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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 Š 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 [3â8].
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 BeĚg [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 [30â33]. 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 nutrientsâ 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 [38â42].
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
đa1
H1C1 C0
c
H2
T1
a2
đ
d1
d2
o
Figure 1: Physical model of the problem.
with the width of đ1+ đ2. The surfaces đť
1and đť
2of the
asymmetric channel are maintained at constant temperaturesđ0and đ
1and the constant concentrations đś
0and đś
1,
respectively (see Figure 1).The porous medium is assumed tobe homogeneous. The motion is induced by sinusoidal wavetrains propagating with constant speed đ along the channelwalls as defined by the following:
đť1= đ1+ đ1cos(2đ
đ(đ â đđĄ)) (right side wall) ,
đť2= âđ2â đ2cos(2đ
đ(đ â đđĄ) + đ) (left side wall) ,
(1)
where đ1and đ2are the wave amplitudes, đ is the wave length,
đ1+ đ2is the channel width, đ is the velocity of propagation,
đĄ is the time, andđ is the direction of wave propagation. Thephase difference đ varies in the range 0 ⤠đ ⤠đ, in which đ =0 corresponds to symmetric channel with waves out of phaseand đ = đ corresponds to waves in phase, and further đ
1, đ2,
đ1,đ2, andđmeet the following relation đ2
1+đ2
2+2đ1đ2cosđ â¤
(đ1+ đ2)2.
The continuity, momentum, energy, and concentrationequations for an MHD incompressible couple stress fluid, inthe absence of body couples, are [8, 16]
â â đ = 0,
đđˇđ
đˇđĄ= ââđ â đâ
2đ â đâ
4đ + đ˝ Ă đľ + đ
+ đđđ˝đ(đ â đ
0) + đđđ˝
đś(đś â đś
0) ,
đđđ
đˇđ
đˇđĄ= đââ2đ + đ
0,
đˇđś
đˇđĄ= đˇâ
2đś +
đˇđžđ
đđ
â2đ,
(2)
in which đˇ/đˇđĄ represents the material derivative and đ isDarcyâs resistance in the porous medium which are given by
đˇ
đˇđĄ=đ
đđĄ+ đ˘
đ
đđĽ+ V
đ
đđŚ, đ = â
đ
đ0
đ, (3)
where đ is the velocity vector, đ is the density, đ is thepressure, đ is the viscosity, đ is material constant associatedwith couple stress, đ˝ is the electric current density, đľ isthe total magnetic field, đ is the acceleration due to thegravity, đ˝
đis the coefficient of thermal expansion, đ˝
đśis the
coefficient of expansion with concentration, đđis the specific
heat at constant pressure, đ is the temperature, đś is themass concentration, đâ is the thermal conductivity, đ
0is the
constant heat generation parameter, đˇ is the coefficient ofmass diffusivity, đž
đis the thermal diffusion ratio, đ
đis the
mean temperature, and đ0is 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
đđ
đđ+đđ
đđ= 0,
đ (đđ
đđĄ+ đ
đđ
đđ+ đ
đđ
đđ)
= âđđ
đđ+ đ(
đ2đ
đđ2+đ2đ
đđ2)
4 Journal of Fluids
â đ(đ4đ
đđ4+ 2
đ4đ
đđ2đđ2+đ4đ
đđ4)
â đđľ2
0đ â
đ
đ0
đ + đđđ˝đ(đ â đ
0) + đđđ˝
đś(đś â đś
0) ,
đ (đđ
đđĄ+ đ
đđ
đđ+ đ
đđ
đđ)
= âđđ
đđ+ đ(
đ2đ
đđ2+đ2đ
đđ2)
â đ(đ4đ
đđ4+ 2
đ4đ
đđ2đđ2+đ4đ
đđ4) â
đ
đ0
đ,
đđđ(đđ
đđĄ+ đ
đđ
đđ+ đ
đđ
đđ) = đ
â(đ2đ
đđ2+đ2đ
đđ2) + đ
0,
(đđś
đđĄ+ đ
đđś
đđ+ đ
đđś
đđ)
= đˇ(đ2đś
đđ2+đ2đś
đđ2) +
đˇđžđ
đđ
(đ2đ
đđ2+đ2đ
đđ2) ,
(4)
in whichđ andđ are the respective velocity components,đ isthe pressure, đ is the temperature and đś is the concentrationin the reference to fixed frame system, đ is the electricalconductivity of the fluid, and đľ
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 (đ, đ) and wave frame (đĽ, đŚ)are related by the following expressions:
đĽ = đ â đđĄ, đŚ = đ, đ˘ = đ â đ, V = đ,
đ (đĽ, đŚ) = đ (đ, đ, đĄ) , đ (đĽ, đŚ) = đ (đ, đ, đĄ) ,
đś (đĽ, đŚ) = đś (đ, đ, đĄ) ,
(5)
in which đ˘, V, đ, đ, and đś 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:
đđ˘
đđĽ+đVđđŚ= 0,
đ (đ˘đđ˘
đđĽ+ Vđđ˘
đđŚ)
= âđđ
đđĽ+ đ(
đ2đ˘
đđĽ2+đ2đ˘
đđŚ2)
â đ(đ4đ˘
đđĽ4+ 2
đ4đ˘
đđĽ2đđŚ2+đ4đ˘
đđŚ4)
â đđľ2
0(đ˘ + đ) â
đ
đ0
(đ˘ + đ) + đđđ˝đ(đ â đ
0)
+ đđđ˝đś(đś â đś
0) ,
đ (đ˘đVđđĽ+ VđVđđŚ)
= âđđ
đđŚ+ đ(
đ2VđđĽ2
+đ2VđđŚ2
)
â đ(đ4VđđĽ4
+ 2đ4V
đđĽ2đđŚ2+đ4VđđŚ4
) âđ
đ0
V,
đđđ(đ˘đđ
đđĽ+ Vđđ
đđŚ) = đ
â(đ2đ
đđĽ2+đ2đ
đđŚ2) + đ
0,
(đ˘đđś
đđĽ+ Vđđś
đđŚ)
= đˇ(đ2đś
đđĽ2+đ2đś
đđŚ2) +
đˇđžđ
đđ
(đ2đ
đđĽ2+đ2đ
đđŚ2) .
