research article analysis of heat and mass transfer on...

Research ArticleAnalysis of Heat and Mass Transfer on MHD Peristaltic Flowthrough a Tapered Asymmetric Channel

M. Kothandapani,1 J. Prakash,2 and V. Pushparaj3

1Department of Mathematics, University College of Engineering Arni (A Constituent College of Anna University, Chennai),Arni, Tamil Nadu 632 326, India2Department of Mathematics, Arulmigu Meenakshi Amman College of Engineering, Vadamavandal, Tamil Nadu 604 410, India3Department of Mathematics, C. Abdul Hakeem College of Engineering & Technology, Melvisharam, Tamil Nadu 632 509, India

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

Received 31 August 2014; Accepted 15 December 2014

Academic Editor: Miguel Onorato

Copyright © 2015 M. Kothandapani et al. 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.

This paper describes the peristaltic flow of an incompressible viscous fluid in a tapered asymmetric channel with heat and masstransfer. The fluid is electrically conducting fluid in the presence of a uniform magnetic field. The propagation of waves on thenonuniform channel walls to have different amplitudes and phase but with the same speed is generated the tapered asymmetricchannel.The assumptions of lowReynolds number and longwavelength approximations have been used to simplify the complicatedproblem into a relatively simple problem. Analytical expressions for velocity, temperature, and concentration have been obtained.Graphically results of the flow characteristics are also sketched for various embedded parameters of interest entering the problemand interpreted.

1. Introduction

The study of peristaltic transport has enjoyed increased inter-est from investigators in several engineering disciplines.From amechanical point of view, peristalsis offers the oppor-tunity of constructing pumps in which the transportedmedium does not come in direct contact with any movingparts such as valves, plungers, and rotors. This could be ofgreat benefit in cases where themedium is either highly abra-sive or decomposable under stress. This has led to the devel-opment of fingers and roller pumps which work accordingto the principle of peristalsis. Applications include dialysismachines, open-heart bypass pump machines, and infusionpumps. After the first investigation reported by Latham[1], several theoretical and experimental investigations [2–5]about the peristaltic flow of Newtonian and non-Newtonianfluids have been made under different conditions with refer-ence to physiological and mechanical situations.

In view of the processes like hemodialysis and oxygena-tion, some progress is shown in the theory of peristalsis

with heat transfer [6–8]. Such analysis of heat transfer isof great value in biological tissues, dilution technique inexamining blood flow, destruction of undesirable cancertissues, metabolic heat generation, and so forth. In addition,the mass transfer effect on the peristaltic flow of viscousfluid has been examined in the studies [9–11]. The effectof magnetic field on a Newtonian fluid has been reportedfor treatment of gastronomic pathologies, constipation, andhypertension.

Recently, few attempts have been made in the peristalticliterature to study the combined effects of heat and masstransfer. Eldabe et al. [10] analyzed the mixed convective heatand mass transfer in a non-Newtonian fluid at a peristalticsurface with temperature-dependent viscosity. The influenceof heat and mass transfer on MHD peristaltic flow througha porous space with compliant walls was studied by Srinivasand Kothandapani [11]. The effects of elasticity of the flexiblewalls on the peristaltic transport of viscous fluid with heattransfer in a two-dimensional uniform channel have beeninvestigated by Srinivas and Kothandapani [12]. Ogulu [13]

Hindawi Publishing CorporationJournal of FluidsVolume 2015, Article ID 561263, 9 pageshttp://dx.doi.org/10.1155/2015/561263

2 Journal of Fluids

Y

B0

0

b2

𝜆

𝜙

d

d

c

X

H2

H1

b1

C = C0

C = C1

T = T1

T = T0

Figure 1: Schematic diagram of the tapered asymmetric channel.

examined heat andmass transfer of blood under the influenceof a uniform magnetic field.

The problem of intrauterine fluid motion in a non-pregnant uterus caused by myometrial contractions is aperistaltic-type fluid motion and the myometrial contrac-tions may occur in both symmetric and asymmetric direc-tions. Further it is observed that the intrauterine fluid flowin a sagittal cross section of the uterus discloses a narrowchannel enclosed by two fairly parallel walls with wave trainshaving different amplitudes and phase difference [14–16].Keeping in view of the abovementioned reasons, a newtheoretical investigation of peristaltic motion of a Newtonianfluid in the presence of heat and mass transfer in the mostgeneralized form of the channel, namely, the tapered asym-metric channel, is carried out. The governing equations ofmotion, energy, and concentration are simplified by using theassumptions of long wavelength and low Reynolds numberapproximations. The exact solutions of velocity, temperature,and concentration of the fluid are generated. Also interestingflow quantities are analyzed by plotting various graphs.

