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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 Β© 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 [2β8].
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
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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 BeΜg [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 [10β12]. 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 authorβs 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
π
π
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]
ππ
ππ+ππ
ππ= 0,
ππ [ππ
ππ‘+ π
ππ
ππ+ π
ππ
ππ]
= βππ
ππ+ (πV + πV) β
2π β πV
ππ
ππ
+ (1 β πΆ0) ππππΌ (π β π0)
+ (ππ β ππ) ππ½(πΆ β πΆ0) ,
ππ [ππ
ππ‘+ π
ππ
ππ+ π
ππ
ππ]
= βππ
ππ+ (πV + πV) β
2π β πV
ππ
ππ,
πππ½ [ππ
ππ‘+ π
ππ
ππ+ π
ππ
ππ]
= πΎVβ2π+ πV (
ππ
ππβππ
ππ) β 2πVπ,
(ππ)π[ππ
ππ‘+ π
ππ
ππ+ π
ππ
ππ]
= π [π2π
ππ2+π2π
ππ2] β (
πππ
ππ+πππ
ππ) + π0 (π β π0)
+ (ππ)ππ·π΅ (
ππΆ
ππ
ππ
ππ+ππΆ
ππ
ππ
ππ)
+
π·π (ππ)π
ππ
[(ππ
ππ)
2
+ (ππ
ππ)
2
] ,
[ππΆ
ππ‘+ π
ππΆ
ππ+ π
ππΆ
ππ]
= π·π΅ [π2πΆ
ππ2+π2πΆ
ππ2] +
π·π
ππ
[π2π
ππ2+π2π
ππ2] ,
(1)
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Applied Bionics and Biomechanics 3
where π, π are the components of velocity along π and πdirections, respectively, π‘ is the dimensional time, the volu-metric volume expansion coefficient is π, ππ is the density ofthe fluid, ππ is the density of the particle, π is the accelerationdue to gravity, π is the pressure, πV, πΎV, and πV are the martialparameters [9β12],π is the temperature,πΆ is the nanoparticleconcentration, πΌ is the thermal expansion coefficient, π/ππ‘represents the material time derivative, π½ is the coefficientof expansion with concentration, π is the microrotation ofthe nanofluid, π is the microgyration parameter, ππ is thefluid mean temperature, π = (ππ)π/(ππ
)π is the ratio of
the effective heat capacity of nanoparticle material and heatcapacity of the fluid with π being the density, π is the thermalconductivity of the nanofluids, π·π΅ is the Brownian diffusioncoefficient,π·π is the thermophoretic diffusion coefficient,π0is the constant heat addition/absorption, and the radioactiveheat flux is ππ.
Hence, for the Rosseland approximation for thermalradiation, we have [31, 32]
ππ = β4πβ
3πβ
ππ4
ππ, (2)
where πβ and πβ are the Stefan-Boltzmann constant and themean absorption coefficient.
Let π = π»1 and π = π»2 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 π1 and nanoparticle volume fraction πΆ1 whilethe left wall has temperature π0 and nanoparticle volumefraction πΆ0. The geometry of the wall surface is defined as
π»1 (π, π‘) = βπ β π
π β π1sin [
2π
π(π β ππ‘
) + π] , (3a)
π»2 (π, π‘) = π + π
π + π2sin [
2π
π(π β ππ‘
)] , (3b)
where π1 and π2 are the amplitudes of left and right walls,respectively, π is the wavelength, π (π βͺ 1) is thenonuniform parameter, the phase difference π varies in therange 0 β€ π β€ π, π = 0 corresponds to symmetric channelwithwaves out of the phase, and further π1, π2, π, andπ satisfythe condition for the divergent channel at the inlet of flow
π21 + π22 + 2π1π2 cos (π) β€ (2π)
2. (4)
The nondimensional parameters are as follows:
π₯ =π
π,
π¦ =π
π,
π‘ =ππ‘
π,
π’ =π
π,
V =π
ππΏ,
πΏ =π
π,
β1 =π»1
π,
β2 =π»2
π,
Ξ© =ππ
π,
π =π½
π2
π =ππ
π,
π =π1
π,
π =π2
π,
π =π β π0
π1 β π0
,
π =πΆ β πΆ0
πΆ1 β πΆ0
,
πΊπ =(1 β πΆ0) ππππΌπ
2(π1 β π0)
ππV,
π½ =π0π2
(π1 β π0) ]ππ,
ππ =πππ
π ,
ππ =ππ·π΅ (πΆ1 β πΆ0)
],
ππ‘ =ππ·π (πΆ1 β πΆ0)
π0],
π π =16πβπ30
3πβπVππ,
Br =(ππ β ππ) ππ½
π2(πΆ1 β πΆ0)
ππ,
Sc = Vπ·π΅
,
π =π2π
πππ,
π =ππππ
πV.
