research article analysis of peristaltic motion of a nanofluid ...research article analysis of...

Research ArticleAnalysis of Peristaltic Motion of a Nanofluid with Wall ShearStress, Microrotation, and Thermal Radiation Effects

C. Dhanapal,1 J. Kamalakkannan,2,3 J. Prakash,4 and M. Kothandapani5

1Department of Science and Humanities, Adhiparasakthi College of Engineering, Kalavai 632506, India2Bharathiar University, Coimbatore 641046, India3Department of Mathematics, Priyadarshini Engineering College, Vaniyambadi, Tamil Nadu 635751, India4Department of Mathematics, Agni College of Technology, Thalambur, Chennai 600130, India5Department of Mathematics, University College of Engineering Arni, Arni, Tamil Nadu 632326, India

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

Received 26 April 2016; Revised 10 July 2016; Accepted 11 July 2016

Academic Editor: Saverio Affatato

Copyright © 2016 C. Dhanapal et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper analyzes the peristaltic flow of an incompressible micropolar nanofluid in a tapered asymmetric channel in the presenceof thermal radiation and heat sources parameters.The rotation of the nanoparticles is incorporated in the flowmodel.The equationsgoverning the nanofluid flow are modeled and exact solutions are managed under long wavelength and flow Reynolds number andlong wavelength approximations. Explicit expressions of axial velocity, stream function, microrotation, nanoparticle temperature,and concentration have been derived.The phenomena of shear stress and trapping have also been discussed. Finally, the influencesof various parameters of interest on flow variables have been discussed numerically and explained graphically. Besides, the resultsobtained in this paper will be helpful to those who are working on the development of various realms like fluidmechanics, the rota-tion, Brownian motion, thermophoresis, coupling number, micropolar parameter, and the nondimensional geometry parameters.

1. Introduction

Peristaltic pumping is one of the keystones for the develop-ment of science and engineering research in modern years.Peristalsis also plays an indispensable role in transportingphysiological fluids inside living bodies, and many biome-chanical and engineering devices have been designed on thebasis of the principle of peristaltic pumping to transportfluids without internal moving parts. The problem of themechanism of peristaltic transport has attracted the attentionofmany investigators since the first exploration of Latham [1].A number of analytical, numerical, and experimental studieson peristaltic motion of different fluids have been describedunder various conditions with reference to physiological andmechanical environment [2–8].

Micropolar fluids have been a subject of great interest toresearch workers and a number of research papers have beenpublished on this flow model. Physically, micropolar fluidsmay represent fluids consisting of rigid, randomly oriented(or spherical) particles suspended in a viscous medium,

where the deformation of fluid particles is ignored. Thisconstitutes a substantial generalization of the Navier-Stokesmodel and opens a new field of potential applications includ-ing a large number of complex fluids. Animal bloods and liq-uid crystals (with dumbbell typemolecules) are few examplesof micropolar fluids. Local conservation laws of mass, linear,and angular momentum and the energy for polar fluids werereceived by Grad [9] by using the method of statistical ther-modynamics. Eringen [10] proposed the theory ofmicropolarfluids in which the microscopic effect arises from localstructure and fluid elements of micromotion are taken intoaccount. Later, Eringen [11] generalized the micropolar fluidstheory to include thermal effects. Using quasi-linearizationfinite difference technique, an impact of temperature depen-dent heat sources and frictional heating on the fully devel-oped free convection micropolar fluid flow between twoporous parallel plates was analyzed by Agarwal andDhanapal[12]. Devi and Devanathan [13] premeditated the peristalticmotion of a micropolar fluid in a cylindrical tube with sinu-soidal waves of small amplitude travelling down in its flexible

Hindawi Publishing CorporationApplied Bionics and BiomechanicsVolume 2016, Article ID 4123741, 15 pageshttp://dx.doi.org/10.1155/2016/4123741

2 Applied Bionics and Biomechanics

wall for the case of low Reynolds number. Srinivasacharyaet al. [14] recently examined the peristaltic transport ofa micropolar fluid in a circular tube using low Reynoldsnumber and long wavelength assumptions.

Nowadays, there is a continuous focus of the researchersin the flow analysis of nanofluids because of its large numberof applications in biomedical and industrial engineering.Choi [15] was the first who initiated this nanofluid tech-nology. A detailed analysis of nanofluids was discussed byBuongiorno [16]. Sheikholeslami et al. [17] studied the naturalconvection in a concentric annulus between a cold outersquare and heated inner circular cylinders in the presenceof static radial magnetic field. After initiating a study ofnanofluids flow under the effect of peristalsis by Akbar andNadeem [18], Akbar et al. [19] discussed the slip effects on theperistaltic transport of nanofluid in an asymmetric channel.Recently, Mustafa et al. [20] examined the influence of wallproperties on the peristaltic flow of a nanofluid. Mixed con-vection peristaltic flows of magnetohydrodynamic (MHD)nanofluids were analyzed by Hayat et al. [21]. The effects ofwall properties on the peristaltic flow of an incompressiblepseudoplastic fluid in a curved channel were investigatedby Hina et al. [22]. Hayat et al. [23] studied the peristaltictransport of viscous nanofluid in an asymmetric channel.The channel walls satisfy the convective conditions and alsoeffects of Brownian motion and thermophoresis have alsobeen taken into account. The influence of nanofluid char-acteristic on peristaltic heat transfer in a two-dimensionalaxisymmetric channel was discussed analytically by Tripathiand Bég [24]. Moreover, the tremendous applications ofnanofluids and the interaction of nanoparticles in peristalticflows have obtained attentions of many researchers [25, 26].

In the recent years, it is well known by physiologists[27, 28] that the intrauterine fluid flow due to myometrialcontractions displays peristalsis andmyometrial contractionsmay occur in both symmetric and asymmetric directions andalso noted that blood behaves like as a non-Newtonian fluidin microcirculation [10–12]. Motivated from the above analy-sis and the importance of peristaltic flows, the purpose of thepresent paper is to investigate the effects of thermal radiationand heat source/sink on the peristaltic flow of micropolarnanofluids in the tapered asymmetric channel. Therefore,such a consideration of peristaltic transport may be used toevaluate intrauterine fluid flow in a nonpregnant uterus [29].To the best of the author’s knowledge, no attempt is availablein the literature which deals with the peristalsis flow ofmicropolar nanofluid in the tapered asymmetric channel.Thepresent analysis of peristaltic flow is confined to large wave-length and low Reynolds number assumptions. Explicit solu-tions are developed for axial velocity, axial pressure gradient,stream function, microrotation of the nanofluids, nanofluidtemperature, and nanoparticle concentration. The numericaldiscussion of the pressure rise, shear stresses, and trapping arealso obtained and the results are discussed through graphs.

2. Mathematical Formulation

Let us consider the motion of peristaltic transport of anincompressible micropolar nanofluid through a tapered

0

C = C0T = T0

c

XH1 H2

C = C1T = T1

Y

d

d

𝜆

𝜙

a1

a2

Figure 1: Geometry of the generalized channel (tapered asymmetricchannel) with peristaltic wave motion of wall.

channel induced by sinusoidal wave trains propagating withconstant speed but with different amplitudes and phases; seeFigure 1. The governing equations of motion for the presentinvestigation are [13, 16, 30]

𝜕𝑈

𝜕𝑋+𝜕𝑉

𝜕𝑌= 0,

𝜌𝑓 [𝜕𝑈

𝜕𝑡+ 𝑈

𝜕𝑈

𝜕𝑋+ 𝑉

𝜕𝑈

𝜕𝑌]

= −𝜕𝑃

𝜕𝑋+ (𝜇V + 𝑘V) ∇

2𝑈 − 𝑘V

𝜕𝑊

𝜕𝑌

+ (1 − 𝐶0) 𝜌𝑓𝑔𝛼 (𝑇 − 𝑇0)

+ (𝜌𝑝 − 𝜌𝑓) 𝑔𝛽(𝐶 − 𝐶0) ,

𝜌𝑓 [𝜕𝑉

𝜕𝑡+ 𝑈

𝜕𝑉

𝜕𝑋+ 𝑉

𝜕𝑉

𝜕𝑌]

= −𝜕𝑃

𝜕𝑌+ (𝜇V + 𝑘V) ∇

2𝑉 − 𝑘V

𝜕𝑊

𝜕𝑋,

𝜌𝑓𝐽 [𝜕𝑊

𝜕𝑡+ 𝑈

𝜕𝑊

𝜕𝑋+ 𝑉

𝜕𝑊

𝜕𝑌]

= 𝛾V∇2𝑊+ 𝑘V (

𝜕𝑉

𝜕𝑋−𝜕𝑈

𝜕𝑌) − 2𝑘V𝑊,

(𝜌𝑐)𝑓[𝜕𝑇

𝜕𝑡+ 𝑈

𝜕𝑇

𝜕𝑋+ 𝑉

𝜕𝑇

𝜕𝑌]