(6)
We introduce the following dimensionless parameters:
đĽ =đĽ
đ, đŚ =
đŚ
đ1
, đ˘ =đ˘
đ, V =
Vđ,
â1=đť1
đ1
, â2=đť2
đ1
, đĄ =đđĄ
đ,
đ =đ2
1
đđđđ,
đż =đ1
đ, đ =
đ2
đ1
, đ =đ1
đ1
, đ =đ2
đ1
,
Re =đđđ1
đ, đ = â
đ
đđľ0đ1, đˇ
đ=đ0
đ21
,
đž = âđ
đđ1, Gr =
đđđ2
1đ˝đ(đ1â đ0)
đđ,
Gc =đđđ2
1đ˝đś(đś1â đś0)
đđ, đ =
đ
đđ1
, Pr =đđđ
đâ,
đ =đ â đ
0
đ1â đ0
, ÎŚ =đś â đś
0
đś1â đś0
, đ˝ =đ0đ2
1
đâ (đ1â đ0),
Sc =đ
đđˇ, Sr =
đđˇđžđ(đ1â đ0)
đđđ(đś1â đś0),
(7)
where đż is the dimensionlesswave number, Re is the Reynoldsnumber, đ is the Hartmann number, đˇ
đis the Darcy
number, đž 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, đ is the dimen-sionless temperature, ÎŚ is the dimensionless concentration,đ˝ 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
đżđđ˘
đđĽ+đVđđŚ= 0, (8)
Re đż(đ˘đđ˘đđĽ+1
đżVđđ˘
đđŚ)
= âđđ
đđĽ+ (đż2 đ2đ˘
đđĽ2+đ2đ˘
đđŚ2)
â1
đž2(đż4 đ4đ˘
đđĽ4+ 2đż2 đ4đ˘
đđĽ2đđŚ2+đ4đ˘
đđŚ4)
â (đ2+1
đˇđ
) (đ˘ + 1) + Grđ + GcÎŚ,
(9)
Re đż2 (đ˘ đVđđĽ+1
đżVđVđđŚ)
= âđđ
đđŚ+ (đż3 đ2VđđĽ2
+ đżđ2VđđŚ2
)
â1
đž2(đż5 đ4VđđĽ4
+ 2đż3 đ4V
đđĽ2đđŚ2+ đż
đ4VđđŚ4
) â1
đˇđ
đżV,
(10)
RePrđż(đ˘đđđđĽ+ Vđđ
đđŚ) = (đż
2 đ2đ
đđĽ2+đ2đ
đđŚ2) + đ˝, (11)
Re đż(đ˘đÎŚđđĽ
+ VđÎŚ
đđŚ)
=1
Sc(đ2ÎŚ
đđĽ2+đ2ÎŚ
đđŚ2) + Sr(đż2 đ
2đ
đđĽ2+đ2đ
đđŚ2) .
(12)
The dimensionless velocity components (đ˘, V) in terms ofstream function đ are related by the following relations:
đ˘ =đđ
đđŚ, V = âđż
đđ
đđĽ. (13)
Using (13), the governing equations (9)â(12) reduced to
Re đż [(đđ
đđŚ
đ
đđĽâđđ
đđĽ
đ
đđŚ)đđ
đđŚ]
= âđđ
đđĽ+ (đż2 đ3đ
đđĽ2đđŚ+đ3đ
đđŚ3)
â1
đž2(đż4 đ5đ
đđĽ4đđŚ+ 2đż2 đ5đ
đđĽ2đđŚ3+đ5đ
đđŚ5)
â (đ2+1
đˇđ
)(đđ
đđŚ+ 1) + Grđ + GcÎŚ,
â Re đż3 [(đđ
đđŚ
đ
đđĽâđđ
đđĽ
đ
đđŚ)đđ
đđĽ]
= âđđ
đđŚâ (đż4 đ3đ
đđĽ3+ đż2 đ3đ
đđĽđđŚ2)
+1
đž2(đż6 đ5đ
đđĽ5+ 2đż4 đ5đ
đđĽ3đđŚ2+ đż2 đ5đ
đđĽđđŚ4)
+1
đˇđ
đż2 đđ
đđĽ,
RePrđż(đđ
đđŚ
đđ
đđĽâđđ
đđĽ
đđ
đđŚ) = (đż
2 đ2đ
đđĽ2+đ2đ
đđŚ2) + đ˝,
Re đż(đđ
đđŚ
đÎŚ
đđĽâđđ
đđĽ
đÎŚ
đđŚ)
=1
Sc(đż2 đ2ÎŚ
đđĽ2+đ2ÎŚ
đđŚ2) + Sr(đż2 đ
2đ
đđĽ2+đ2đ
đđŚ2) .
(14)
The dimensionless boundary conditions can be put in theforms
đ =đš
2,
đđ
đđŚ= â1,
đ3đ
đđŚ3= 0,
đ = 0, ÎŚ = 0
at đŚ = â1= 1 + đ cos (2đđĽ) ,
đ = âđš
2,
đđ
đđŚ= â1,
đ3đ
đđŚ3= 0,
đ = 1, ÎŚ = 1
at đŚ = â2= âđ â đ cos (2đđĽ + đ) .
(15)
The dimensionless mean flow rate Î in fixed frame is relatedto the nondimensional mean flow rate đš in wave frame by
Î = đš + 1 + đ, (16)
in which
đš = âŤ
â1
â2
đđ
đđŚđđŚ = đ (â
1(đĽ)) â đ (â
2(đĽ)) . (17)
We note that â1and â
2represent the dimensionless forms of
the peristaltic walls
â1(đĽ) = 1 + đ cos (2đđĽ) ,
â2(đĽ) = âđ â đ cos (2đđĽ + đ) ,
(18)
where đ, đ, đ, and đ satisfy the relation đ2 + đ2 + 2đđ cosđ â¤(1 + đ)
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
đż becomes very small.This assumption is known as longwavelength approximation. Since đż is very small, all the higherpowers of đż are also very small. Therefore, neglecting termscontaining đż and its higher powers from (14), we get
đđ
đđĽ=đ3đ
đđŚ3â1
đž2
đ5đ
đđŚ5
â (đ2+1
đˇđ
)(đđ
đđŚ+ 1) + Grđ + GcÎŚ,
(19)
đđ
đđŚ= 0, (20)
đ2đ
đđŚ2+ đ˝ = 0, (21)
1
Scđ2ÎŚ
đđŚ2+ Srđ
2đ
đđŚ2= 0. (22)
Elimination of pressure form from (19) and (20) yields
đ4đ
đđŚ4â1
đž2
đ6đ
đđŚ6â (đ
2+1
đˇđ
)đ2đ
đđŚ2+ Grđđ
đđŚ+ GcđÎŚ
đđŚ= 0.
(23)
Solving (21) and (22) with the boundary conditions (15), thetemperature and concentration are obtained as
đ = âđ˝đŚ2
2+ đ´1đŚ + đ´
2,
ÎŚ = SrScđ˝đŚ2
2+ đľ1đŚ + đľ2,
(24)
where
đ´1=đ˝ (â2
1â â2
2) â 2
2 (â1â â2); đ´
2=đ˝â2
1â 2đ´1â1
2;
đľ1= â
ScSrđ˝ (â21â â2
2) + 2
2 (â1â â2)
; đľ2= â
ScSrđ˝â21+ 2đľ1â1
2.
(25)
Inserting (24) in (23), with the help of boundary conditions(15), we obtain the stream function as
đ = đˇ1đŚ3+ đˇ2đŚ2+ đś3đŚ + đś
4+ đś5cosh (đ
1đŚ)
+ đś6sinh (đ
1đŚ) + đś
7cosh (đ
2đŚ) + đś
8sinh (đ
2đŚ) .