2. Mathematical Formulation

Consider the unsteady, combined convective heat and masstransfer, MHD flow of an electrically conducting viscousfluid in a two-dimensional tapered asymmetric channel. Let𝑌 = 𝐻

1and 𝑌 = 𝐻

2, respectively, the lower and upper

wall boundaries of the tapered asymmetric channel. Theflow is generated by sinusoidal wave trains propagating withconstant speed 𝑐 along the tapered asymmetric channel(Figure 1).The geometry of the wall surface [15, 16] is definedas

𝐻2(𝑋, 𝑡) = 𝑑 + 𝑚

𝑋 + 𝑏2sin [2𝜋

𝜆(𝑋 − 𝑐𝑡)] .

upper wall,(1a)

𝐻1(𝑋, 𝑡) = −𝑑 − 𝑚

𝑋 − 𝑏1sin [2𝜋

𝜆(𝑋 − 𝑐𝑡) + 𝜙] .

lower wall,(1b)

where 𝑑 is the half width of the channel at the inlet, 𝑏1and 𝑏2

are the amplitudes of lower and upper walls, respectively, 𝑐 isthe phase speed of the wave,𝑚 (𝑚 ≪ 1) is the nonuniform

parameter, 𝜆 is the wavelength, the phase difference 𝜙 variesin the range 0 ≤ 𝜙 ≤ 𝜋, 𝜙 = 0 corresponds to symmetricchannel with waves out of phase (i.e., both walls movetowards outward or inward simultaneously), and further 𝑏

1,

𝑏2, and 𝜙 satisfy the following conditions for the divergent

channel at the inlet:

𝑏2

1+ 𝑏2

2+ 2𝑏1𝑏2cos (𝜙) ≤ (2𝑑)2 . (2)

It is assumed that the temperature and concentration atlower wall are 𝑇

0and 𝐶

0, respectively, while the temper-

ature and concentration at the upper wall are 𝑇1and 𝐶

1,

respectively (𝑇0< 𝑇1& 𝐶0< 𝐶1). The equations of conti-

nuity, momentum, energy, and concentration are describedas follows [11, 12]:

𝜕𝑈

𝜕𝑋

+𝜕𝑉

𝜕𝑌

= 0,

𝜌 [𝜕𝑈

𝜕𝑡+ 𝑈

𝜕𝑈

𝜕𝑋

+ 𝑉𝜕𝑈

𝜕𝑌

]

= −𝜕𝑃

𝜕𝑋

+ 𝜇[𝜕2𝑈

𝜕𝑋2+𝜕2𝑈

𝜕𝑌2] − 𝜎𝐵

2

0𝑈,

𝜌 [𝜕𝑉

𝜕𝑡+ 𝑈

𝜕𝑉

𝜕𝑋

+ 𝑉𝜕𝑉

𝜕𝑌

] = −𝜕𝑃

𝜕𝑌

+ 𝜇[𝜕2𝑉

𝜕𝑋2+𝜕2𝑉

𝜕𝑌2] ,

𝜌𝜁 [𝜕𝑇

𝜕𝑡+ 𝑈

𝜕𝑇

𝜕𝑋

+ 𝑉𝜕𝑇

𝜕𝑌

]

= 𝜅[𝜕2𝑇

𝜕𝑋2+𝜕2𝑇

𝜕𝑌2]

+ 𝜇{2[(𝜕𝑈

𝜕𝑋

)

2

+ (𝜕𝑉

𝜕𝑌

)

2

] + (𝜕𝑈

𝜕𝑌

+𝜕𝑉

𝜕𝑋

)

2

} ,

[𝜕𝐶

𝜕𝑡+ 𝑈

𝜕𝐶

𝜕𝑋

+ 𝑉𝜕𝐶

𝜕𝑌

]

= 𝐷𝑚[𝜕2𝐶

𝜕𝑋2+𝜕2𝐶

𝜕𝑌2] +

𝐷𝑚𝐾𝑇

𝑇𝑚

[𝜕2𝑇

𝜕𝑋2+𝜕2𝑇

𝜕𝑌2] ,

(3)

where 𝑈, 𝑉 are the components of velocity along 𝑋 and𝑌 directions, respectively, 𝑡 is the dimensional time, 𝜇 isthe coefficient of viscosity, 𝜎 is the electrical conductivity ofthe fluid, 𝐵

0is the uniform applied magnetic field, 𝜌 is the

fluid density, 𝜁 is the specific heat at constant volume, 𝑃 isthe pressure, 𝑇 is the temperature, 𝐶 is the concentrationof the fluid, 𝑇

𝑚is the mean temperature, 𝜅 is the thermal

conductivity,𝐷𝑚is the coefficient of mass diffusivity, and𝐾

𝑇

is the thermal diffusion ratio.