(5)
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4 Applied Bionics and Biomechanics
Using the above nondimensional quantities in (1)β(4), theresulting equations are
π πΏ(π
ππ‘+ π’
π
ππ₯+ V
π
ππ¦)π’
= βππ
ππ₯+
1
1 β π(π
πΞ©
ππ¦+ πΏ2 π2π’
ππ₯2+π2π’
ππ¦2) + Grπ
+ Brπ,
π πΏ3(π
ππ‘+ π’
π
ππ₯+ V
π
ππ¦) V
= βππ
ππ¦+
πΏ2
1 β π(βπ
ππ€
ππ₯+ πΏ2 π2Vππ₯2
+π2Vππ¦2
) ,
π πΏπ (1 β π)
π(π
ππ‘+ π’
π
ππ₯+ V
π
ππ¦)Ξ©
= β2Ξ© + (πΏ2 πVππ₯
βππ’
ππ¦)
+2 β π
π2(πΏ2 π2Ξ©
ππ₯2+π2Ξ©
ππ¦2) ,
π πΏ [ππ
ππ‘+ π’
ππ
ππ₯+ πΏV
ππ
ππ¦]
=1
Pr[πΏ2 π2π
ππ₯2+π2π
ππ¦2] + π π
π2π
ππ¦2
+ ππ [πΏ2 ππ
ππ₯
ππ
ππ₯+ππ
ππ¦
ππ
ππ¦] + π½π
+ ππ‘ [πΏ2(ππ
ππ₯)
2
+ (ππ
ππ¦)
2
] ,
π πΏSc [ππππ‘
+ π’ππ
ππ₯+ πΏV
ππ
ππ¦]
= πΏ2 π2π
ππ₯2+π2π
ππ¦2+ππ‘
ππ
[πΏ2 π2π
ππ₯2+π2π
ππ¦2] .
(6)
inwhichπ = πV/(πV+πV) is the coupling number (0 β€ π β€ 1)and π2 = π2πV(2πV+πV)/(πΎV(πV+πV)) is themicropolar param-eter. We introduce the nondimensional variables such as πis dimensionless pressure, π and π are amplitudes of left andright walls, respectively, πΏ is wave number, π is the nonuni-form parameter, π is the Reynolds number, ] is the nanofluidkinematic viscosity, Ξ© is the dimensionless microrotation,π is the dimensionless temperature, π 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, ππ is the Brownian motion parameter, ππ‘ is thethermophoresis parameter, and π π is the radiation parameteras follows. The above equations can reduce to the classicalNavier-Stokes equation when πV β 0.
In several previous attempts [19β24], we employ the longwavelength and low Reynolds number approximations andthus (6) that
ππ
ππ₯=
1
1 β π
π2π’
ππ¦2+
π
1 β π
πΞ©
ππ¦+ Grπ + Brπ, (7)
ππ
ππ¦= 0, (8)
β 2Ξ© βππ’
ππ¦+ (
2 β π
π2)π2Ξ©
ππ¦2= 0, (9)
(1 + π πππ
ππ
)π2π
ππ¦2+ ππ (
ππ
ππ¦
ππ
ππ¦) + ππ‘ (
ππ
ππ¦)
2
+ π½π
= 0,
(10)
π2π
ππ¦2+ππ‘
ππ
π2π
ππ¦2= 0. (11)
The appropriate boundary conditions are
π’ = 0,
Ξ© = 0,
π = 0,
π = 0
at π¦ = β1 = β1 β ππ₯ β π sin (2π (π₯ β π‘) + π) ,
(12a)
π’ = 0,
Ξ© = 0,
π = 1,
π = 1
at π¦ = β2 = 1 + ππ₯ + π sin (2π (π₯ β π‘)) ,
(12b)
which satisfy, at the inlet of channel,
π2+ π2+ 2ππ cos (π) β€ 4. (13)
3. Exact Solution
By integration of (11) with respect to π¦ and implementationin (10) and boundary conditions of (12a) and (12b), thenanoparticles temperature field is obtained as
π
=sinh (β1π1) cosh (π1π¦) β cosh (β1π1) sinh (π2π¦)sinh (β1π1) cosh (π1β2) β cosh (β1π1) sinh (π2β2)
.