(26)
in which
đ1= âđ2 +
1
đˇđ
; đ1=âđž2+ đžâđž2 â 4đ2
1
2;
đ2=âđž2â đžâđž2 â 4đ2
1
2;
đś1= đ˝đž2(GcScSr â Gr) ; đś
2= đž2(Grđ´
1+ Gcđľ
1) ;
đś3= đˇ19+ đś8đˇ20;
đś4=đš
2â đˇ1â3
1â đˇ2â2
1â đś3â1â đś5cosh (đ
1â1)
â đś6sinh (đ
1â1) â đś7cosh (đ
2â1) â đś8sinh (đ
2â1) ;
đś5= đˇ13+ đś8đˇ14; đś
6= đˇ15+ đś8đˇ16;
đś7= đˇ17+ đś8đˇ18; đś
8=đˇ21
đˇ22
;
đˇ1=
đś1
6đ21đž2; đˇ
2=
đś2
2đ21đž2;
đˇ3= sinh (đ
1â1) â sinh (đ
1â2) ;
đˇ4= cosh (đ
1â1) â cosh (đ
1â2) ;
đˇ5= sinh (đ
2â1) â sinh (đ
2â2) ;
đˇ6= cosh (đ
2â1) â cosh (đ
2â2) ;
đˇ7= đ3
1(sinh (đ
1â1) â
đˇ3sinh (đ
2â1)
đˇ5
) ;
đˇ8= đ3
1(â cosh (đ
1â1) +
đˇ4sinh (đ
2â1)
đˇ5
) ;
đˇ9= đ3
2(cosh (đ
2â1) +
đˇ6sinh (đ
2â1)
đˇ5
) ;
đˇ10= 3đˇ1(â2
2â â2
1) + 2đˇ
2(â2â â1)
â6đ1đˇ1đˇ4
đˇ8
+6đ3
1đˇ1đˇ4
đ22đˇ8
;
đˇ11= đ1đˇ3+đ1đˇ4đˇ7
đˇ8
âđ3
1đˇ3
đ22
âđ3
1đˇ4đˇ7
đ22đˇ8
;
đˇ12=đ1đˇ4đˇ9
đˇ8
âđ3
1đˇ4đˇ9
đ22đˇ8
;
đˇ13=đˇ10
đˇ11
; đˇ14=đˇ12
đˇ11
;
đˇ15=đˇ7đˇ13+ 6đˇ1
đˇ8
; đˇ16=đˇ7đˇ14â đˇ9
đˇ8
;
đˇ17= â
đ3
1(đˇ3đˇ13+ đˇ4đˇ15)
đ32đˇ5
;
đˇ18= â
đ3
1(đˇ3đˇ14+ đˇ4đˇ16) + đ3
2đˇ6
đ32đˇ5
;
đˇ19= â 1 â 3đˇ
1â2
1â 2đˇ2â1â đ1đˇ13sinh (đ
1â1)
â đ1đˇ15cosh (đ
1â1) â đ2đˇ17sinh (đ
2â1) ;
đˇ20= â đ
1đˇ14sinh (đ
1â1) â đ1đˇ16cosh (đ
1â1)
â đ2đˇ18sinh (đ
2â1) â đ2cosh (đ
2â1) ;
đˇ21= đš â đˇ
1(â3
1â â3
2) â đˇ2(â2
1â â2
2) â đˇ19(â1â â2)
â đˇ4đˇ13â đˇ3đˇ15â đˇ6đˇ17;
đˇ22= đˇ20(â1â â2) + đˇ4đˇ14+ đˇ3đˇ16+ đˇ6đˇ18+ đˇ5.
(27)
Journal of Fluids 7
0 0.5 1 1.5
0
0.5
â1â1
â0.5
â0.5
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
â1 â0.5
u
y
â0.2
â0.4
â0.6
â0.8
â1
Sc = 0Sc = 1
Sc = 2Sc = 3
(b)
0 0.5 1 1.5
0
0.5
â1â1
â0.5
â0.5
u
y
đž = 4
đž = 5
đž = 7đž = 11
(c)
0 0.5 1 1.5
0
0.5
â1â1
â0.5
â0.5
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
â1 â0.5
u
y
â0.2
â0.4
â0.6
â0.8
â1
đ˝ = 0đ˝ = 3
đ˝ = 6
đ˝ = 9
(e)
0 0.5 1 1.5
0
0.5
â1â1
â0.5
â0.5
u
y
Gr = 0Gr = 1
Gr = 2Gr = 3
(f)
Figure 2: Velocity profile for (a) đĽ = 1, đ = 0.7, đ = 0.5, đ = 1, đˇđ= 0.5, đž = 5, đ = đ/3, Gr = 0.5, Gc = 0.5, Sr = 0.4, Sc = 0.4, Î = 2, and
đ˝ = 0.2; (b) đĽ = 1, đ = 0.7, đ = 0.5, đ = 1,đ = 1,đˇđ= 0.5, đž = 5, đ = đ/3, Gr = 0.5, Gc = 0.5, Sr = 0.4, Î = 2, and đ˝ = 7; (c) đĽ = 1, đ = 0.7,
đ = 0.5, đ = 1,đ = 1,đˇđ= 0.5, đ = đ/3, Gr = 0.5, Gc = 0.5, Sr = 0.4, Sc = 0.4,Î = 2, and đ˝ = 0.2; (d) đĽ = 1, đ = 0.7, đ = 0.5, đ = 1,đ = 1,
đž = 5, đ = đ/3, Gr = 0.5, Gc = 0.5, Sr = 0.4, Sc = 0.4, Î = 2, and đ˝ = 0.2; (e) đĽ = 1, đ = 0.7, đ = 0.5, đ = 1,đ = 1,đˇđ= 0.5, đž = 5, đ = đ/3,
Gr = 0.5, Gc = 0.5, Sr = 0.8, Sc = 0.8, and Î = 2; (f) đĽ = 1, đ = 0.7, đ = 0.5, đ = 1,đ = 1, đˇđ= 0.5, đž = 5, đ = đ/3, Gc = 0.5, Sr = 0.4,
Sc = 0.4, Î = 2, and đ˝ = 0.2.