Journal of Fluids 3

The corresponding boundary conditions are

𝑈 = 0, 𝑇 = 𝑇0, 𝐶 = 𝐶

0at 𝑌 = 𝐻

1, (4a)

𝑈 = 0, 𝑇 = 𝑇1, 𝐶 = 𝐶

1, at 𝑌 = 𝐻

2. (4b)

By introducing the following set of nondimensional variablesin (3),

𝑥=𝑋

𝜆, 𝑦=𝑌

𝑑, 𝑡=𝑐𝑡

𝜆,

𝑢=𝑈

𝑐, V =

𝑉

𝑐, 𝛿 =

𝑑

𝜆,

ℎ1=𝐻1

𝑑, ℎ2=𝐻2

𝑑, 𝑝=𝑑2𝑃

𝑐𝜆𝜇,

𝜃 =𝑇 − 𝑇0

𝑇1− 𝑇0

, Θ =𝐶 − 𝐶

0

𝐶1− 𝐶0

,

(5)

we obtain

𝑅𝛿[𝜕𝑢

𝜕𝑡+ 𝑢 𝜕𝑢

𝜕𝑥+ V

𝜕𝑢

𝜕𝑦]

= −𝜕𝑝

𝜕𝑥+ 𝛿2 𝜕2𝑢

𝜕𝑥2+𝜕2𝑢

𝜕𝑦2−𝑀2𝑢,

𝑅𝛿3[𝜕V

𝜕𝑡+ 𝑢 𝜕V

𝜕𝑥+ V

𝜕V

𝜕𝑦]

= −𝜕𝑝

𝜕𝑦+ 𝛿2[𝜕2V

𝜕𝑥2+𝜕2V

𝜕𝑦2] ,

𝑅𝛿 [𝜕𝜃

𝜕𝑡+ 𝑢 𝜕𝜃

𝜕𝑥+ V

𝜕𝜃

𝜕𝑦]

=1

Pr[𝛿2 𝜕2𝜃

𝜕𝑥2+𝜕2𝜃

𝜕𝑦2]

+ 𝐸[2[𝛿2((

𝜕𝑢

𝜕𝑥)

2

+ (𝜕V

𝜕𝑦)

2

)]

+[𝜕𝑢

𝜕𝑦+ 𝛿

𝜕V

𝜕𝑥]

2

] ,

𝑅𝛿 [𝜕Θ

𝜕𝑡+ 𝑢 𝜕Θ

𝜕𝑥+ V

𝜕Θ

𝜕𝑦]

=1

Sc[𝛿2 𝜕2Θ

𝜕𝑥2+𝜕2Θ

𝜕𝑦2] + Sr[𝛿2 𝜕

2𝜃

𝜕𝑥2+𝜕2𝜃

𝜕𝑦2] ,

(6)

where 𝑎 (= 𝑏1/𝑑) and 𝑏 (= 𝑏

2/𝑑) are nondimensional ampli-

tudes of lower and upper walls, respectively, 𝑘1= (𝜆𝑚

/𝑑) is

nonuniformparameter,𝑅 (= 𝑐𝑑𝜌/𝜇) is the Reynolds number,Sc = 𝜇/𝐷

𝑚𝜌 is the Schmidt number, Sr = 𝜌𝐷

𝑚𝐾𝑇(𝑇1−

𝑇0)/𝜇(𝐶

1− 𝐶0)𝑇𝑚is the Soret number,𝑀 = √𝜎/𝜇𝐵

0𝑑 is the

Hartmann number, Pr = 𝜌]𝜁/𝜅 is the Prandtl number, and𝐸 = 𝑐2/𝜁(𝑇1− 𝑇0) is the Eckert number.