= 𝜅 [𝜕2𝑇

𝜕𝑋2+𝜕2𝑇

𝜕𝑌2] − (

𝜕𝑞𝑟

𝜕𝑋+𝜕𝑞𝑟

𝜕𝑌) + 𝑄0 (𝑇 − 𝑇0)

+ (𝜌𝑐)𝑝𝐷𝐵 (

𝜕𝐶

𝜕𝑋

𝜕𝑇

𝜕𝑋+𝜕𝐶

𝜕𝑌

𝜕𝑇

𝜕𝑌)

+

𝐷𝑇 (𝜌𝑐)𝑝

𝑇𝑚

[(𝜕𝑇

𝜕𝑋)

2

+ (𝜕𝑇

𝜕𝑌)

2

] ,

[𝜕𝐶

𝜕𝑡+ 𝑈

𝜕𝐶

𝜕𝑋+ 𝑉

𝜕𝐶

𝜕𝑌]

= 𝐷𝐵 [𝜕2𝐶

𝜕𝑋2+𝜕2𝐶

𝜕𝑌2] +

𝐷𝑇

𝑇𝑚

[𝜕2𝑇

𝜕𝑋2+𝜕2𝑇

𝜕𝑌2] ,

(1)

Applied Bionics and Biomechanics 3

where 𝑈, 𝑉 are the components of velocity along 𝑋 and 𝑌directions, respectively, 𝑡 is the dimensional time, the volu-metric volume expansion coefficient is 𝑐, 𝜌𝑓 is the density ofthe fluid, 𝜌𝑝 is the density of the particle, 𝑔 is the accelerationdue to gravity, 𝑃 is the pressure, 𝜇V, 𝛾V, and 𝑘V are the martialparameters [9–12],𝑇 is the temperature,𝐶 is the nanoparticleconcentration, 𝛼 is the thermal expansion coefficient, 𝜕/𝜕𝑡represents the material time derivative, 𝛽 is the coefficientof expansion with concentration, 𝑊 is the microrotation ofthe nanofluid, 𝑗 is the microgyration parameter, 𝑇𝑚 is thefluid mean temperature, 𝜏 = (𝜌𝑐)𝑝/(𝜌𝑐

)𝑓 is the ratio of

the effective heat capacity of nanoparticle material and heatcapacity of the fluid with 𝜌 being the density, 𝜅 is the thermalconductivity of the nanofluids, 𝐷𝐵 is the Brownian diffusioncoefficient,𝐷𝑇 is the thermophoretic diffusion coefficient,𝑄0is the constant heat addition/absorption, and the radioactiveheat flux is 𝑞𝑟.

Hence, for the Rosseland approximation for thermalradiation, we have [31, 32]

𝑞𝑟 = −4𝜎∗

3𝑘∗

𝜕𝑇4

𝜕𝑌, (2)

where 𝜎∗ and 𝑘∗ are the Stefan-Boltzmann constant and themean absorption coefficient.

Let 𝑌 = 𝐻1 and 𝑌 = 𝐻2 be, respectively, the left and rightwall boundaries of the tapered asymmetric channel. Heat andmass transfer along with nanoparticle phenomena have beentaken into account. The right wall of the channel is sustainedat temperature 𝑇1 and nanoparticle volume fraction 𝐶1 whilethe left wall has temperature 𝑇0 and nanoparticle volumefraction 𝐶0. The geometry of the wall surface is defined as

𝐻1 (𝑋, 𝑡) = −𝑑 − 𝑚

𝑋 − 𝑎1sin [

2𝜋

𝜆(𝑋 − 𝑐𝑡

) + 𝜙] , (3a)

𝐻2 (𝑋, 𝑡) = 𝑑 + 𝑚

𝑋 + 𝑎2sin [

2𝜋

𝜆(𝑋 − 𝑐𝑡

)] , (3b)

where 𝑎1 and 𝑎2 are the amplitudes of left and right walls,respectively, 𝜆 is the wavelength, 𝑚 (𝑚 ≪ 1) is thenonuniform parameter, the phase difference 𝜙 varies in therange 0 ≤ 𝜙 ≤ 𝜋, 𝜙 = 0 corresponds to symmetric channelwithwaves out of the phase, and further 𝑎1, 𝑎2, 𝑑, and𝜙 satisfythe condition for the divergent channel at the inlet of flow

𝑎21 + 𝑎22 + 2𝑎1𝑎2 cos (𝜙) ≤ (2𝑑)

2. (4)

The nondimensional parameters are as follows:

𝑥 =𝑋

𝜆,

𝑦 =𝑌

𝑑,

𝑡 =𝑐𝑡

𝜆,

𝑢 =𝑈

𝑐,

V =𝑉

𝑐𝛿,

𝛿 =𝑑

𝜆,

ℎ1 =𝐻1

𝑑,

ℎ2 =𝐻2

𝑑,

Ω =𝑑𝑊

𝑐,

𝑗 =𝐽

𝑑2

𝑚 =𝑚𝜆

𝑑,

𝑎 =𝑎1

𝑑,

𝑏 =𝑎2

𝑑,

𝜃 =𝑇 − 𝑇0

𝑇1 − 𝑇0

,

𝜎 =𝐶 − 𝐶0

𝐶1 − 𝐶0

,

𝐺𝑟 =(1 − 𝐶0) 𝜌𝑓𝑔𝛼𝑑

2(𝑇1 − 𝑇0)

𝑐𝜇V,

𝛽 =𝑄0𝑑2

(𝑇1 − 𝑇0) ]𝑐𝑝,

𝑃𝑟 =𝜇𝑐𝑓

𝜅,

𝑁𝑏 =𝜏𝐷𝐵 (𝐶1 − 𝐶0)

],

𝑁𝑡 =𝜏𝐷𝑇 (𝐶1 − 𝐶0)

𝑇0],

𝑅𝑛 =16𝜎∗𝑇30

3𝑘∗𝜇V𝑐𝑓,

Br =(𝜌𝑝 − 𝜌𝑓) 𝑔𝛽

𝑑2(𝐶1 − 𝐶0)

𝑐𝜇,

Sc = V𝐷𝐵

,

𝑝 =𝑑2𝑃

𝑐𝜆𝜇,

𝑅 =𝑐𝑑𝜌𝑓

𝜇V.

(5)


Using the above nondimensional quantities in (1)–(4), theresulting equations are

𝑅𝛿(𝜕

𝜕𝑡+ 𝑢

𝜕

𝜕𝑥+ V

𝜕

𝜕𝑦)𝑢

= −𝜕𝑝

𝜕𝑥+

1

1 − 𝑁(𝑁

𝜕Ω

𝜕𝑦+ 𝛿2 𝜕2𝑢

𝜕𝑥2+𝜕2𝑢

𝜕𝑦2) + Gr𝜃

+ Br𝜎,

𝑅𝛿3(𝜕

𝜕𝑡+ 𝑢

𝜕

𝜕𝑥+ V

𝜕

𝜕𝑦) V

= −𝜕𝑝

𝜕𝑦+

𝛿2

1 − 𝑁(−𝑁

𝜕𝑤

𝜕𝑥+ 𝛿2 𝜕2V𝜕𝑥2

+𝜕2V𝜕𝑦2

) ,

𝑅𝛿𝑗 (1 − 𝑁)

𝑁(𝜕

𝜕𝑡+ 𝑢

𝜕

𝜕𝑥+ V

𝜕

𝜕𝑦)Ω

= −2Ω + (𝛿2 𝜕V𝜕𝑥

−𝜕𝑢

𝜕𝑦)

+2 − 𝑁

𝑛2(𝛿2 𝜕2Ω

𝜕𝑥2+𝜕2Ω

𝜕𝑦2) ,

𝑅𝛿 [𝜕𝜃

𝜕𝑡+ 𝑢

𝜕𝜃

𝜕𝑥+ 𝛿V

𝜕𝜃

𝜕𝑦]

=1

Pr[𝛿2 𝜕2𝜃

𝜕𝑥2+𝜕2𝜃

𝜕𝑦2] + 𝑅𝑛

𝜕2𝜃

𝜕𝑦2

+ 𝑁𝑏 [𝛿2 𝜕𝜎

𝜕𝑥

𝜕𝜃

𝜕𝑥+𝜕𝜎

𝜕𝑦

𝜕𝜃

𝜕𝑦] + 𝛽𝜃

+ 𝑁𝑡 [𝛿2(𝜕𝜃

𝜕𝑥)

2

+ (𝜕𝜃

𝜕𝑦)

2

] ,

𝑅𝛿Sc [𝜕𝜎𝜕𝑡

+ 𝑢𝜕𝜎

𝜕𝑥+ 𝛿V

𝜕𝜎

𝜕𝑦]

= 𝛿2 𝜕2𝜎

𝜕𝑥2+𝜕2𝜎

𝜕𝑦2+𝑁𝑡

𝑁𝑏

[𝛿2 𝜕2𝜃

𝜕𝑥2+𝜕2𝜃

𝜕𝑦2] .