Using (24) and (26) in (19), the pressure gradient is given by
đđ
đđĽ= đ¸1đŚ2+ đ¸2đŚ + đ¸3+ đ¸4cosh (đ
1đŚ) + đ¸
5sinh (đ
1đŚ)
+ đ¸6cosh (đ
2đŚ) + đ¸
7sinh (đ
2đŚ) ,
(28)
where
đ¸1=GcScSrđ˝ â Grđ˝ â 6đ2
1đˇ1
2;
đ¸2= Grđ´
1+ Gcđľ
1â 2đ2
1đˇ2;
đ¸3= Grđ´
2+ Gcđľ
2+ 6đˇ1â đ2
1(1 + đś
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
đ˝ = 0đ˝ = 3
đ˝ = 6
đ˝ = 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
đž = 4
đž = 5
đž = 7đž = 11
(f)
Figure 3: Pressure gradient for (a) đŚ = 0, đ = 0.7, đ = 0.5, đ = 1, đˇđ= 0.5, đž = 5, đ = đ/3, Gr = 0.5, Gc = 0.5, Sr = 0.4, Sc = 0.4, Î = â1,
and đ˝ = 0.2; (b) đŚ = 0, đ = 0.7, đ = 0.5, đ = 1,đ = 1,đˇđ= 0.5, đž = 5, đ = đ/3, Gr = 0.5, Gc = 0.5, Sr = 0.4, Sc = 0.4, and Î = â1; (c) đŚ = 0,
đ = 0.7, đ = 0.5, đ = 1,đ = 1, đˇđ= 0.5, đž = 5, đ = đ/2, Gc = 0.5, Sr = 0.4, Sc = 0.4, Î = â1, and đ˝ = 0.2; (d) đŚ = 0, đ = 0.7, đ = 0.5, đ = 1,
đ = 1, đž = 5, đ = đ/3, Gr = 0.5, Gc = 0.5, Sr = 0.4, Sc = 0.4, Î = â1, and đ˝ = 0.2; (e) đŚ = 0, đ = 0.7, đ = 0.5, đ = 1,đ = 1,đˇđ= 0.5, đž = 5,
đ = đ/2, Gr = 0.5, Gc = 0.5, Sr = 0.4, Î = â1, and đ˝ = 0.2; (f) đŚ = 0, đ = 0.7, đ = 0.5, đ = 1,đ = 1, đˇđ= 0.5, đ = đ/3, Gr = 0.5, Gc = 0.5,
Sr = 0.4, Sc = 0.4, Î = â1, and đ˝ = 0.2.
đ¸4= đś6(đ3
1â đ2
1đ1âđ5
1
đž2) ;
đ¸5= đś5(đ3
1â đ2
1đ1âđ5
1
đž2) ;
đ¸6= đś8(đ3
2â đ2
1đ2âđ5
2
đž2) ;
đ¸7= đś7(đ3
2â đ2
1đ2âđ5
2
đž2) .
(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
â0.2
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 đŚ = 0, đ = 0.7, đ = 0.5, đˇđ= 0.5,đ = 1, đž = 5, đ = 0, Gr = 0.5,
Gc = 0.5, Sr = 0.4, Sc = 0.4, and đ˝ = 0.2.
The nondimensional expressions for the pressure differencefor one wave lengthÎ
đđ
, the frictional forces at both wallsđšđ1
at đŚ = â1and đš
đ2
at đŚ = â2, and the heat transfer coefficients
đâ1
and đâ2
at the right and left walls are defined as follows[32]:
Îđđ= âŤ
1
0
(đđ
đđĽ)đđĽ, đš
đ1
= âŤ
1
0
â2
1(âđđ
đđĽ)đđĽ,
đšđ2
= âŤ
1
0
â2
2(âđđ
đđĽ)đđĽ,
đâ1
=đâ1
đđĽ
đđ
đđŚ, đ
â2
=đâ2
đđĽ
đđ
đđŚ.
(30)
Using (30), the heat transfer coefficients at the right and leftwalls, respectively, are obtained as
đâ1
= 2đđ (đ˝đŚ â đ´1) sin (2đđĽ) ,
đâ2
= â2đđ (đ˝đŚ â đ´1) sin (2đđĽ + đ) .
(31)
The expressions for pressure rise Îđđand frictional forces
at both walls đšđ1
at đŚ = â1and đš
đ2
at đŚ = â2involve the
integration of đđ/đđĽ. Due to the complexity of đđ/đđĽ, 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:â1(đĽ) = 1 + đ sin (2đđĽ) ,
â2(đĽ) = âđ â đ sin (2đđĽ + đ) .
(32)
(2) Triangular wave:
â1(đĽ) = 1 + đ(
8
đ3
â
â
đ=1
(â1)đ+1
(2đ â 1)2sin (2đ (2đ â 1) đĽ)) ,
â2(đĽ) = âđ â đ(
8
đ3
â
â
đ=1
(â1)đ+1
(2đ â 1)2sin (2đ (2đ â 1) đĽ + đ)) .
(33)
(3) Square wave:
â1(đĽ) = 1 + đ(
4
đ
â
â
đ=1
(â1)đ+1
2đ â 1cos (2đ (2đ â 1) đĽ)) ,
â2(đĽ) = âđ â đ(
4
đ
â
â
đ=1
(â1)đ+1
2đ â 1cos (2đ (2đ â 1) đĽ + đ)) .
(34)
10 Journal of Fluids
0 0.5 1 1.5 2
05
10152025
Î
Îpđ
â2 â1.5 â1 â0.5
â5
â10
â15
M = 0.0
M = 1.0
M = 1.5
M = 2.0
(a)
0 0.5 1 1.5 2
05
10152025
Î
Îpđ
â2 â1.5 â1 â0.5
â5
â10
â15
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
Î
Îpđ
â2 â1.5 â1 â0.5
â5
â10
đž = 4đž = 5 đž = 9
đž = 6
(c)
0
0 0.5 1 1.5 2
5
10
15
20
Î
Îpđ
â5
â2 â1.5 â1 â0.5â10
Sr = 0Sr = 1
Sr = 2Sr = 3
(d)
0
0 0.5 1 1.5 2
5
10
15
20
Î
Îpđ
â5
â2 â1.5 â1 â0.5â10
đ˝ = 0đ˝ = 3
đ˝ = 6
đ˝ = 9
(e)
0
0 0.5 1 1.5 2
5
10
15
20
Î
Îpđ
â5
â2 â1.5 â1 â0.5â10
Gc = 0Gc = 1
Gc = 2Gc = 3
(f)
Figure 5: Pressure difference for (a) đŚ = 0, đ = 0.7, đ = 0.5, đ = 1, đˇđ= 0.5, đž = 5, đ = đ/3, Gr = 0.5, Gc = 0.5, Sr = 0.4, Sc = 0.4, Î = 2,
and đ˝ = 0.2; (b) đŚ = 0, đ = 0.7, đ = 0.5, đ = 1,đ = 1, đž = 5, đ = đ/3, Gr = 0.5, Gc = 0.5, Sr = 0.4, Sc = 0.4, Î = 2, and đ˝ = 0.2; (c) đŚ = 0,đ = 0.7, đ = 0.5, đ = 1,đ = 1,đˇ
đ= 0.5, đ = đ/3, Gr = 0.5, Gc = 0.5, Sr = 0.4, Sc = 0.4,Î = 2, and đ˝ = 0.2; (d) đŚ = 0, đ = 0.7, đ = 0.5, đ = 1,
đ = 1, đˇđ= 0.5, đž = 5, đ = đ/2, Gr = 0.5, Gc = 0.5, Sc = 0.4, Î = 2, and đ˝ = 0.2; (e) đŚ = 0, đ = 0.7, đ = 0.5, đ = 1,đ = 1, đˇ
đ= 0.5, đž = 5,
đ = đ/3, Gr = 0.5, Gc = 0.5, Sr = 0.4, Sc = 0.4, and Î = 2; (f) đŚ = 0, đ = 0.7, đ = 0.5, đ = 1,đ = 1, đˇđ= 0.5, đž = 5, đ = đ/3, Gr = 0.5,
Sr = 0.4, Sc = 0.4, Î = 2, and đ˝ = 0.2.