In order to discuss the results quantitatively, we assumethe instantaneous volume rate of the flow 𝐹(𝑥, 𝑡), periodic in(𝑥 − 𝑡) [17–19], as

𝐹 (𝑥, 𝑡) = 𝑄 + 𝑎 sin [2𝜋 (𝑥 − 𝑡) + 𝜙] + 𝑏 sin 2𝜋 (𝑥 − 𝑡) , (7)

where𝑄 is the time-average of the flow over one period of thewave and

𝐹 = ∫

ℎ2

ℎ1

𝑢 𝑑𝑦. (8)

3. Exact Analytical Solution

Using the long wavelength approximation and neglecting thewave number along with low Reynolds number and omittingprime, one can find from (6) that

0 = −𝜕𝑝

𝜕𝑥+𝜕2𝑢

𝜕𝑦2−𝑀2𝑢, (9)

0 = −𝜕𝑝

𝜕𝑦, (10)

0 =1

Pr𝜕2𝜃

𝜕𝑦2+ 𝐸(

𝜕𝑢

𝜕𝑦)

2

, (11)

0 =𝜕2Θ

𝜕𝑦2+ ScSr𝜕

2𝜃

𝜕𝑦2. (12)

Equation (10) shows that 𝑝 is not function of 𝑦.The corresponding boundary conditions are

𝑢 = 0, 𝜃 = 1, and Θ = 1

at 𝑦 = ℎ2= 1 + 𝑘

1𝑥 + 𝑏 sin (2𝜋 (𝑥 − 𝑡)) ,

(13a)

𝑢 = 0, 𝜃 = 0, and Θ = 0

at 𝑦 = ℎ1= −1 − 𝑘

1𝑥 − 𝑎 sin (2𝜋 (𝑥 − 𝑡) + 𝜙)

(13b)

which satisfy, at the inlet of channel,

𝑎2+ 𝑏2+ 2𝑎𝑏 cos (𝜙) ≤ 4. (14)

The set of (9)–(12), subject to the conditions (13a) and (13b),are solved exactly for 𝑢, 𝜃, and Θ, and we have

𝑢 =(𝜕𝑝/𝜕𝑥)

𝑀2

× (1 −(cosh (𝑀ℎ

2) − cosh (𝑀ℎ

1)) sinh (𝑀ℎ

1)

𝑀2 sinh (𝑀 (ℎ1− ℎ2))

)

×cosh (𝑀𝑦)cosh (𝑀ℎ

1)+(𝜕𝑝/𝜕𝑥)

𝑀2

× ((cosh (𝑀ℎ


1)) sinh (𝑀𝑦)


− 1) ,

4 Journal of Fluids

𝜃 = 𝐴1+ 𝐵1𝑦 −

Pr𝐸𝐴2

8[cosh (2𝑀𝑦) − 2𝑦2𝑀2]

−Pr𝐸𝐵2

8[cosh (2𝑀𝑦) + 2𝑦2𝑀2]

−Pr𝐸𝐴𝐵

8sinh (2𝑀𝑦) ,

Θ = 4𝐷3ScSr𝑀2ℎ2

1− 𝐷4ℎ1− 𝐷1ScSr𝑒2𝑀ℎ1

− 𝐷2ScSr𝑒−2𝑀ℎ1 + 𝐷

4𝑦 + 𝐷

1ScSr𝑒2𝑀𝑦

+ 𝐷2ScSr𝑒−2𝑀𝑦 − 4𝐷

3ScSr𝑀2𝑦2,

𝜕𝑝

𝜕𝑥= 𝐹𝑀

3 cosh (𝑀ℎ1) sinh (𝑀 (ℎ

1− ℎ2))

× ((sinh (𝑀ℎ2) − sinh (𝑀ℎ

1))

× [sinh (𝑀 (ℎ1− ℎ2))

− [cosh (𝑀ℎ2) − cosh (𝑀ℎ

1)] sinh (𝑀ℎ

1)])−1

+𝐹𝑀3 sinh (𝑀 (ℎ

1− ℎ2))

[cosh (𝑀ℎ2) − cosh (𝑀ℎ

1)]2−

𝐹𝑀2

ℎ2− ℎ1

,

(15)

where

𝐴 =

((𝜕𝑝/𝜕𝑥) − 𝐵𝑀2 sinh (𝑀ℎ

1))

𝑀2 cosh (𝑀ℎ1)

,

𝐵 =(𝜕𝑝/𝜕𝑥) (cosh (𝑀ℎ


1))


,

𝐴1=Pr𝐸𝐴2

8[cosh (2𝑀ℎ

1) − 2ℎ

2

1𝑀2]

+Pr𝐸𝐵2

8[cosh (2𝑀ℎ

1) + 2ℎ

2

1𝑀2]

+Pr𝐸𝐴𝐵

8sinh [2𝑀ℎ

1] − 𝐷ℎ

1,

𝐵1=

1

ℎ2− ℎ1

[1 +Pr𝐸𝐴2

8[cosh (2𝑀ℎ

2) − 2ℎ

2

2𝑀2

− cosh (2𝑀ℎ1) + 2ℎ

2

1𝑀2]