(6)

inwhich𝑁 = 𝑘V/(𝜇V+𝑘V) is the coupling number (0 ≤ 𝑁 ≤ 1)and 𝑛2 = 𝑑2𝑘V(2𝜇V+𝑘V)/(𝛾V(𝜇V+𝑘V)) is themicropolar param-eter. We introduce the nondimensional variables such as 𝑝is dimensionless pressure, 𝑎 and 𝑏 are amplitudes of left andright walls, respectively, 𝛿 is wave number, 𝑚 is the nonuni-form parameter, 𝑅 is the Reynolds number, ] is the nanofluidkinematic viscosity, Ω is the dimensionless microrotation,𝜃 is the dimensionless temperature, 𝜎 is the dimensionlessrescaled nanoparticle volume fraction, Pr is the Prandtlnumber, Gr is the local temperature Grashof number, Br isthe local nanoparticle Grashof number, Sc is the Schmidtnumber, 𝑁𝑏 is the Brownian motion parameter, 𝑁𝑡 is thethermophoresis parameter, and 𝑅𝑛 is the radiation parameteras follows. The above equations can reduce to the classicalNavier-Stokes equation when 𝑘V → 0.

In several previous attempts [19–24], we employ the longwavelength and low Reynolds number approximations andthus (6) that

𝜕𝑝

𝜕𝑥=

1

1 − 𝑁

𝜕2𝑢

𝜕𝑦2+

𝑁

1 − 𝑁

𝜕Ω

𝜕𝑦+ Gr𝜃 + Br𝜎, (7)

𝜕𝑝

𝜕𝑦= 0, (8)

− 2Ω −𝜕𝑢

𝜕𝑦+ (

2 − 𝑁

𝑛2)𝜕2Ω

𝜕𝑦2= 0, (9)

(1 + 𝑅𝑛𝑃𝑟

𝑃𝑟

)𝜕2𝜃

𝜕𝑦2+ 𝑁𝑏 (

𝜕𝜎

𝜕𝑦

𝜕𝜃

𝜕𝑦) + 𝑁𝑡 (

𝜕𝜃

𝜕𝑦)

2

+ 𝛽𝜃

= 0,

(10)

𝜕2𝜎

𝜕𝑦2+𝑁𝑡

𝑁𝑏

𝜕2𝜃

𝜕𝑦2= 0. (11)

The appropriate boundary conditions are

𝑢 = 0,

Ω = 0,

𝜃 = 0,

𝜎 = 0

at 𝑦 = ℎ1 = −1 − 𝑚𝑥 − 𝑎 sin (2𝜋 (𝑥 − 𝑡) + 𝜙) ,

(12a)

𝑢 = 0,

Ω = 0,

𝜃 = 1,

𝜎 = 1

at 𝑦 = ℎ2 = 1 + 𝑚𝑥 + 𝑏 sin (2𝜋 (𝑥 − 𝑡)) ,

(12b)

which satisfy, at the inlet of channel,

𝑎2+ 𝑏2+ 2𝑎𝑏 cos (𝜙) ≤ 4. (13)

3. Exact Solution

By integration of (11) with respect to 𝑦 and implementationin (10) and boundary conditions of (12a) and (12b), thenanoparticles temperature field is obtained as

𝜃

=sinh (ℎ1𝜃1) cosh (𝜃1𝑦) − cosh (ℎ1𝜃1) sinh (𝜃2𝑦)sinh (ℎ1𝜃1) cosh (𝜃1ℎ2) − cosh (ℎ1𝜃1) sinh (𝜃2ℎ2)

.

(14)

Substituting (14) into (11), moreover integrating (11) withrespect to 𝑦 and using proper boundary conditions of (12a)and (12b), the nanoparticle concentration field is received as


𝜎 =(ℎ − 𝑦) (𝑁𝑏 + 𝐵𝑁𝑡 sinh (ℎ2𝜃2) + 𝐴𝑁𝑡 cosh (ℎ2𝜃1)) + (𝑦 − ℎ2)𝑁𝑡 (𝐵 sinh (ℎ1𝜃2) + 𝐴 cosh (ℎ1𝜃1))

𝑁𝑏 (ℎ1 − ℎ2)

−𝑁𝑡

𝑁𝑏

(𝐴 cosh (𝜃1𝑦) + 𝐵 sinh (𝜃2𝑦)) .(15)

Equation (7) can be written in the following form:

𝜕2𝑢

𝜕𝑦2=

𝜕

𝜕𝑦[(1 − 𝑁)(

𝜕𝑝

𝜕𝑥)𝑦 − 𝑁𝜛] − (1 − 𝑁)Gr𝜃

− (1 − 𝑁)Br𝜎.(16)

Integration of the above equation yields

𝜕𝑢

𝜕𝑦= (1 − 𝑁)

𝜕𝑝

𝜕𝑥𝑦 − 𝑁Ω

−(1 − 𝑁)𝐺𝑟

𝜃1𝜃2

(𝜃2𝐴 sinh 𝜃1𝑦 + 𝜃1𝐵 cosh 𝜃2𝑦) − (1

− 𝑁)𝐵𝑟 [2𝑐𝑦 + 𝐷𝑦

2

2

−𝑁𝑡

𝑁𝑏𝜃1𝜃2

(𝐴𝜃2 sinh 𝜃1𝑦 + 𝐵𝜃1 cosh 𝜃2𝑦)] + 𝐺 (𝑥) .

(17)

From (9) and (17), one can write

𝜕2Ω

𝜕𝑦2− 𝑛2Ω =

𝑛2(1 − 𝑁)

2 − 𝑁(𝜕𝑝

𝜕𝑥)𝑦 + 𝐴4 sinh (𝜃1𝑦)

+ 𝐴5 cosh (𝜃2𝑦) + 𝐴6𝑦 + 𝐴7𝑦2

+𝑚2𝐺 (𝑥)

2 − 𝑁.

(18)

The general solution of the above equation can be written as

Ω = 𝐸 cosh (𝑛𝑦) + 𝐹 sinh (𝑛𝑦) − 1 − 𝑁2 − 𝑁

(𝜕𝑝

𝜕𝑥)𝑦

+𝐴4 sinh (𝜃1𝑦)

𝜃21 − 𝑛2

+𝐴5 cosh (𝜃2𝑦)

𝜃22 − 𝑛2

−𝐴6𝑦

𝑛2

− 𝐴7 (𝑛2𝑦2+ 2

𝑛4) −

𝐺 (𝑥)

2 − 𝑁.

(19)

Making the above equation into (17), one obtains

𝑢 = (𝜕𝑝

𝜕𝑥)(

𝑦2

(2 − 𝑁)−𝐴8𝑁 sinh (𝑛𝑦)

𝑛

−𝐴11𝑁 cosh (𝑛𝑦)

𝑛) −

𝐴9𝑁 sinh (𝑛𝑦)𝑛

−𝐴12𝑁 cosh (𝑛𝑦)

𝑛+ cosh (𝜃1𝑦)(

(1 − 𝑁)Br𝑁𝑡𝐴𝑁𝑏𝜃21

−(1 − 𝑁)Gr𝐴

𝜃21

−𝑁𝐴4

𝜃1 (𝜃21 − 𝑛2)) + (

𝑁𝐴7

3𝑛2

−(1 − 𝑁)Br𝐷

6)𝑦3+ sinh (𝜃2𝑦)(

(1 − 𝑁)Br𝑁𝑡𝐵𝑁𝑏𝜃22

−(1 − 𝑁)Gr𝐵

𝜃22

−𝑁𝐴5

𝜃2 (𝜃22 − 𝑛2)) +

2𝐴7𝑦𝑁

𝑛4

+𝑁𝐴6𝑦

2

2𝑛2+ 𝐺 (𝑥)(

2𝑦

(2 − 𝑁)+𝐴13𝑁 cosh (𝑛𝑦)

𝑛

+𝐴10𝑁sinh (𝑛𝑦)

𝑛) + 𝐻 (𝑥) .

(20)

The volume flux through each cross section in the wave frameis given by

𝐹 = ∫

ℎ2

ℎ1

𝑢 𝑑𝑦. (21)

Using (21), we find that

𝜕𝑝

𝜕𝑥= (𝐹 −

𝐴21

𝜃1

(sinh (𝜃1ℎ2) − sinh (𝜃1ℎ1))

−𝐴22

𝜃2

(cosh (𝜃2ℎ2) − cosh (𝜃2ℎ1))

−𝐴32 (ℎ

32 − ℎ31)

3−𝐴29

𝑛(sinh (𝑛ℎ2) − sinh (𝑛ℎ1))

−𝐴30

𝑛(cosh (𝑛ℎ2) − cosh (𝑛ℎ1)) −

𝐴24 (ℎ42 − ℎ41)

4

−𝐴31 (ℎ

22 − ℎ21)

4− 𝐴25 (ℎ2 − ℎ1))

⋅ (𝐴20 (ℎ

22 − ℎ21)

2 − 𝑁+ℎ32 − ℎ31

6 − 3𝑁


+𝐴27

𝑛(cosh (𝑛ℎ2) − cosh (𝑛ℎ1))

+𝐴28

𝑛(sinh (𝑛ℎ2) − cosh (𝑛ℎ1))

+ 𝐴26 (ℎ2 − ℎ1))

−1

.