(4) Trapezoidal wave:
â1(đĽ)
= 1 + đ(32
đ2
â
â
đ=1
sin ((đ/8) (2đ â 1))(2đ â 1)
2sin (2đ (2đ â 1) đĽ)) ,
â2(đĽ)
= âđ
â đ(32
đ2
â
â
đ=1
sin ((đ/8) (2đ â 1))(2đ â 1)
2sin (2đ (2đ â 1) đĽ + đ)) .
(35)
Journal of Fluids 11
0 0.5 1 1.5 2
0
5
10
â5
â10
â2 â1.5 â1 â0.5
Î
Fđ1
M = 0.0
M = 1.0
M = 1.5
M = 2.0
(a)
0 0.5 1 1.5 2
0
5
10
â5
â10
â2 â1.5 â1 â0.5
Î
Fđ1
Da = 0.2
Da = 0.3
Da = 0.4
Da = 0.6
(b)
0 0.5 1 1.5 2
0246
â2
â4
â6
â8
â10â2 â1.5 â1 â0.5
Î
Fđ1
đž = 4đž = 5 đž = 9
đž = 6
(c)
0 0.5 1 1.5 2â2 â1.5 â1 â0.5Î
02468
â2
â4
â6
â8
â10
Fđ1
Sr = 0Sr = 1
Sr = 2Sr = 3
(d)
0 0.5 1 1.5 2â2 â1.5 â1 â0.5Î
0246
â2
â4
â6
â8
â10
Fđ1
đ˝ = 0đ˝ = 3
đ˝ = 6
đ˝ = 9
(e)
0 0.5 1 1.5 2â2 â1.5 â1 â0.5Î
02468
â2â4â6â8â10â12
Fđ1
Gc = 0Gc = 1
Gc = 2Gc = 3
(f)
Figure 6: Frictional force at the right wall for (a) đŚ = 0, đ = 0.7, đ = 0.5, đ = 1, đˇđ= 0.5, đž = 5, đ = đ/3, Gr = 0.5, Gc = 0.5, Sr = 0.4,
Sc = 0.4, Î = 2, and đ˝ = 0.2; (b) đŚ = 0, đ = 0.7, đ = 0.5, đ = 1,đ = 1, đž = 5, đ = đ/3, Gr = 0.5, Gc = 0.5, Sr = 0.4, Sc = 0.4, Î = 2, andđ˝ = 0.2; (c) đŚ = 0, đ = 0.7, đ = 0.5, đ = 1,đ = 1, đˇ
đ= 0.5, đ = đ/3, Gr = 0.5, Gc = 0.5, Sr = 0.4, Sc = 0.4, Î = 2, and đ˝ = 0.2; (d) đŚ = 0,
đ = 0.7, đ = 0.5, đ = 1,đ = 1, đˇđ= 0.5, đž = 5, đ = đ/2, Gr = 0.5, Gc = 0.5, Sc = 0.4, Î = 2, and đ˝ = 0.2; (e) đŚ = 0, đ = 0.7, đ = 0.5, đ = 1,
đ = 1,đˇđ= 0.5, đž = 5, đ = đ/3, Gr = 0.5, Gc = 0.5, Sr = 0.4, Sc = 0.4, and Î = 2; (f) đŚ = 0, đ = 0.7, đ = 0.5, đ = 1,đ = 1,đˇ
đ= 0.5, đž = 5,
đ = đ/3, Gr = 0.5, Sr = 0.4, Sc = 0.4, Î = 2, and đ˝ = 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)â2(f) illustrate theinfluence of Hartmann number đ, Schmidt number Sc,couple stress parameter đž, Darcy numberđˇ
đ, heat generation
parameter đ˝, 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
12 Journal of Fluids
0 0.5 1 1.5 2
01020304050607080
Trapezoidal waveSquare wave
Sinusoidal waveTriangular wave
â20
â10
â2 â1.5 â1 â0.5
Î
Îpđ
(a)
0 0.5 1 1.5 2
02468
â2
â4
â6
â8
â10â2 â1.5 â1 â0.5
Î
Trapezoidal waveSquare wave
Sinusoidal waveTriangular wave
Fđ1
(b)
0
5
10
0 0.5 1 1.5 2â2 â1.5 â1 â0.5Î
â5
â10
â15
â20
Trapezoidal waveSquare wave
Sinusoidal waveTriangular wave
Fđ2
(c)
Figure 7: (a) Pressure difference, (b) frictional force at right wall, and (c) frictional force at left wall for different wave forms when đŚ = 0,đ = 0.7, đ = 0.5, đ = 1,đˇ
đ= 0.5,đ = 1, đž = 5, đ = 0, Gr = 0.5, Gc = 0.5, Sr = 0.4, Sc = 0.4, Î = 2, and đ˝ = 0.2.
of the channel and the velocity profiles are nearly parabolicin all cases. It is noted from Figure 2(a) that as Hartmannnumber đ 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 đž (see Figures 2(b) and 2(c)). It isobserved from Figure 2(d) that increasing of Darcy numberđˇđincreases 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 đ˝ (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 đĽ â [0, 1].The effects ofđ, đ˝, and Gr on pressure gradient are displayedin Figures 3(a)â3(c). It can be seen from Figure 3(a) thatincreasing of Hartmann number đ 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 đ˝ and Gr increases the pressuregradient. From Figures 3(d)â3(f), it is noted that with theincreasing of đˇ
đ, Sc, and đž the pressure gradient decreases.
It is noticed that, in the wider part of the channels đĽ â[0, 0.2] and đĽ â [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 channelđĽ â [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)â4(d) that, in all the wave forms, increase inđ decreases pressure gradient.
