

−0.2

−0.3

−0.4

−0.5−1.5 −1 −0.5

(d) 𝜙 = 2𝜋/3

0 0.5 1 1.5

00.10.20.30.40.5

x

y

−0.1

−0.2

−0.3

−0.4

−0.5−1.5 −1 −0.5

(e) 𝜙 = 5𝜋/6

0 0.5 1 1.5

00.10.20.30.40.5

x

y

−0.1

−0.2

−0.3

−0.4

−0.5−1.5 −1 −0.5

(f) 𝜙 = 𝜋

Figure 11: Streamlines for 𝑎 = 0.5, 𝑏 = 0.5, 𝑑 = 1,𝑀 = 1,𝐷𝑎= 0.5, Gr = 0.5, Gc = 0.5, 𝛾 = 5, Sc = 0.4, Sr = 0.4, Θ = 2, and 𝛽 = 0.2.

𝑋,𝑌: Coordinates of fixed frame𝑃: Pressure in the fixed frame𝑞: Velocity vector𝑅: Darcy’s resistance in the porous medium𝜌: Density𝜇: Viscosity𝜂: Material constant associated with couple stress𝐽: Electric current density𝐵: Total magnetic field𝑔: Acceleration due to the gravity𝛽𝑇: Coefficient of thermal expansion

𝛽𝐶: Coefficient of expansion with concentration

𝑐𝑝: Specific heat at constant pressure

𝑇: Temperature𝐶: Mass concentration

𝑘∗: Thermal conductivity𝑄0: Heat generation parameter

𝐷: Coefficient of mass diffusivity𝐾𝑇: Thermal diffusion ratio

𝑇𝑚: Mean temperature

𝑘0: Permeability parameter

𝑈,𝑉: Velocity components in the fixed frame𝑃: Pressure in the fixed frame𝜎: Electrical conductivity of the fluid𝐵0: Uniform applied magnetic field

𝛿: Dimensionless wave length𝑥, 𝑦: Coordinates of wave frame𝑝: Pressure in the wave frame𝑢, V: Velocity components in the wave frameRe: Reynolds number


0 0.5 1 1.5

00.10.20.30.40.50.60.7

x

y

−0.1

−0.2−1.5 −1 −0.5

(a) Sinusoidal wave

0 0.5 1 1.5

00.10.20.30.40.50.60.7

x

y

−0.1

−0.2−1.5 −1 −0.5

(b) Triangular wave

0 0.5 1 1.5

00.10.20.30.40.5

x

y

−0.1

−0.2

−0.3

−0.4

−0.5−1.5 −1 −0.5

(c) Square wave

0 0.5 1 1.5

00.10.20.30.40.50.60.7

x

y

−0.1

−0.2−1.5 −1 −0.5


Figure 12: Streamlines for various wave forms for fixed values of 𝑎 = 0.7, 𝑏 = 0.5, 𝑑 = 1,𝐷𝑎= 0.5,𝑀 = 1, 𝛾 = 5, 𝜙 = 𝜋/3, Gr = 0.5, Gc = 0.5,

Sr = 0.4, Sc = 0.4, Θ = 2, and 𝛽 = 0.

0 0.5 1 1.5

00.10.20.30.40.50.60.7

x

y

−0.1

−0.2−1.5 −1 −0.5

(a) Sinusoidal wave

0 0.5 1 1.5

00.10.20.30.40.50.60.7

x

y

−0.1

−0.2−1.5 −1 −0.5

(b) Triangular wave

0 0.5 1 1.5

00.10.20.30.40.5

x

y

−0.1

−0.2

−0.3

−0.4

−0.5−1.5 −1 −0.5

(c) Square wave

0 0.5 1 1.5

00.10.20.30.40.50.60.7

x

y

−0.1

−0.2−1.5 −1 −0.5


Figure 13: Streamlines for various wave forms for fixed values of (a) 𝑎 = 0.7, 𝑏 = 0.5, 𝑑 = 1, 𝐷𝑎= 0.5,𝑀 = 1, 𝛾 = 5, 𝜙 = 𝜋/3, Gr = 0.5,

Gc = 0.5, Sr = 0.4, Sc = 0.4, Θ = 2, and 𝛽 = 5.


𝑀: Hartman number𝐷𝑎: Darcy number

𝛾: Couple stress parameterGr: Local temperature Grashof numberGc: Local concentration Grashof numberPr: Prandtl number𝜃: Dimensionless temperatureΦ: Dimensionless concentration𝛽: Dimensionless heat generation parameterSc: Schmidt numberSr: Soret number𝜓: Stream functionΘ: Time mean flow rate in the fixed frame𝐹: Time mean flow rate in the wave frame𝑄: Volume flow rate in the fixed frame𝑞: Volume flow rate in the wave frame.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

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