(22)The corresponding stream function from (20) is

𝜓 = (𝜕𝑝

𝜕𝑥)(

𝑦2

(2 − 𝑁)−𝐴8𝑁(cosh (𝑛𝑦) − 1)

𝑛2

−𝐴11𝑁 sinh (𝑛𝑦)

𝑛2) −

𝐴9𝑁 cosh (𝑛𝑦)𝑛2

−𝐴12𝑁 sinh (𝑛𝑦)

𝑛2+ (

sinh (𝜃1𝑦)𝜃1

)

⋅ ((1 − 𝑁)Br𝑁𝑡𝐴

𝑁𝑏𝜃21

−(1 − 𝑁)Gr𝐴

𝜃21

−𝑁𝐴4

𝜃1 (𝜃21 − 𝑛2)) + (

𝑁𝐴7

12𝑛2−(1 − 𝑁)Br𝐷

24)𝑦4.

(23)

The constant values appeared are listed out in Appendix.The nondimensional expression for the pressure rise per

wavelength Δ𝑝𝜆 is

Δ𝑃𝜆 = ∫

1

0∫

1

0(𝜕𝑝

𝜕𝑥)

𝑦=0

𝑑𝑥 𝑑𝑡. (24)

Interestingly, we note that the stress tensor in micropolarfluid is not symmetric in behavior. For that reason, thedimensionless form of the shear stress implicated in thepresent problem under consideration is given by [30]

𝜏𝑥𝑦 =𝜕𝑢

𝜕𝑦−

𝑁

(1 − 𝑁)Ω, (25)

𝜏𝑥𝑦 = 3𝑦2(Br𝐷 (𝑁 − 1)

6+𝐴7𝑁

3𝑛2) − (

𝜕𝑝

𝜕𝑥)

⋅ (𝑦 (𝑁 − 1) + 𝐴8𝑁 cosh (𝑛𝑦) + 𝐴11𝑁 sinh (𝑛𝑦)

−2𝑁𝑦 (𝑁 − 1)

2 (𝑁 − 2)) + 𝐺(𝐴10𝑁 cosh (𝑛𝑦) −

𝑁

𝑁 − 2

+ 𝐴13𝑁 sinh (𝑛𝑦) + 1) + cosh (𝜃2𝑦)

⋅ (𝐵Gr (𝑁 − 1)

𝜃2

+𝐴5𝑁

𝑛2 − 𝜃22

−𝐵Br𝑁𝑡 (𝑁 − 11)

𝑁𝑏𝜃2

)

− (𝑁

𝑁 − 1)(

𝐴5 cosh (𝜃2𝑦)𝑛2 − 𝜃22

− 𝐹 sinh (𝑛𝑦)

−𝐺

𝑁 − 2− 𝐸 cosh (𝑛𝑦) +

𝐴4 sinh (𝜃1𝑦)𝑛2 − 𝜃21

+𝐴7 (𝑛

2𝑦2+ 2)

𝑛4+𝐴6𝑦

𝑛2+(𝜕𝑝/𝜕𝑥) (𝑁 − 1)

𝑁 − 2)

+ 𝜃1 sinh (𝜃1𝑦)(𝐴Gr (𝑁 − 1)

𝜃21

+𝐴4𝑁

𝜃1 (𝑛2 − 𝜃21)

−𝐴Br𝑁𝑡 (𝑁 − 1)

𝑁𝑏𝜃21

) − 𝐴9𝑁 cosh (𝑛𝑦)

− 𝐴12𝑁 sinh (𝑛𝑦) +2𝐴7𝑁

𝑛4+𝐴6𝑁𝑦

𝑛2,

𝜏𝑦𝑥 =1

(1 − 𝑁)

𝜕𝑢

𝜕𝑦+

𝑁

(1 − 𝑁)Ω,

𝜏𝑦𝑥 = (−1

𝑁 − 1)(3𝑦

2(Br𝐷 (𝑁 − 1)

6+𝐴7𝑁

3𝑛2)

− (𝜕𝑝

𝜕𝑥) (𝑦 (𝑁 − 1) + 𝐴8𝑁 cosh (𝑛𝑦)

+ 𝐴11𝑁 sinh (𝑛𝑦) −2𝑁𝑦 (𝑁 − 1)

2 (𝑁 − 2))

+ 𝐺(𝐴10𝑁 cosh (𝑛𝑦) −𝑁

𝑁 − 2+ 𝐴13𝑁 sinh (𝑛𝑦)

+ 1) + cosh (𝜃2𝑦)(𝐵Gr (𝑁 − 1)

𝜃2

+𝐴5𝑁

𝑛2 − 𝜃22

−𝐵Br𝑁𝑡 (𝑁 − 11)

𝑁𝑏𝜃2

) + 𝜃1 sinh (𝜃1𝑦)

⋅ (𝐴Gr (𝑁 − 1)

𝜃21

+𝐴4𝑁

𝜃1 (𝑛2 − 𝜃21)

−𝐴Br𝑁𝑡 (𝑁 − 1)

𝑁𝑏𝜃21

) +2𝐴7𝑁

𝑛4+𝐴6𝑁𝑦

𝑛2

− 𝐴9𝑁 cosh (𝑛𝑦) − 𝐴12𝑁 sinh (𝑛𝑦)) − (𝑁

𝑁 − 1)

⋅ (𝐴5cosh (𝜃2𝑦)

𝑛2 − 𝜃22

− 𝐹 sinh (𝑛𝑦) − 𝐺𝑁 − 2

− 𝐸 cosh (𝑛𝑦) +𝐴4 sinh (𝜃1𝑦)

𝑛2 − 𝜃21

+𝐴7 (𝑛

2𝑦2+ 2)

𝑛4

+𝐴6𝑦

𝑛2+(𝜕𝑝/𝜕𝑥) (𝑁 − 1)

𝑁 − 2) .

(26)

The numerical computation for the shear stress 𝜏𝑥𝑦 isobtained at left wall of the channel and whose graphicalrepresentation is presented in the next section.

4. Numerical Results and Discussion

In view of the fact that the constant value of rate of volumeflow 𝐹 gives the pressure rise (Δ𝑃𝜆) always negative and


0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

1.2

u

y

−1 −0.5

u: 0.6134

y: 0.8749u: 0.7441

m = 0.0m = 0.2

m = 0.4

m = 0.6

a = 0.1; b = 0.3; Θ = 1.5; n = 0.1;; Nt = 0.3; Nb = 0.1;

; Rn = 0.3;𝜙 = 𝜋/6; 𝛽 = 0.3; N = 0.4

Pr = 0.5

Gr = 0.7; Br = 0.9

y: −0.6781

Figure 2: Variation of𝑚 on the velocity 𝑢 with respect to 𝑦.

hence pumping action cannot be perceived. To discuss thefound results quantitatively, we assume the form of theinstantaneous volume rate of the flow 𝐹(𝑥, 𝑡) [30, 33–37],

𝐹 (𝑥, 𝑡) = Θ + 𝑎 sin 2𝜋 (𝑥 − 𝑡) + 𝑏 sin [2𝜋 (𝑥 − 𝑡) + 𝜙] (27)

in which 𝐹 = 𝑄/𝑐𝑑, Θ = 𝑞/𝑐𝑑, 𝐹 = ∫ℎ2ℎ1

𝑢 𝑑𝑦 = 𝜓(ℎ2) − 𝜓(ℎ1),and Θ is the time-average of flow over one period of wave.

In order to get nearby into the given substantial prob-lem, we observe physical characteristics of average rise inpressure, axial velocity, microrotation velocity, shear stress,nanoparticles temperature, and concentration with respect ofvarious values to the parameters appearing in the problem bysketching Figures 2–20 with the constant values 𝑥 = 0.5 and𝑡 = 0.2. It is found that, in the absence of nonuniformparame-ter, local temperatureGrashof number, and local nanoparticleGrashof number, the present analysis reduces to approximateanalytical solution of peristaltic flow of a Newtonian fluid inasymmetric channel [32].Thus, the results introducedmay beapplied in the real life problems associated with nanoparticlesmovement in the gastrointestinal tract, intrauterine fluidmotion induced by uterine contraction, and flow throughsmall blood vessels and intrapleural membranes.