The dimensionless pressure difference per unit wavelength versus time mean flow rate Î has been plotted inFigure 5. We split the whole region into four segments asfollows: peristaltic pumping region where Îđ
đ> 0 and
Î > 0 and augmented pumping region when Îđđ< 0 and
Î > 0. There is retrograde pumping region when Îđđ> 0
Journal of Fluids 13
0 0.5 1 1.50123456789
1011
đ
y
â0.5â1
đ˝ = 0đ˝ = 3
đ˝ = 6
đ˝ = 9
(a)
0 0.5 1 1.5
00.20.40.60.8
1
y
â0.5â1
â0.2â0.4â0.6â0.8â1
ÎŚ
đ˝ = 0đ˝ = 3
đ˝ = 6
đ˝ = 9
(b)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
010203040
â10
â20
â30
â40
â50
â60
x
Zh1
đ˝ = 0đ˝ = 3
đ˝ = 6
đ˝ = 9
(c)
Figure 8: (a) Temperature profile, (b) concentration profile, and (c) heat transfer coefficient at the right wall for đ = 0.7, đ = 0.5, đ = 1,đˇđ= 0.5,đ = 1, đž = 5, đ = đ/3, Gr = 0.5, Gc = 0.5, Sr = 0.4, Sc = 0.4, and Î = 2.
and Î < 0. Free pumping region corresponds to Îđđ=
0. The region where Îđđ> 0 and Î > 0 is known as
peristaltic pumping region. In this region, the positive valueof Î is entirely due to the peristalsis after overcoming thepressure difference. The region where Îđ
đ< 0 and Î > 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 Îđ
đ> 0
and Î < 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 betweenÎđ
đandÎ. From
Figure 5(a), it is clear that with the increasing of đ, in theaugmented pumping and free pumping regions, the pumpingdecreases, in the peristaltic pumping region, the pumpingincreases up to a critical value of Î and decreases afterthe critical value, and in the retrograde pumping region thepumping increases. It is observed from Figure 5(b) that, withthe increasing ofđˇ
đ, the behaviour is quite opposite withđ.
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 đ˝ and Gc.
Figure 6 describes the variation of frictional forces againstflow rate Î for different values of đ, đˇ
đ, đž, Sr, đ˝, and Gc.
It is observed that there is a direct linear relation betweenfrictional forces and Î. 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 ofđ˝. We can observe that the temperature and concentrationprofiles are almost parabolic except whenđ˝ = 0. It is observedfrom Figure 8(a) that the temperature increases with đ˝. Itis clear from Figure 8(b) that the concentration profile hasquite opposite behaviour of temperature profile. It is noticed
14 Journal of Fluids
0 0.5 1 1.5
00.10.20.30.40.5
x
y
â0.1â0.2â0.3â0.4â0.5
â1.5 â1 â0.5
(a) đˇđ = 0.3, đ = 0
0 0.5 1 1.5
00.10.20.30.40.5
x
y
â0.1â0.2â0.3â0.4â0.5
â1.5 â1 â0.5
(b) đˇđ = 0.3, đ = đ/3
0 0.5 1 1.5
00.10.20.30.40.5
x
y
â0.1â0.2â0.3â0.4â0.5
â1.5 â1 â0.5
(c) đˇđ = 0.35, đ = 0
0 0.5 1 1.5
00.10.20.30.40.5
x
y
â0.1â0.2â0.3â0.4â0.5
â1.5 â1 â0.5
(d) đˇđ = 0.35, đ = đ/3
0 0.5 1 1.5
00.10.20.30.40.5
x
y
â0.1â0.2â0.3â0.4â0.5
â1.5 â1 â0.5
(e) đˇđ = 0.4, đ = 0
0 0.5 1 1.5
00.10.20.30.40.5
x
y
â0.1â0.2â0.3â0.4â0.5
â1.5 â1 â0.5
(f) đˇđ = 0.4, đ = đ/3
Figure 9: Streamlines for đ = 0.5, đ = 0.5, đ = 1,đ = 1, đž = 5, Gr = 0.5, Gc = 0.5, Sr = 0.4, Sc = 0.4, Î = 2, and đ˝ = 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 đ˝.
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 đˇ
đ, đž, and đ in the symmetric and asymmetric
channels we have plotted Figures 9â11. 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 whenđˇđincreases, 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 đˇ
đincreases the
trapped bolus near both walls when Gc = 0 and Gr = 0.The effects of đž on the trapping phenomena are displayedin Figure 10. It is shown that with the increasing đž decreasethe trapping phenomena for the symmetric and asymmetricchannels. Since đž = âđ/đđ
1, đž decreases as đ increases
and hence increasing of couple stresses increases the sizeof trapped bolus. Figure 11 gives the trapping behaviour forvarious values of đ. It is evident that as đ increases, the trap-ping bolus decreases and when it reaches to đ the trappingdisappears. Moreover, with the increase of đ the bolus movesupward with decreasing effect. Figures 12 and 13 providethe variations of đ˝ 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 đ˝ in all the wave forms.
Journal of Fluids 15
0 0.5 1 1.5
00.10.20.30.40.5
x
y
â0.1â0.2â0.3â0.4â0.5
â1.5 â1 â0.5
(a) đž = 5, đ = 0
0 0.5 1 1.5
00.10.20.30.40.5
x
y
â0.1â0.2â0.3â0.4â0.5
â1.5 â1 â0.5
(b) đž = 5, đ = đ/3
0 0.5 1 1.5
00.10.20.30.40.5
x
y
â0.1â0.2â0.3â0.4â0.5
â1.5 â1 â0.5
(c) đž = 5.05, đ = 0
0 0.5 1 1.5
00.10.20.30.40.5
x
y
â0.1
â0.2
â0.3
â0.4
â0.5â1.5 â1 â0.5
(d) đž = 5.05, đ = đ/3
0 0.5 1 1.5
00.10.20.30.40.5
x
y
â0.1
â0.2
â0.3
â0.4
â0.5â1.5 â1 â0.5
(e) đž = 5.1, đ = 0
0 0.5 1 1.5
00.10.20.30.40.5
x
y
â0.1
â0.2
â0.3
â0.4
â0.5â1.5 â1 â0.5
(f) đž = 5.1, đ = đ/3
Figure 10: Streamlines for đ = 0.5, đ = 0.5, đ = 1,đ = 1,đˇđ= 0.5, Gr = 0.5, Gc = 0.5, Sr = 0.4, Sc = 0.4, Î = 2, and đ˝ = 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
đ1, đ2: Wave amplitudes
đ: Wave lengthđ: Propagation velocityđĄ: Time
16 Journal of Fluids
0 0.5 1 1.5
00.10.20.30.40.5
x
y
â0.1
â0.2
â0.3
â0.4
â0.5â1.5 â1 â0.5
(a) đ = 0
0 0.5 1 1.5
00.10.20.30.40.5
x
y
â0.1
â0.2
â0.3
â0.4
â0.5â1.5 â1 â0.5
(b) đ = đ/4
0 0.5 1 1.5
00.10.20.30.40.5
x
y
â0.1
â0.2
â0.3
â0.4
â0.5â1.5 â1 â0.5
(c) đ = đ/2
0 0.5 1 1.5
00.10.20.30.40.5
x
y
â0.1
â0.2
â0.3
â0.4
â0.5â1.5 â1 â0.5
(d) đ = 2đ/3
0 0.5 1 1.5
00.10.20.30.40.5
x
y
â0.1
â0.2
â0.3
â0.4
â0.5â1.5 â1 â0.5
(e) đ = 5đ/6
0 0.5 1 1.5
00.10.20.30.40.5
x
y
â0.1
â0.2
â0.3
â0.4
â0.5â1.5 â1 â0.5
(f) đ = đ
Figure 11: Streamlines for đ = 0.5, đ = 0.5, đ = 1,đ = 1,đˇđ= 0.5, Gr = 0.5, Gc = 0.5, đž = 5, Sc = 0.4, Sr = 0.4, Î = 2, and đ˝ = 0.2.