+Pr𝐸𝐵2

8[cosh (2𝑀ℎ

2) + 2ℎ

2

2𝑀2

− cosh (2𝑀ℎ1) − 2ℎ

2

1𝑀2]

+Pr𝐸𝐴𝐵

8[sinh (2𝑀ℎ

2) − sinh (2𝑀ℎ

1)]] ,

𝐷1=Pr𝐸𝐴2

16+Pr𝐸𝐵2

16+Pr𝐸𝐴𝐵

8,

𝐷2=Pr𝐸𝐴2

16+Pr𝐸𝐵2

16−Pr𝐸𝐴𝐵

8,

𝐷3=Pr𝐸𝐴2

16−Pr𝐸𝐵2

16,

𝐷4=

1

ℎ1− ℎ2

(4𝐷3ScSr𝑀2 (ℎ2

1− ℎ2

2)

− 𝐷1ScSr (𝑒2𝑀ℎ1 − 𝑒2𝑀ℎ2)

− 𝐷2ScSr (𝑒−2𝑀ℎ1 − 𝑒−2𝑀ℎ2) − 1) .

(16)

The coefficient of heat transfer at the lower wall is given by

𝑍 = ℎ1𝑥𝜃𝑦. (17)

4. Numerical Results and Discussion

The effect of various flow parameters on temperature isplotted in Figure 2 for the fixed values of 𝑥 = 0.6 and 𝑡 = 0.4.The influence of the nonuniform parameter (𝑘

1) on 𝜃 is

depicted in Figure 2(a). It is noticed that the temperature(𝜃) increases nearer to the lower wall of the tapered channelwhile the situation is reversed as nonuniform parameter (𝑘

1)

increases. Figure 2(b) reveals that the temperature profileincreases with increase of Eckert number. It is consideredfrom Figure 2(c) that the temperature profile increases as theamplitude of lower tapered channel increases. The variationof the Prandtl number (Pr) on 𝜃 is shown in Figure 2(d).This figure indicates that an increase in Prandtl numberresults in increase in the temperature of the fluid. Figure 2(e)depicts the effect of temperature for the various values of𝑄. It is observed that the temperature 𝜃 increases with anincrease in the time-average flow rate 𝑄 in the entire taperedchannel. The variations of the Hartmann number𝑀 against𝜃 are shown in Figure 2(f). This figure indicates that byincreasing the Hartmann number the temperature decreases.The result presented in Figure 3 indicates the behavior of 𝐸,𝑎, 𝑄,𝑀, 𝜙, and Pr on the heat transfer coefficient (𝑍). Thesefigures display the typical oscillatory behavior of heat transferwhich may be due to the phenomenon of peristalsis. Figures3(a)–3(f) reveal that the absolute value of the heat transfercoefficient increases by increasing 𝐸, 𝑎, 𝑄,𝑀, 𝜙, and Pr.

The effect of various physical parameters on the con-centration of the fluid (Θ) is shown in Figure 4. Figure 4(a)exposes that the fluid concentration decreases with anincrease of 𝐸. It is noticed from Figure 4(b) that the concen-tration of the fluid increases as 𝑎 increases. Figure 4(c) showsthat the absolute value of concentration distribution increasesat the central part of channelwhen 𝑘

1is increased. Figure 4(d)

is plotted to see the influence of𝑀 on the concentration. Wenotice that an increase in𝑀 increases Θ. The concentrationfor the phase difference is shown in Figure 4(e). It is observedthat an increase in 𝜙 causes increase in Θ. In Figure 4(f),the cause of Prandtl number Pr on Θ is captured. It is

Journal of Fluids 5

−1.5 −1 −0.5 0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

y

k1 = 0.1

k1 = 0.2

k1 = 0.4

𝜃

(a)

−1.5 −1 −0.5 0 0.5 1 1.50

0.20.40.60.8

11.21.41.61.8

2

y

E = 0.5E = 2

E = 4

𝜃

(b)

y

−1.5 −1 −0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

a = 0.1a = 0.3

a = 0.5

𝜃

(c)

−1 −0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

y

Pr = 0.75Pr = 2

Pr = 5

𝜃

(d)

−1 −0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

y

Q = 1.5Q = 1.7

Q = 1.9

𝜃

(e)

−1 −0.5 0 0.5 10

0.5

1

1.5

2

y

M = 0.5M = 1

M = 3

𝜃

(f)