4.1. Flow Characteristics. Figures 2–7 express that the varia-tion of axial velocity 𝑢with respect to 𝑦 for different values ofthe nonuniform parameter (𝑚), amplitude of right wall (𝑏),local temperature Grashof number (Gr), Brownian motionparameter (𝑁𝑏), micropolar parameter (𝑛), and couplingnumber (𝑁). The dimensionless axial velocity profiles (𝑢)satisfy the boundary conditions and are varied by a smoothcurve for different (𝑚) values in Figure 2. This nonuniformparameter effect has a tendency to slow down the motion ofthe fluid which results in decreasing the axial velocity profilesat the core of the channel. Figure 3 represents the variation of

0

0.2

0.4

0.6

0.8

1

1.2

u

0 0.5 1y

−1 −0.5

b = 0

b = 0.1

b = 0.2b = 0.3

a = 0.4; m = 0.3; Θ = 2; n = 0.1;; Nt = 0.3; Nb = 0.2;

; Rn = 0.1;𝜙 = 𝜋/2; 𝛽 = 0.3; N = 0.8

Pr = 1.5

Gr = 0.4; Br = 0.2

Figure 3: Variation of 𝑏 on the velocity 𝑢 with respect to 𝑦.

0

0.2

0.4

0.6

0.8

1

1.2

u

0 0.5 1y

−1 −0.5

u: 1.2

a = 0.4; b = 0.2; Θ = 1.6; m = 0.1;; Nt = 0.2; Nb = 0.7;

; Rn = 0.2;𝜙 = 𝜋/4; 𝛽 = 0.2; N = 0.25

Pr = 0.4

n = 0.1; Br = 0.4

y: 0.004308

Gr = 0

Gr = 2

Gr = 4Gr = 6

Figure 4: Variation of Gr on the velocity 𝑢 with respect to 𝑦.

𝑢with𝑦 for various values of the amplitude of right wall (𝑏). Itis found that an increase of 𝑏 results in increase of the velocityof the fluid near the right part of the channel. The effects ofGr and𝑁𝑏 on the axial velocity pattern are shown in Figures4 and 5. When Gr and 𝑁𝑏 values are larger and 𝑢 curves arelarger too at the central part of the channel, so the velocityprofiles become larger with right wall. This is due to the factthat buoyancy force gives rise to fluid flow. This force has atendency to accelerate themotion of the fluid which results inincreasing the axial velocity profiles at right wall; otherwiseit gets decreased. The effect of coupling number 𝑁 on axialvelocity is depicted in Figure 6. It is observed that the flow


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

u

0 0.5 1y

−1 −0.5

u: 0.986

Nb = 0.01Nb = 0.02

Nb = 0.03

Nb = 0.1

a = 0.2; b = 0.3; Θ = 1.4; m = 0.2;; Nt = 0.5; n = 0.2;

; Rn = 0.2;𝜙 = 𝜋/8; 𝛽 = 0.4; N = 0.3

Pr = 0.75

Gr = 0.5; Br = 0.4

y: −0.1971

Figure 5: Variation of𝑁𝑏 on the velocity 𝑢 with respect to 𝑦.

0

0.2

0.4

0.6

0.8

1

1.2

u

0 0.5 1 1.5y

−1 −0.5

u: 1.213

N NewtonianN = 0.4

N = 0.8

N→1

a = 0.4; b = 0.5; Θ = 1.8; m = 0.4;; Nt = 0.5; Nb = 0.4;

; Rn = 0.3;𝜙 = 𝜋/5; 𝛽 = 0.2; n = 0.2

Pr = 0.8

Gr = 1; Br = 0.5

y: 0.1202

Figure 6: Variation of𝑁 on the velocity 𝑢 with respect to 𝑦.

reversal near the left wall of the channel increases with theincrease of coupling number, while the reversal trend occursin the vicinity of the right channel wall. But similar significantchange is found for the variation of micropolar parameter 𝑛in Figure 7.

4.2. Heat Transfer and Nanoparticle Mass Transfer Distri-butions. The dimensionless nanoparticle temperature andconcentration profiles (𝜃, 𝜎) satisfy the boundary conditionsand are varied by a smooth curve for different 𝛽,𝑁𝑏, and 𝑅𝑛values. In the absence of amplitude of left wall 𝑎 = 0, heat

0 0.5 10

0.2

0.4

0.6

0.8

1

1.2

u

1.5y

−1 −0.5

u: 1.146

n Newtoniann = 0.3

n = 0.6

n = 1

a = 0.3; b = 0.4; Θ = 1.6; m = 0.2;; Nt = 0.5; Nb = 0.1;

; Rn = 0.2;𝜙 = 𝜋/3; 𝛽 = 0.2; N = 0.9

Pr = 0.1

Gr = 0.7; Br = 0.3

y: 0.2471

Figure 7: Variation of 𝑛 on the velocity 𝑢 with respect to 𝑦.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5y

−1 −0.5

𝜃

𝛽 = 0𝛽 = 0.25

𝛽 = 0.5

𝛽 = 0.75

a = 0.3; b = 0.4; m = 0.4;Pr = 0.5; Nt = 0.2; Nb = 0.4;𝜙 = 𝜋/4; Rn = 0.1;

Figure 8: Temperature profile 𝜃(𝑦) for 𝛽.

source/sink 𝛽 = 0, Prandtl number 𝑃𝑟 = 1, thermal radiationparameter 𝑅𝑛 = 0, and nonuniform parameter 𝑚 = 0 ournanoparticle temperature and concentration distributionsresults are in close agreement with earlier works of Tripathiand Bég [24]. The effects of heat source/sink parameter (𝛽)on the nanoparticle temperature and concentration distribu-tions are displayed in Figures 8 and 9. That is, increase in theheat source strength amounts to increase in energy supplyto the tapered asymmetric channel and opposite behavioris noticed form nanoparticle mass transfer. The effects ofBrownianmotion parameter on the nanoparticle temperatureand concentration distributions are considered in Figures 10


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5y

−1 −0.5

𝛽 = 0𝛽 = 0.5

𝛽 = 1

𝛽 = 1.5

a = 0.3; b = 0.4; m = 0.4;Pr = 0.5; Nt = 0.2; Nb = 0.4;𝜙 = 𝜋/4; Rn = 0.1;

𝜎

Figure 9: Nanoparticle volume fraction 𝜎(𝑦) for 𝛽.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5y

−1 −0.5

𝜃

a = 0.2; b = 0.4; m = 0.3;Pr = 0.8; Nt = 0.7; 𝛽 = 0.8;

𝜙 = 𝜋/3; Rn = 1

Nb → 0

Nb = 1

Nb = 2

Nb = 3

Figure 10: Temperature profile 𝜃(𝑦) for𝑁𝑏.

and 11. When 𝑁𝑏 is larger and 𝜎 curve is lower, so the masstransfer effect is higher for a larger𝑁𝑏 and opposite behaviorfound in heat transfer. In nanofluid system, the size of thenanoparticle generates Brownian motion which affects theheat and mass transfer properties. As the particle size scaleapproaches to the nanometer scale, the Brownian motion ofparticles and its diffusion effect on the liquid play a significantrole in heat transfer. For prominent values of𝑁𝑏, the Brown-ian diffusion effect is large compared to the thermal diffusioneffect. Figure 12 shows the variation of thermal radiationparameter over the temperature field.Therefore higher valuesof thermal radiation parameter imply higher surface heat

0 0.5 1 1.5y

−1 −0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

𝜎

Nb = 1

Nb = 2

Nb = 3

a = 0.2; b = 0.4; m = 0.3;Pr = 0.8; Nt = 0.7; 𝛽 = 0.8;

𝜙 = 𝜋/3; Rn = 1

Nb = 0.5

Figure 11: Nanoparticle volume fraction 𝜎(𝑦) for𝑁𝑏.

0

0.2

0.4

0.6

0.8

1

0 0.5 1y

−1 −0.5

𝜃

a = 0.2; b = 0.3; m = 0.2;Pr = 0.6; Nt = 0.5; Nb = 0.5;

𝜙 = 𝜋/4; 𝛽 = 1

Rn = 0Rn = 0.3

Rn = 0.6

Rn = 1

Figure 12: Temperature profile 𝜃(𝑦) for 𝑅𝑛.

flux and so, it decreases the temperature within the taperedasymmetric channel, whereas opposite effects are observed ina nanoparticle mass transfer as shown in Figure 13.

4.3. Spin Velocity Distribution. The variation of microrota-tion velocity (Ω) with respect to 𝑦 for various values of cou-pling number (𝑁), Brownian motion (𝑁𝑏), and nonuniformparameter (𝑚) are shown in Figures 14–16. Parabolic micro-rotation velocity profile is observed for the present flow prob-lem.The velocity is maximumorminimumnear the center ofchannel. The cause of coupling number on the microrotation


0 0.5 1y

−1 −0.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

𝜎

a = 0.2; b = 0.3; m = 0.2;Pr = 0.6; Nt = 0.5; Nb = 0.5;

𝜙 = 𝜋/4; 𝛽 = 1

Rn = 0Rn = 0.3

Rn = 0.6

Rn = 1

Figure 13: Nanoparticle volume fraction 𝜎(𝑦) for 𝑅𝑛.