đ,đ: Coordinates of fixed frameđ: Pressure in the fixed frameđ: Velocity vectorđ : Darcyâs resistance in the porous mediumđ: Densityđ: Viscosityđ: Material constant associated with couple stressđ˝: Electric current densityđľ: Total magnetic fieldđ: Acceleration due to the gravityđ˝đ: Coefficient of thermal expansion
đ˝đś: Coefficient of expansion with concentration
đđ: Specific heat at constant pressure
đ: Temperatuređś: Mass concentration
đâ: Thermal conductivityđ0: Heat generation parameter
đˇ: Coefficient of mass diffusivityđžđ: Thermal diffusion ratio
đđ: Mean temperature
đ0: Permeability parameter
đ,đ: Velocity components in the fixed frameđ: Pressure in the fixed frameđ: Electrical conductivity of the fluidđľ0: Uniform applied magnetic field
đż: Dimensionless wave lengthđĽ, đŚ: Coordinates of wave frameđ: Pressure in the wave frameđ˘, V: Velocity components in the wave frameRe: Reynolds number
Journal of Fluids 17
0 0.5 1 1.5
00.10.20.30.40.50.60.7
x
y
â0.1
â0.2â1.5 â1 â0.5
(a) Sinusoidal wave
0 0.5 1 1.5
00.10.20.30.40.50.60.7
x
y
â0.1
â0.2â1.5 â1 â0.5
(b) Triangular wave
0 0.5 1 1.5
00.10.20.30.40.5
x
y
â0.1
â0.2
â0.3
â0.4
â0.5â1.5 â1 â0.5
(c) Square wave
0 0.5 1 1.5
00.10.20.30.40.50.60.7
x
y
â0.1
â0.2â1.5 â1 â0.5
(d) Trapezoidal wave
Figure 12: Streamlines for various wave forms for fixed values of đ = 0.7, đ = 0.5, đ = 1,đˇđ= 0.5,đ = 1, đž = 5, đ = đ/3, Gr = 0.5, Gc = 0.5,
Sr = 0.4, Sc = 0.4, Î = 2, and đ˝ = 0.
0 0.5 1 1.5
00.10.20.30.40.50.60.7
x
y
â0.1
â0.2â1.5 â1 â0.5
(a) Sinusoidal wave
0 0.5 1 1.5
00.10.20.30.40.50.60.7
x
y
â0.1
â0.2â1.5 â1 â0.5
(b) Triangular wave
0 0.5 1 1.5
00.10.20.30.40.5
x
y
â0.1
â0.2
â0.3
â0.4
â0.5â1.5 â1 â0.5
(c) Square wave
0 0.5 1 1.5
00.10.20.30.40.50.60.7
x
y
â0.1
â0.2â1.5 â1 â0.5
(d) Trapezoidal wave
Figure 13: Streamlines for various wave forms for fixed values of (a) đ = 0.7, đ = 0.5, đ = 1, đˇđ= 0.5,đ = 1, đž = 5, đ = đ/3, Gr = 0.5,
Gc = 0.5, Sr = 0.4, Sc = 0.4, Î = 2, and đ˝ = 5.
18 Journal of Fluids
đ: Hartman numberđˇđ: Darcy number
đž: Couple stress parameterGr: Local temperature Grashof numberGc: Local concentration Grashof numberPr: Prandtl numberđ: Dimensionless temperatureÎŚ: Dimensionless concentrationđ˝: Dimensionless heat generation parameterSc: Schmidt numberSr: Soret numberđ: Stream functionÎ: Time mean flow rate in the fixed frameđš: Time mean flow rate in the wave frameđ: Volume flow rate in the fixed frameđ: 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.
References
[1] V. K. Stokes, âCouple stresses in fluids,â Physics of Fluids, vol. 9,no. 9, pp. 1709â1715, 1966.
[2] V. K. Stokes, Theories of Fluids with Microstructure, Springer,New York, NY, USA, 1984.
[3] M. Devakar and T. K. V. Iyengar, âRun up flow of a couple stressfluid between parallel plates,âNonlinear Analysis: Modelling andControl, vol. 15, no. 1, pp. 29â37, 2010.
[4] M. Devakar and T. K. V. Iyengar, âStokesâ problems for anincompressible couple stress fluid,â Nonlinear Analysis: Mod-elling and Control, vol. 1, no. 2, pp. 181â190, 2008.
[5] T. Hayat, M. Mustafa, Z. Iqbal, and A. Alsaedi, âStagnation-point flow of couple stress fluid with melting heat transfer,âApplied Mathematics and Mechanics, vol. 34, no. 2, pp. 167â176,2013.
[6] N. S. Akbar and S. Nadeem, âIntestinal flow of a couple stressnanofluid in arteries,â IEEETransactions onNanoBioscience, vol.12, no. 4, pp. 332â339, 2013.
[7] D. Srinivasacharya, N. Srinivasacharyulu, and O. Odelu, âFlowand heat transfer of couple stress fluid in a porous channel withexpanding and contracting walls,â International Communica-tions in Heat andMass Transfer, vol. 36, no. 2, pp. 180â185, 2009.
[8] R. Muthuraj, S. Srinivas, and D. Lourdu Immaculate, âHeat andmass transfer effects on MHD fully developed flow of a couplestress fluid in a vertical channel with viscous dissipation andoscillating wall temperature,â International Journal of AppliedMathematics and Mechanics, vol. 9, no. 7, pp. 95â117, 2013.
[9] V. P. Srivastava and M. Saxena, âA two-fluid model of non-Newtonian blood flow induced by peristaltic waves,â RheologicaActa, vol. 34, no. 4, pp. 406â414, 1995.
[10] M. Mishra and A. Ramachandra Rao, âPeristaltic transport ofa Newtonian fluid in an asymmetric channel,â Zeitschrift fuĚrangewandte Mathematik und Physik, vol. 54, no. 3, pp. 532â550,2003.
[11] S. K. Pandey and M. K. Chaube, âStudy of wall properties onperistaltic transport of a couple stress fluid,âMeccanica, vol. 46,no. 6, pp. 1319â1330, 2011.
[12] N. Ali and T. Hayat, âPeristaltic flow of a micropolar fluidin an asymmetric channel,â Computers & Mathematics withApplications, vol. 55, no. 4, pp. 589â608, 2008.
[13] P. Naga Rani and G. Sarojamma, âPeristaltic transport of aCasson fluid in an asymmetric channel,â Australasian Physicaland Engineering Sciences in Medicine, vol. 27, no. 2, pp. 49â59,2004.
[14] T. Hayat, A. Afsar, and N. Ali, âPeristaltic transport of aJohnson-Segalman fluid in an asymmetric channel,âMathemat-ical and ComputerModelling, vol. 47, no. 3-4, pp. 380â400, 2008.
[15] T. Hayat andM. Javed, âExact solution to peristaltic transport ofpower-law fluid in asymmetric channel with compliant walls,âApplied Mathematics and Mechanics, vol. 31, no. 10, pp. 1231â1240, 2010.