Figure 2: Temperature distribution (for fixed value of 𝑡 = 0.4 and 𝑥 = 0.6): (a) 𝑄 = 1.7, 𝜙 = 𝜋/3,𝑀 = 1, 𝐸 = 0.5, Pr = 1.7, 𝑎 = 0.3, and𝑏 = 0.4; (b) 𝐸 = 0.5, 𝑘

1= 0.2, 𝑄 = 1.85,𝑀 = 2, 𝜙 = 𝜋/4, 𝑎 = 0.35, and 𝑏 = 0.3; (c) 𝐸 = 1, 𝜙 = 𝜋/3, Pr = 2,𝑀 = 0.5, 𝑘

1= 0.3, 𝑄 = 1.75, and

𝑏 = 0.3; (d) 𝐸 = 0.5, 𝑘1= 0.2, 𝑄 = 1.85,𝑀 = 2, 𝜙 = 𝜋/4, 𝑎 = 0.35, and 𝑏 = 0.3; (e) 𝐸 = 0.7, 𝜙 = 𝜋/6, Pr = 0.6, 𝑘

1= 0.25,𝑀 = 0.8, 𝑎 = 0.1,

and 𝑏 = 0.2; (f) 𝜙 = 𝜋/2, 𝐸 = 2, Pr = 1.5, 𝑏 = 0.3, 𝑎 = 0.2, 𝑘1= 0.3, and 𝑄 = 1.6.

6 Journal of Fluids

−7 −6 −5 −4 −3 −2 −1 0 10

0.10.20.30.40.50.60.70.80.9

1

Z

x

E = 0.5E = 2

E = 5

(a)

−10 −5 0 50

0.10.20.30.40.50.60.70.80.9

1

Z

x

a = 0.1a = 0.3

a = 0.6

(b)

−25 −20 −15 −10 −5 0 5 10 150

0.10.20.30.40.50.60.70.80.9

1

Z

x

Q = 1.6Q = 1.7

Q = 1.8

(c)

−15 −10 −5 0 5 10 150

0.10.20.30.40.50.60.70.80.9

1

Z

x

M = 1M = 2

M = 5

(d)

−4 −3 −2 −1 0 1 2 30

0.10.20.30.40.50.60.70.80.9

1

Z

x

𝜙 = 𝜋/6

𝜙 = 𝜋/4

𝜙 = 𝜋/2

(e)

−8 −6 −4 −2 0 2 40

0.10.20.30.40.50.60.70.80.9

1

Z

x

Pr = 0.5Pr = 1

Pr = 1.5

(f)

Figure 3: Coefficient of heat transfer distribution: (a) 𝑄 = 1.8, Pr = 0.5, 𝑘1= 0.5,𝑀 = 3, 𝑡 = 0.5, 𝜙 = 𝜋/2, 𝑏 = 0.2, and 𝑎 = 0.1; (b) 𝑄 = 1.6,

𝑀 = 0.8, 𝑡 = 0.5, 𝜙 = 𝜋, 𝑘1= 0.2, 𝐸 = 1, 𝑏 = 0.2, and Pr = 1.1; (c) 𝑡 = 0.35, 𝐸 = 2, 𝑎 = 0.4, 𝑘

1= 0.25,𝑀 = 2.5, Pr = 1.5, 𝜙 = 𝜋/3, 𝑘

1= 0.1,

and 𝑏 = 0.3; (d) 𝑄 = 1.6, 𝑡 = 0.4, 𝜙 = 𝜋/4, 𝐸 = 2, 𝑏 = 0.35, Pr = 1, and 𝑎 = 0.4; (e) 𝑄 = 1.78, 𝑥 = 0.48, 𝐸 = 1, Pr = 0.7, 𝑎 = 0.26, 𝑏 = 0.3,𝑘1= 0.29, and𝑀 = 1.6; (f) 𝑄 = 1.72, 𝑥 = 0.42, 𝐸 = 1, 𝑎 = 0.2, 𝑘

1= 0.23,𝑀 = 1.2, 𝜙 = 𝜋/4, and 𝑏 = 0.3.

Journal of Fluids 7

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.81

y

E = 1E = 3

E = 6

Θ

(a)

−1 −0.5 0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.91

y

a = 0.1a = 0.3

a = 0.5

Θ

(b)

−1 −0.5 0 0.5−0.2

0

0.2

0.4

0.6

0.8

1

y

k1 = 0.1

k1 = 0.2

k1 = 0.3

Θ

(c)

−1 −0.5 0 0.5

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

y

M = 1M = 3

M = 5

Θ

(d)

−1 −0.5 0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y

Θ

𝜙 = 0

𝜙 = 𝜋/3

𝜙 = 𝜋/2

(e)

−1 −0.5 0 0.5 1−0.5

0

0.5

1

y

Pr = 0.75Pr = 1

Pr = 2

Θ

(f)

Figure 4: Continued.