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

1.2

u

y

−1 −0.5

y: 0.1202u: 1.213

N newtonianN = 0.4

N = 0.8

N→ 1

a = 0.4; b = 0.5; Θ = 1.8;m = 0.4;Pr = 0.8;N

t= 0.5;N

b= 0.4;

Gr = 1; Br = 0.5; Rn= 0.3;

= /5; = 0.2; n = 0.2

Figure 14: Microrotation velocity profile Ω(𝑦) for different valuesof𝑁.

velocity is illustrated in Figure 14. It is seen that in the micro-rotation velocity increases by increasing the coupling number𝑁 (i.e., microrotation velocity for the micropolar nanofluidis wider than that of the region for Newtonian nanofluid).In the case of 𝑁 = 0, there is no appreciable differencebetween Newtonian nanofluids and micropolar nanofluids.The microrotation velocity for the Brownian motion param-eter (𝑁𝑏) is plotted in Figure 15. It is seen that, with theincrease in the Brownian motion parameter, microrotationvelocity profile decreases andmaximummicrorotation veloc-ity occurs at 𝑁𝑏 → 0. Figure 16 depicts the microrotationvelocity field for different values of nonuniform parameter. It

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 0.5 1y

−1 −0.5

Nb = 0.02

Nb = 0.03

Nb = 0.04Nb = 0.01

Ω

−0.02

a = 0.2; b = 0.3; Θ = 1.4;m = 0.2;n = 0.2; Gr = 0.5; Br = 0.4; Rn = 0.2; 𝜙 = 𝜋/8;

Pr = 0.75;Nt = 0.5; 𝛽 = 0.4;N = 0.3

Figure 15:Microrotation velocity profileΩ(𝑦) for different values of𝑁𝑏.

0

1

2

3

4

5

6

7

8

9

10

0 0.5 1 1.5y

−1 −0.5

m = 0.0m = 0.2

m = 0.4

m = 0.6

a = 0.1; b = 0.3; Θ = 1.5;

n = 0.1; 𝜙 = 𝜋/6; 𝛽 = 0.3;

Pr = 0.5;Nt = 0.3;Nb = 0.1; Br = 0.9, Rn = 0.3;

N = 0.4

×10−3

Ω

Gr = 0.7

Figure 16: Microrotation velocity profile Ω(𝑦) for different valuesof𝑚.

is also viewed that the microrotation velocity for a divergentchannel (𝑚 > 0) is higher compared to its value for a uniformchannel (𝑚 = 0).

4.4. Shear Stress Distribution. It is well known that the stresstensor is not symmetric in micropolar nanofluid. In Figures17–19, we have plotted the shear stresses 𝜏𝑥𝑦 at the left wall forvalues of the nonuniform parameter, micropolar parameter,and coupling number. One can observe from Figure 17 that


1.4 1.6 1.8 2 2.2 2.4 2.60

0.5

1

1.5

2

m = 0m = 0.05

m = 0.1

m = 0.15

x

𝜏xy

a = b = 0.2;N = 0.8,Θ = 2.5; n = 0.1;Pr = 1.5;Nt = 0.3;Nb = 0.2; Gr = 0.4;Br = 0.2; Rn = 0.1;𝜙 = 𝜋/4; 𝛽 = 0.3

y = ℎ1

Figure 17: Shear stress profile 𝜏𝑥𝑦(𝑥) for different values of𝑚.

0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x

𝜏xy

n→ 0n = 0.4

n = 0.8

n = 1

a = 0.5; b = 0.3; Θ = 1.8;m = 0.5;Pr = 0.8;Nt = 0.2;Nb = Rn = 0.1;Gr = 3; Br = 2; 𝜙 = 𝜋/2; 𝛽 = 0.4;

N = 0.9

Figure 18: Shear stress profile 𝜏𝑥𝑦(𝑥) for different values of 𝑛.

the wall shear stress decreases with an increase in the nonuni-form parameter. From Figure 18, we notice that shear stress isin oscillation behavior, which may be due to the creation ofcontraction and expansion walls. It is indicated that the shearstress decreases with an increase in themicropolar parameter𝑛, while it increases as the coupling number 𝑁 increases inFigure 19. Hence, we observed that the shear stress for a New-tonian nanofluid is less than that for a micropolar nanofluid.

4.5. Trapping Phenomena. The effects of nonuniform param-eter, coupling parameter, and micropolar parameter on thestreamlines are shown in Figure 20. Besides, a comparisonbetween uniform channel and the nonuniform channel is

0.6 0.8 1 1.2 1.4 1.6 1.80

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x

𝜏xy

N = 0

N = 0.4

N = 0.8

N→ 1

a = 0.2; b = 0.3; Θ = 1.2;n = 0.3; Pr = 1;Nt = 0.4;Nb = 0.2; Gr = 0.5; Br = 0.5;

Rn = 0.2; 𝜙 = 𝜋/4; 𝛽 = 0.1;m = 0.2

Figure 19: Shear stress profile 𝜏𝑥𝑦(𝑥) for different values of𝑁.

made in Figures 20(a) and 20(b). We note that the sizeof the trapping bolus increases with increasing nonuniformparameter and symmetry nature is also noticed with respectto uniform symmetric channel. The effects of𝑁 on trappingare presented in Figures 20(b) and 20(c). This figure revealsthat the size of lower bolus decreases with an increase in 𝑁.FromFigures 20(c) and 20(d), it is clear that the trapped bolusincreases in size as 𝑛 increases.

5. Concluding Remarks

A mathematical model is presented to study wall inducedflow of a micropolar nanofluid in the most generalized(tapered asymmetric) channel with the presence of heatsource and thermal radiation parameters. The flow model inthe rotation of nanoparticles was included. Long wavelengthand low Reynolds number assumptions are used in themath-ematical modeling. In this investigation, special emphasishas been paid to study the flow features, the axial velocity,the nanoparticles temperature and concentration, the shearstress, and the trapping phenomena. The study leads to thefollowing conclusions:

(i) The axial velocity of fluid decreases at the core part ofchannel when𝑚 is increased as anticipated.

(ii) The axial velocity increases near the left wall ofchannel and decreases near the right wall of thechannel with increase of the coupling number andmicropolar parameter.

(iii) The nanoparticles mass transfer 𝜎 has reverse behav-ior when compared to heat transfer.

(iv) Coupling number and Brownian motion parametershave opposite effects on the microrotation velocity.


0

0

0.1

0.1

0.2

0.2

0.4

0.4

0.4

0.5

0.5

0.6

0.6

0.7

0.7

0.8

0.8

0.9

0.9

𝜓(x, y)2.0

1.5

1.0

0.5

0.0

x

−2 −1 0 1 2

y

−0.2

−0.1

−0.1

−0.6

−0.9

−0.7

−0.7

−0.8

−0.3

−0.6

−0.2−0.8−0.9

(a)

0

0.1

0.1 0.2

0.2

0.4

0.4

0.5

0.5

0.7

0.7

0.8

0.8

0.9

0.9

𝜓(x, y)2.0

1.5

1.0

0.5

0.0

−2 −1−1.4

0 1 2

y

x

−0.8

−0.8

−0.6

−0.6

−0.6

−0.9

−0.1

−0.1

−0.2

−0.2−0.3

0.6

0.6

−0.7

−0.7

(b)

0

0

𝜓(x, y)2.0

1.5

1.0

0.5

0.0

0.9

0.9

0.5

0.8

0.8

0.5

0.6

0.6

0.1

0.7

0.7

0.1

0.4

0.4

0.4

0.2

0.2

−2

−0.5

−0.8

−0.8

−0.7

−0.7

−0.9

−0.9

−0.6

−0.6

−0.2

−0.3

−0.1

−0.1

−1 0 1 2

y

x

(c)

0𝜓(x, y)

2.0

1.5

1.0

0.5

0.0

0.9

0.9

0.8

0.8

0.5

0.6

0.7

0.1

0.7

0.4

0.4

0.4

0.2

0.2

−2

−0.2

−0.5

−0.8

−0.8

−0.9

−0.9

−0.1−0.7

−0.7

−0.6

−0.6

−0.6

−1 0 1 2

y

x

(d)

Figure 20: Streamlines for 𝑚 = 0, 𝑁 = 0.01, 𝑛 = 0.1 (a), 𝑚 = 0.3, 𝑁 = 0.01, 𝑛 = 0.1 (b), 𝑚 = 0.3, 𝑁 → 1, 𝑛 = 0.1 (c), and 𝑚 = 0.3,𝑁 → 1, 𝑛 = 0.9 (d). The other parameters chosen are 𝑎 = 0.2, 𝑏 = 0.3, 𝑚 = 0.3, Θ = 1.5, 𝜙 = 𝜋/2, 𝑁𝑡 = 0.2, 𝑁𝑏 = 0.3, 𝑅𝑛 = 0.9, Pr = 0.5,Gr = 0.6, Br = 0.2, 𝛽 = 0.2, and 𝑡 = 0.2.

(v) The walls shear stress 𝜏𝑥𝑦 decreases with the increaseof nonuniform parameter 𝑚 at the left wall but theopposite behavior is identified for coupling number.

(vi) It may be interesting to note that the size of trappedbolus gets increased with increasing of nonuniformparameter.