[16] D. Tripathi, âStudy of transient peristaltic heat flow through afinite porous channel,â Mathematical and Computer Modelling,vol. 57, no. 5-6, pp. 1270â1283, 2013.
[17] D. Tripathi, âPeristaltic hemodynamic flow of couple-stressfluids through a porous medium with slip effect,â Transport inPorous Media, vol. 92, no. 3, pp. 559â572, 2012.
[18] D. Tripathi andO.A. BeĚg, âAnumerical study of oscillating peri-staltic flow of generalized Maxwell viscoelastic fluids through aporous medium,â Transport in Porous Media, vol. 95, no. 2, pp.337â348, 2012.
[19] Y. Abd elmaboud andKh. S.Mekheimer, âNon-linear peristaltictransport of a second-order fluid through a porous medium,âApplied Mathematical Modelling, vol. 35, no. 6, pp. 2695â2710,2011.
[20] S. Noreen, âMixed convection peristaltic flow with slip condi-tion and inducedmagnetic field,âTheEuropean Physical JournalPlus, vol. 129, no. 2, article 33, 2014.
[21] T. Hayat, F. M. Mahomed, and S. Asghar, âPeristaltic flow ofa magnetohydrodynamic Johnson-Segalman fluid,â NonlinearDynamics, vol. 40, no. 4, pp. 375â385, 2005.
[22] Y.Wang, N. Ali, and T. Hayat, âPeristaltic motion of a magneto-hydrodynamic generalized second-order fluid in an asymmetricchannel,â Numerical Methods for Partial Differential Equations,vol. 27, no. 2, pp. 415â435, 2011.
[23] S.Nadeemand S.Akram, âPeristaltic flowof a couple stress fluidunder the effect of induced magnetic field in an asymmetricchannel,â Archive of Applied Mechanics, vol. 81, no. 1, pp. 97â109,2011.
[24] S. Nadeem and S. Akram, âInfluence of inclined magneticfield on peristaltic flow of a Williamson fluid model in aninclined symmetric or asymmetric channel,âMathematical andComputer Modelling, vol. 52, no. 1-2, pp. 107â119, 2010.
[25] T. Hayat, M. Umar Qureshi, and Q. Hussain, âEffect of heattransfer on the peristaltic flowof an electrically conducting fluidin a porous space,â Applied Mathematical Modelling, vol. 33, no.4, pp. 1862â1873, 2009.
[26] S. Srinivas and R. Muthuraj, âEffects of chemical reactionand space porosity on MHD mixed convective flow in avertical asymmetric channel with peristalsis,âMathematical andComputer Modelling, vol. 54, no. 5-6, pp. 1213â1227, 2011.
[27] M.Mustafa, S. Hina, T. Hayat, and A. Alsaedi, âInfluence of wallproperties on the peristaltic flow of a nanofluid: analytic andnumerical solutions,â International Journal of Heat and MassTransfer, vol. 55, no. 17-18, pp. 4871â4877, 2012.
[28] Y. Abd elmaboud, Kh. S. Mekheimer, and A. I. Abdellateef,âThermal properties of couple-stress fluid flow in an asymmet-ric channel with peristalsis,â Journal of Heat Transfer, vol. 135,no. 4, Article ID 044502, 2013.
Journal of Fluids 19
[29] S. Nadeem and N. S. Akbar, âEffects of induced magneticfield on peristaltic flow of Johnson-Segalman fluid in a verticalsymmetric channel,â Applied Mathematics and MechanicsâEnglish Edition, vol. 31, no. 8, pp. 969â978, 2010.
[30] S. Srinivas and M. Kothandapani, âPeristaltic transport in anasymmetric channel with heat transferâa note,â InternationalCommunications in Heat and Mass Transfer, vol. 35, no. 4, pp.514â522, 2008.
[31] S. Nadeem and S. Akram, âMagnetohydrodynamic peristalticflow of a hyperbolic tangent fluid in a vertical asymmetricchannel with heat transfer,â Acta Mechanica Sinica, vol. 27, no.2, pp. 237â250, 2011.
[32] O. U. Mehmood, N. Mustapha, and S. Shafie, âHeat transferon peristaltic flow of fourth grade fluid in inclined asymmetricchannel with partial slip,â Applied Mathematics and Mechanics.English Edition, vol. 33, no. 10, pp. 1313â1328, 2012.
[33] N. S. Akbar, T. Hayat, S. Nadeem, and S. Obaidat, âPeristalticflow of a Williamson fluid in an inclined asymmetric channelwith partial slip and heat transfer,â International Journal of Heatand Mass Transfer, vol. 55, no. 7-8, pp. 1855â1862, 2012.
[34] S. Srinivas, R. Gayathri, and M. Kothandapani, âThe influenceof slip conditions, wall properties and heat transfer on MHDperistaltic transport,â Computer Physics Communications, vol.180, no. 11, pp. 2115â2122, 2009.
[35] R. Ellahi, M. Mubashir Bhatti, and K. Vafai, âEffects of heat andmass transfer on peristaltic flow in a non-uniform rectangularduct,â International Journal of Heat and Mass Transfer, vol. 71,pp. 706â719, 2014.
[36] S. Noreen, âMixed convection peristaltic flow of third ordernanofluid with an induced magnetic field,â PLoS ONE, vol. 8,no. 11, Article ID e78770, 2013.
[37] M. Saleem and A. Haider, âHeat and mass transfer on theperistaltic transport of non-Newtonian fluid with creepingflow,â International Journal of Heat and Mass Transfer, vol. 68,pp. 514â526, 2014.
[38] S. Nadeem andN. S. Akbar, âInfluence of heat andmass transferon the peristaltic flow of a Johnson Segalman fluid in a verticalasymmetric channel with inducedMHD,â Journal of the TaiwanInstitute of Chemical Engineers, vol. 42, no. 1, pp. 58â66, 2011.
[39] T. Hayat and S. Hina, âThe influence of wall properties on theMHD peristaltic flow of a Maxwell fluid with heat and masstransfer,âNonlinearAnalysis: RealWorldApplications, vol. 11, no.4, pp. 3155â3169, 2010.
[40] T. Hayat, S. Noreen, M. S. Alhothuali, S. Asghar, and A.Alhomaidan, âPeristaltic flow under the effects of an inducedmagnetic field and heat andmass transfer,â International Journalof Heat and Mass Transfer, vol. 55, no. 1â3, pp. 443â452, 2012.
[41] S. Hina, T. Hayat, S. Asghar, and A. A. Hendi, âInfluence ofcompliant walls on peristaltic motion with heat/mass transferand chemical reaction,â International Journal of Heat and MassTransfer, vol. 55, no. 13-14, pp. 3386â3394, 2012.
[42] T. Hayat, S. Noreen, and M. Qasim, âInfluence of heat andmass transfer on the peristaltic transport of a phan-thien-tannerfluid,â Zeitschrift fur Naturforschung A, vol. 68, no. 12, pp. 751â758, 2013.
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