8 Journal of Fluids

−1 −0.5 0 0.5

0

0.2

0.4

0.6

0.8

1

y

Q = 1.5Q = 1.7

Q = 1.9

Θ

(g)

−1 −0.5 0 0.5−0.2

0

0.2

0.4

0.6

0.8

1

y

Sc = 0.3Sc = 0.4

Sc = 0.5

Θ

(h)

Figure 4: Mass transfer distribution (fixed value of 𝑥 = 0.5 and 𝑡 = 0.6): (a) 𝑄 = 1.7, 𝑎 = 0.2, 𝑏 = 0.1, 𝜙 = 0, Pr = 0.7, 𝑘1= 0.5,𝑀 = 0.8,

Sc = 0.6, and Sr = 1.6; (b) 𝐸 = 0.5, 𝑎 = 0.2, 𝑄 = 1.6, 𝜙 = 𝜋/2, Pr = 0.5, 𝑘1= 0.1,𝑀 = 0.5, Sc = 0.5, and Sr = 1.4; (c) 𝑎 = 0.4, 𝑏 = 0.3,

𝐸 = 1.25, 𝑄 = 1.65, 𝜙 = 𝜋/3, Pr = 1.5,𝑀 = 0.8, Sc = 0.38, and Sr = 1.25; (d) 𝑏 = 0.5, 𝑎 = 0.4, 𝐸 = 2, 𝑄 = 1.55, 𝜙 = 𝜋/6, Pr = 2, 𝑘1= 0.15,

Sc = 0.52, and Sr = 1.42; (e) 𝑏 = 0.2, 𝑎 = 0.1, 𝐸 = 0.9, 𝑄 = 1.8, Pr = 0.75, 𝑘1= 0.3,𝑀 = 0.2, Sc = 0.4, and Sr = 1.22; (f) 𝑏 = 0.25, 𝑎 = 0.2,

𝐸 = 1.5, 𝑄 = 1.75, 𝜙 = 𝜋, 𝑘1= 0.25,𝑀 = 0.5, Sc = 0.44, and Sr = 1.3; (g) 𝑎 = 0.6, 𝑏 = 0.5, 𝐸 = 0.75, 𝜙 = 𝜋/3, Pr = 2, 𝑘

1= 0.4,𝑀 = 3,

Sc = 0.3, and Sr = 1.5; (h) 𝑏 = 0.4, 𝑎 = 0.3, 𝐸 = 1.2, 𝑄 = 1.65, 𝜙 = 𝜋/4, Pr = 1.6,𝑀 = 2.5, and Sr = 1.53.

detected that with an increase in Pr the concentration of thefluid decreases. Figure 4(g) depicts the concentration profilecorresponding to various values of 𝑄. It is seen that theconcentration distribution decreases with an increase in themean flow rate 𝑄. From Figure 4(h), one can view that theconcentration profile decreases with increase of Sc.

5. Conclusion

The present analysis can serve as a model which may helpin understanding the mechanism of physiological flow ofa Newtonian fluid in tapered asymmetric channel. Theanalytical solutions for the temperature, concentration, andcoefficient of heat transfer have been obtained under longwavelength and low Reynolds number approximations. Thefeatures of the flow characteristics are analyzed by plottinggraphs and discussed in detail. The main observations foundfrom the present study are given as follows.

(1) There is an increase in the temperature (𝜃) when thePrandtl number Pr, time-average flow rate 𝑄, occlu-sion parameter 𝑎, and Eckert number are increasedwhile it decreases when𝑀 is increased.

(2) Mass transfer (Θ) increases with the increase of 𝑎,𝑀,and 𝜙 while it decreases with the increase of 𝐸, Pr, 𝑄,and Sc.

(3) The temperature distribution of the fluid increases atthe core part of channel when nonuniform parameteris increased and the reverse situation is observed inrespect of the concentration of the fluid.

(4) The absolute value of the heat transfer coefficientincreases when the𝐸, 𝑎,𝑄,𝑀,𝜙, and Pr are increased.

Conflict of Interests

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

References

[1] T. W. Latham, Fluid Motion in a Peristaltic Pump, MIT Press,Cambridge, Mass, USA, 1966.

[2] A. H. Shapiro, M. Y. Jaffrin, and S. L. Weinberg, “Peristalticpumping with long wave lengths at low Reynolds number,”Journal of Fluid Mechanics, vol. 37, no. 4, pp. 799–825, 1969.