We ultimately conclude that our theoretical analysis bears thepotential to be useful in the field of biomedical and industrialengineering.

Appendix

The following constants are used in solution section:

𝜃1 =

−𝐴2 + √𝐴22 − 4𝐴3

2,

𝜃2 =

−𝐴2 − √𝐴22 − 4𝐴3

2,


𝐴 =−sinh (𝜃1ℎ1)

sinh (𝜃2ℎ2) cosh (𝜃1ℎ1) − sinh (𝜃1ℎ1) cosh (𝜃1ℎ2),

𝐵 =cosh (𝜃1ℎ1)

sinh (𝜃2ℎ2) cosh (𝜃1ℎ1) − sinh (𝜃1ℎ1) cosh (𝜃1ℎ2),

𝐶 = −𝐷ℎ1 +𝑁𝑡

𝑁𝑏

𝐴 cosh (𝜃1ℎ1) +𝑁𝑡

𝑁𝑏

𝐵 sinh (𝜃2ℎ1) ,

𝐷 =1

ℎ2 − ℎ1

+𝑁𝑡

𝑁𝑏 (ℎ2 − ℎ1)𝐴 (cosh (𝜃1ℎ2)

− cosh (𝜃1ℎ1)) +𝑁𝑡

𝑁𝑏 (ℎ2 − ℎ1)𝐵 (sinh (𝜃2ℎ2)

− sinh (𝜃2ℎ1)) ,

𝐸 = 𝐴8 (𝜕𝑝

𝜕𝑥) + 𝐴9 + 𝐺𝐴10,

𝐹 = 𝐴11 (𝜕𝑝

𝜕𝑥) + 𝐴12 + 𝐺𝐴13,

𝐺 = (𝐴17 − 𝐴14

𝐴16 − 𝐴19

)(𝜕𝑝

𝜕𝑥) + (

𝐴18 − 𝐴15

𝐴16 − 𝐴19

) ,

𝐻 = −(𝐴14 +𝐴16 (𝐴17 − 𝐴14)

𝐴16 − 𝐴19

)(𝜕𝑝

𝜕𝑥) − 𝐴15

− (𝐴16 (𝐴18 − 𝐴15)

𝐴16 − 𝐴19

) ,

𝐴1 =𝑁𝑡 + 𝑁𝑏

𝑁𝑏 (ℎ2 − ℎ1),

𝐴2 =𝑁𝑏𝑃𝑟𝐴1

1 + 𝑅𝑛𝑃𝑟

,

𝐴3 =𝑃𝑟𝛽

1 + 𝑅𝑛𝑃𝑟

,

𝐴4 =𝑛2(1 − 𝑁)𝐴

(2 − 𝑁) 𝜃1

(Br𝑁𝑡𝑁𝑏

− Gr) ,

𝐴5 =𝑛2(1 − 𝑁)𝐵

(2 − 𝑁) 𝜃2

(Br𝑁𝑡𝑁𝑏

− Gr) ,

𝐴6 = −(1 − 𝑁)Br𝐶𝑛2

2 − 𝑁,

𝐴7 = −(1 − 𝑁)Br𝐷𝑛2

2 − 𝑁,

𝐴8 =(1 − 𝑁) ℎ1

(2 − 𝑁) cosh (𝑛ℎ1)

−(1 − 𝑁) sinh (𝑛ℎ1) (ℎ1 cosh (𝑛ℎ2) − ℎ2 cosh (𝑛ℎ1))

sinh (𝑛 (ℎ1 − ℎ2)) cosh (𝑛ℎ1),

𝐴9 = −(𝐴33

cosh (𝑛ℎ1)

+(cosh (𝑛ℎ1) − cosh (𝑛ℎ2)) sinh (𝑛ℎ1)

sinh (𝑛 (ℎ1 − ℎ2)) cosh (𝑛ℎ1)) ,

𝐴10 =1

(2 − 𝑁) cosh (𝑛ℎ1)

−sinh (𝑛ℎ1) (cosh (𝑛ℎ2) − cosh (𝑛ℎ1))(2 − 𝑁) sinh (𝑛 (ℎ1 − ℎ2)) cosh (𝑛ℎ1)

,

𝐴11 =(1 − 𝑁) (ℎ1 cosh (𝑛ℎ2) − ℎ2 cosh (𝑛ℎ1))

sinh (𝑛 (ℎ1 − ℎ2)),

𝐴12 =(𝐴34 cosh (𝑛ℎ1) − 𝐴33 cosh (𝑛ℎ2))

sinh (𝑛 (ℎ1 − ℎ2)),

𝐴13 =(cosh (𝑛ℎ2) − cosh (𝑛ℎ1))(2 − 𝑁) sinh (𝑛 (ℎ1 − ℎ2))

,

𝐴14 =ℎ21

2 − 𝑁−𝐴8 sinh (𝑛ℎ1) + 𝐴11 cosh (𝑛ℎ1)

𝑛,

𝐴15 = −𝑁 (𝐴9 sinh 𝑛ℎ1 + 𝐴12 cosh (𝑛ℎ1))

𝑛

+ cosh (𝜃1ℎ1) ((1 − 𝑁)𝐴 (Br𝑁𝑡 − Gr𝑁𝑏)

𝑁𝑏𝜃21

−𝑁𝐴4

𝜃1 (𝜃21 − 𝑛2)) + sinh (𝜃2ℎ1)

⋅ ((1 − 𝑁)𝐵 (Br𝑁𝑡 − Gr𝑁𝑏)

𝑁𝑏𝜃22

−𝑁𝐴5

𝜃2 (𝜃22 − 𝑛2))

+𝑁𝐴6ℎ

21

2𝑛2+ (

𝑁𝐴7

3𝑛2−(1 − 𝑁)Br𝐷

6) ℎ31 +

2𝑁𝐴7ℎ1

𝑛4,

𝐴16 =2ℎ1

2 − 𝑁+𝑁(𝐴13 cosh (𝑛ℎ1) + 𝐴10 sinh (𝑛ℎ1))

𝑛,

𝐴17 =ℎ22

2 − 𝑁−𝐴8 sinh (𝑛ℎ2) + 𝐴11 cosh (𝑛ℎ2)

𝑛,

𝐴18 = −𝑁 (𝐴9 sinh 𝑛ℎ2 + 𝐴12 cosh (𝑛ℎ2))

𝑛

+ cosh (𝜃1ℎ2) ((1 − 𝑁)𝐴 (Br𝑁𝑡 − Gr𝑁𝑏)

𝑁𝑏𝜃21

−𝑁𝐴4

𝜃1 (𝜃21 − 𝑛2)) + sinh (𝜃2ℎ2)

⋅ ((1 − 𝑁)𝐵 (Br𝑁𝑡 − Gr𝑁𝑏)

𝑁𝑏𝜃22

−𝑁𝐴5

𝜃2 (𝜃22 − 𝑛2))

+𝑁𝐴6ℎ

22

2𝑛2+ (

𝑁𝐴7

3𝑛2−(1 − 𝑁)Br𝐷

6) ℎ32 +

2𝑁𝐴7ℎ2

𝑛4,

𝐴19 =2ℎ2

2 − 𝑁+𝑁(𝐴13 cosh (𝑛ℎ2) + 𝐴10 sinh (𝑛ℎ2))

𝑛,

𝐴20 =𝐴17 − 𝐴14

𝐴16 − 𝐴19

,

𝐴21 =(1 − 𝑁)Br𝑁𝑡𝐴

𝑁𝑏𝜃21

−(1 − 𝑁)Gr𝐴

𝜃21

−𝑁𝐴4

𝜃1 (𝜃21 − 𝑛2),

𝐴22 =(1 − 𝑁)Br𝑁𝑡𝐵

𝑁𝑏𝜃22

−(1 − 𝑁)Gr𝐵

𝜃22

−𝑁𝐴5

𝜃2 (𝜃22 − 𝑛2),


𝐴23 =𝑁𝐴7

3𝑛2−(1 − 𝑁)Br𝐷

6,

𝐴24 =𝐴18 − 𝐴15

𝐴16 − 𝐴19

,

𝐴25 = −𝐴15 − 𝐴16𝐴24,

𝐴26 = −𝐴4 − 𝐴16𝐴20,

𝐴27 =𝑁 (𝐴10 − 𝐴8)

𝑛,

𝐴28 =𝑁 (𝐴13 − 𝐴11)

𝑛,

𝐴29 =𝑁 (𝐴24𝐴13 − 𝐴12)

𝑛,

𝐴30 =𝑁 (𝐴24𝐴10 − 𝐴9)

𝑛,

𝐴31 =2𝐴7𝑁

𝑛4+2𝐴24

2 − 𝑁,

𝐴32 =𝑁𝐴6

2𝑛2,

𝐴33 =𝐴4 sinh (𝜃1ℎ1)

𝜃21 − 𝑛2

+𝐴5 cosh (𝜃1ℎ1)

𝜃22 − 𝑛2

−𝐴6ℎ1

𝑛2

− 𝐴7 (𝑛2ℎ21 + 2

𝑛4) ,

𝐴34 =𝐴4 sinh (𝜃1ℎ2)

𝜃21 − 𝑛2

+𝐴5 cosh (𝜃1ℎ2)

𝜃22 − 𝑛2

−𝐴6ℎ2

𝑛2

− 𝐴7 (𝑛2ℎ22 + 2

𝑛4) .

(A.1)

Disclosure

The University College of Engineering Arni is a constituentcollege of Anna University, Chennai.

Competing Interests

The authors declare that they have no competing interests.

References

[1] T. W. Latham, Fluid motions in a peristaltic pump [M.S. thesis],MIT, 1966.

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

[3] M. Y. Jaffrin and A. H. Shapiro, “Peristaltic pumping,” AnnualReview of Fluid Mechanics, vol. 3, pp. 13–36, 1971.

[4] L. M. Srivastava and V. P. Srivastava, “Peristaltic transport ofblood: casson model-II,” Journal of Biomechanics, vol. 17, no. 11,pp. 821–829, 1984.

[5] S. K. Pandey and D. Tripathi, “A mathematical model forperistaltic transport of micro-polar fluids,” Applied Bionics andBiomechanics, vol. 8, no. 3-4, pp. 279–293, 2011.

[6] K. S. Mekheimer and Y. Abd Elmaboud, “The influence of amicropolar fluid on peristaltic transport in an annulus: applica-tion of the clot model,” Applied Bionics and Biomechanics, vol.5, no. 1, pp. 13–23, 2008.

[7] M.Kothandapani, J. Prakash, andV. Pushparaj, “Analysis of heatand mass transfer on MHD peristaltic flow through a taperedasymmetric channel,” Journal of Fluids, vol. 2015, Article ID561263, 9 pages, 2015.

[8] M. Kothandapani, J. Prakash, and V. Pushparaj, “Effects of heattransfer, magnetic field and space porosity on peristaltic flow ofa newtonian fluid in a tapered channel,” Applied Mechanics andMaterials, vol. 813-814, pp. 679–684, 2015.

[9] H. Grad, “Statistical mechanics, thermodynamics, and fluiddynamics of systems with an arbitrary number of integrals,”Communications on Pure and Applied Mathematics, vol. 5, pp.455–494, 1952.

[10] A. C. Eringen, “Simple microfluids,” International Journal ofEngineering Science, vol. 2, no. 2, pp. 205–217, 1964.

[11] A. C. Eringen, “Theory of thermomicrofluids,” Journal ofMathematical Analysis and Applications, vol. 38, no. 2, pp. 480–496, 1972.

[12] R. S. Agarwal and C. Dhanapal, “Numerical solution of freeconvection micropolar fluid flow between two parallel porousvertical plates,” International Journal of Engineering Science, vol.26, no. 12, pp. 1247–1255, 1988.

[13] R.G.Devi andR.Devanathan, “Peristalticmotion of amicropo-lar fluid,”Proceedings of the IndianAcademy of Sciences—SectionA, vol. 81, no. 4, pp. 149–163, 1975.

[14] D. Srinivasacharya, M. Mishra, and A. R. Rao, “Peristalticpumping of a micropolar fluid in a tube,” Acta Mechanica, vol.161, no. 3, pp. 165–178, 2003.

[15] S. U. S. Choi, “Enhancing thermal conductivity of fluidswith nanoparticles,” in Developments and Applications of Non-Newtonian Flows, D. A. Siginer and H. P. Wang, Eds., vol. 66,pp. 99–105, ASME, New York, NY, USA, 1995.

[16] J. Buongiorno, “Convective transport in nanofluids,” Journal ofHeat Transfer, vol. 128, no. 3, pp. 240–250, 2006.

[17] M. Sheikholeslami, M. Gorji-Bandpay, and D. D. Ganji, “Mag-netic field effects on natural convection around a horizontal cir-cular cylinder inside a square enclosure filled with nanofluid,”International Communications in Heat and Mass Transfer, vol.39, no. 7, pp. 978–986, 2012.

[18] N. S. Akbar and S. Nadeem, “Endoscopic effects on peristalticflow of a nanofluid,”Communications inTheoretical Physics, vol.56, no. 4, pp. 761–768, 2011.

[19] N. S. Akbar, S. Nadeem, T. Hayat, and A. A. Hendi, “Peristalticflow of a nanofluid with slip effects,” Meccanica, vol. 47, no. 5,pp. 1283–1294, 2012.

[20] M.Mustafa, S. Hina, T. Hayat, and A. Alsaedi, “Influence of wallproperties on the peristaltic flow of a nanofluid: analytic andnumerical solutions,” International Journal of Heat and MassTransfer, vol. 55, no. 17-18, pp. 4871–4877, 2012.

[21] T. Hayat, F. M. Abbasi, M. Al-Yami, and S. Monaquel, “Slip andJoule heating effects inmixed convection peristaltic transport ofnanofluid with Soret and Dufour effects,” Journal of MolecularLiquids, vol. 194, pp. 93–99, 2014.

[22] S. Hina, M. Mustafa, T. Hayat, and A. Alsaedi, “Peristaltic flowof pseudoplastic fluid in a curved channel with wall properties,”


Journal of Applied Mechanics, vol. 80, no. 2, Article ID 024501,2013.

[23] T. Hayat, H. Yasmin, B. Ahmad, and B. Chen, “Simultaneouseffects of convective conditions and nanoparticles on peristalticmotion,” Journal of Molecular Liquids, vol. 193, pp. 74–82, 2014.

[24] D. Tripathi and O. A. Bég, “A study on peristaltic flow ofnanofluids: application in drug delivery systems,” InternationalJournal of Heat and Mass Transfer, vol. 70, pp. 61–70, 2014.

[25] O.AnwarBég andD.Tripathi, “Mathematica simulation of peri-staltic pumping with double-diffusive convection in nanofluids:a bio-nano-engineering model,” Proceedings of the Institution ofMechanical Engineers, Part N: Journal of Nanoengineering andNanosystems, vol. 225, no. 3, pp. 99–114, 2011.

[26] N. S. Akbar, D. Tripathi, and O. A. Bég, “Modeling nanoparticlegeometry effects on peristaltic pumping of medical magnetohy-drodynamic nanofluids with heat transfer,” Journal ofMechanicsin Medicine and Biology, 2016.

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

[28] K. De Vries, E. A. Lyons, G. Ballard, C. S. Levi, andD. J. Lindsay,“Contractions of the inner third of the myometrium,”AmericanJournal ofObstetrics andGynecology, vol. 162, no. 3, pp. 679–682,1990.

[29] O. Eytan and D. Elad, “Analysis of intra-uterine fluid motioninduced by uterine contractions,” Bulletin of MathematicalBiology, vol. 61, no. 2, pp. 221–238, 1999.

[30] N. Ali and T. Hayat, “Peristaltic flow of a micropolar fluidin an asymmetric channel,” Computers & Mathematics withApplications, vol. 55, no. 4, pp. 589–608, 2008.

[31] M. Kothandapani and J. Prakash, “Effects of thermal radiationand chemical reactions on peristaltic flow of a Newtoniannanofluid under inclinedmagnetic field in a generalized verticalchannel using homotopy perturbation method,” Asia-PacificJournal of Chemical Engineering, vol. 10, no. 2, pp. 259–272, 2015.

[32] M. Kothandapani and J. Prakash, “Influence of thermal radia-tion and magnetic field on peristaltic transport of a Newtoniannanofluid in a tapered asymmetric porous channel,” Journal ofNanofluids, vol. 5, no. 3, pp. 363–374, 2016.

[33] M. Kothandapani and J. Prakash, “Convective boundary con-ditions effect on peristaltic flow of a MHD Jeffery nanofluid,”Applied Nanoscience, vol. 6, no. 3, pp. 323–335, 2016.

[34] 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.

[35] T. Hayat, S. Irfan Shah, B. Ahmad, and M. Mustafa, “Effect ofslip on peristaltic flow of Powell-Eyring fluid in a symmetricchannel,” Applied Bionics and Biomechanics, vol. 11, no. 1-2, pp.69–79, 2014.

[36] K. S. Mekheimer, “Peristaltic flow of blood under effect of amagnetic field in a non-uniform channels,” Applied Mathemat-ics and Computation, vol. 153, no. 3, pp. 763–777, 2004.

[37] T. Hayat, R. Iqbal, A. Tanveer, and A. Alsaedi, “Influence ofconvective conditions in radiative peristaltic flow of pseudo-plastic nanofluid in a tapered asymmetric channel,” Journal ofMagnetism and Magnetic Materials, vol. 408, no. 2, pp. 168–176,2016.

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation http://www.hindawi.com

Journal ofEngineeringVolume 2014

Submit your manuscripts athttp://www.hindawi.com

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014

SensorsJournal of


Modelling & Simulation in EngineeringHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks


research article analysis of peristaltic motion of a nanofluid ...research article analysis of...

Documents