[3] G. Radhakrishnamacharya and V. Radhakrishna Murty, “Heattransfer to peristaltic transport in a non-uniform channel,”Defence Science Journal, vol. 43, no. 3, pp. 275–280, 1993.

[4] T. Hayat, Y. Wang, K. Hutter, S. Asghar, and A. M. Siddiqui,“Peristaltic transport of anOldroyd-B fluid in a planar channel,”Mathematical Problems in Engineering, vol. 2004, no. 4, pp. 347–376, 2004.

[5] K. Vajravelu, S. Sreenadh, and V. R. Babu, “Peristaltic transportof a Herschel-Bulkley fluid in an inclined tube,” InternationalJournal of Non-Linear Mechanics, vol. 40, no. 1, pp. 83–90, 2005.

[6] S. Nadeem, T. Hayat, N. S. Akbar, and M. Y. Malik, “On theinfluence of heat transfer in peristalsis with variable viscosity,”International Journal of Heat and Mass Transfer, vol. 52, no. 21-22, pp. 4722–4730, 2009.

[7] T. Hayat, M. U. Qureshi, and Q. Hussain, “Effect of heat transferon the peristaltic flow of an electrically conducting fluid in

Journal of Fluids 9

a porous space,” Applied Mathematical Modelling, vol. 33, no. 4,pp. 1862–1873, 2009.

[8] K. S. Mekheimer, S. Z. A. Husseny, and Y. A. Elmaboud, “Effectsof heat transfer and space porosity on peristaltic flow in avertical asymmetric channel,” Numerical Methods for PartialDifferential Equations, vol. 26, no. 4, pp. 747–770, 2010.

[9] A. Ogulu, “Effect of heat generation on low Reynolds numberfluid andmass transport in a single lymphatic blood vessel withuniformmagnetic field,” International Communications in Heatand Mass Transfer, vol. 33, no. 6, pp. 790–799, 2006.

[10] N. T. M. Eldabe, M. F. El-Sayed, A. Y. Ghaly, and H. M. Sayed,“Mixed convective heat and mass transfer in a non-Newtonianfluid at a peristaltic surfacewith temperature-dependent viscos-ity,” Archive of Applied Mechanics, vol. 78, no. 8, pp. 599–624,2008.

[11] S. Srinivas and M. Kothandapani, “The influence of heat andmass transfer on MHD peristaltic flow through a porous spacewith compliant walls,” Applied Mathematics and Computation,vol. 213, no. 1, pp. 197–208, 2009.

[12] S. Srinivas and M. Kothandapani, “Peristaltic transport in anasymmetric channel with heat transfer—a note,” InternationalCommunications in Heat and Mass Transfer, vol. 35, no. 4, pp.514–522, 2008.

[13] T. A. Ogulu, “Effect of heat generation on low Reynolds numberfluid andmass transport in a single lymphatic blood vessel withuniformmagnetic field,” International Communications in Heatand Mass Transfer, vol. 33, no. 6, pp. 790–799, 2006.

[14] O. Eytan, A. J. Jaffa, and D. Elad, “Peristaltic flow in a taperedchannel: application to embryo transport within the uterinecavity,”Medical Engineering and Physics, vol. 23, no. 7, pp. 473–482, 2001.

[15] M. Kothandapani and J. Prakash, “The peristaltic transport ofCarreau nanofluids under effect of a magnetic field in a taperedasymmetric channel: application of the cancer therapy,” Journalof Mechanics in Medicine and Biology, vol. 15, Article ID1550030, 2014.

[16] M. Kothandapani and J. Prakash, “Effect of radiation and mag-netic field on peristaltic transport of nanofluids through aporous space in a tapered asymmetric channel,” Journal of Mag-netism and Magnetic Materials, vol. 378, pp. 152–163, 2015.

[17] L. M. Srivastava, V. P. Srivastava, and S. N. Sinha, “Peristaltictransport of a physiological fluid. Part-I. Flow in non-uniformgeometry,” Biorheology, vol. 20, no. 2, pp. 153–166, 1983.

[18] M. Kothandapani and J. Prakash, “Influence of heat source,thermal radiation and inclined magnetic field on peristalticflow of a Hyperbolic tangent nanofluid in a tapered asymmetricchannel,” IEEE Transactions on NanoBioscience, 2014.

[19] M. Kothandapani and J. Prakash, “Effects of thermal radiationparameter and magnetic field on the peristaltic motion of Will-iamson nanofluids in a tapered asymmetric channel,” Interna-tional Journal of Heat and Mass Transfer, vol. 81, pp. 234–245,2015.

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