(14)
Substituting (14) into (11), moreover integrating (11) withrespect to π¦ and using proper boundary conditions of (12a)and (12b), the nanoparticle concentration field is received as
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Applied Bionics and Biomechanics 5
π =(β β π¦) (ππ + π΅ππ‘ sinh (β2π2) + π΄ππ‘ cosh (β2π1)) + (π¦ β β2)ππ‘ (π΅ sinh (β1π2) + π΄ cosh (β1π1))
ππ (β1 β β2)
βππ‘
ππ
(π΄ cosh (π1π¦) + π΅ sinh (π2π¦)) .(15)
Equation (7) can be written in the following form:
π2π’
ππ¦2=
π
ππ¦[(1 β π)(
ππ
ππ₯)π¦ β ππ] β (1 β π)Grπ
β (1 β π)Brπ.(16)
Integration of the above equation yields
ππ’
ππ¦= (1 β π)
ππ
ππ₯π¦ β πΞ©
β(1 β π)πΊπ
π1π2
(π2π΄ sinh π1π¦ + π1π΅ cosh π2π¦) β (1
β π)π΅π [2ππ¦ + π·π¦
2
2
βππ‘
πππ1π2
(π΄π2 sinh π1π¦ + π΅π1 cosh π2π¦)] + πΊ (π₯) .
(17)
From (9) and (17), one can write
π2Ξ©
ππ¦2β π2Ξ© =
π2(1 β π)
2 β π(ππ
ππ₯)π¦ + π΄4 sinh (π1π¦)
+ π΄5 cosh (π2π¦) + π΄6π¦ + π΄7π¦2
+π2πΊ (π₯)
2 β π.
(18)
The general solution of the above equation can be written as
Ξ© = πΈ cosh (ππ¦) + πΉ sinh (ππ¦) β 1 β π2 β π
(ππ
ππ₯)π¦
+π΄4 sinh (π1π¦)
π21 β π2
+π΄5 cosh (π2π¦)
π22 β π2
βπ΄6π¦
π2
β π΄7 (π2π¦2+ 2
π4) β
πΊ (π₯)
2 β π.
(19)
Making the above equation into (17), one obtains
π’ = (ππ
ππ₯)(
π¦2
(2 β π)βπ΄8π sinh (ππ¦)
π
βπ΄11π cosh (ππ¦)
π) β
π΄9π sinh (ππ¦)π
βπ΄12π cosh (ππ¦)
π+ cosh (π1π¦)(
(1 β π)Brππ‘π΄πππ21
β(1 β π)Grπ΄
π21
βππ΄4
π1 (π21 β π2)) + (
ππ΄7
3π2
β(1 β π)Brπ·
6)π¦3+ sinh (π2π¦)(
(1 β π)Brππ‘π΅πππ22
β(1 β π)Grπ΅
π22
βππ΄5
π2 (π22 β π2)) +
2π΄7π¦π
π4
+ππ΄6π¦
2
2π2+ πΊ (π₯)(
2π¦
(2 β π)+π΄13π cosh (ππ¦)
π
+π΄10πsinh (ππ¦)
π) + π» (π₯) .
(20)
The volume flux through each cross section in the wave frameis given by
πΉ = β«
β2
β1
π’ ππ¦. (21)
Using (21), we find that
ππ
ππ₯= (πΉ β
π΄21
π1
(sinh (π1β2) β sinh (π1β1))
βπ΄22
π2
(cosh (π2β2) β cosh (π2β1))
βπ΄32 (β
32 β β31)
3βπ΄29
π(sinh (πβ2) β sinh (πβ1))
βπ΄30
π(cosh (πβ2) β cosh (πβ1)) β
π΄24 (β42 β β41)
4
βπ΄31 (β
22 β β21)
4β π΄25 (β2 β β1))
β (π΄20 (β
22 β β21)
2 β π+β32 β β31
6 β 3π
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6 Applied Bionics and Biomechanics
+π΄27
π(cosh (πβ2) β cosh (πβ1))
+π΄28
π(sinh (πβ2) β cosh (πβ1))
+ π΄26 (β2 β β1))
β1
.
(22)The corresponding stream function from (20) is
π = (ππ
ππ₯)(
π¦2
(2 β π)βπ΄8π(cosh (ππ¦) β 1)
π2
βπ΄11π sinh (ππ¦)
π2) β
π΄9π cosh (ππ¦)π2
βπ΄12π sinh (ππ¦)
π2+ (
sinh (π1π¦)π1
)
β ((1 β π)Brππ‘π΄
πππ21
β(1 β π)Grπ΄
π21
βππ΄4
π1 (π21 β π2)) + (
ππ΄7
12π2β(1 β π)Brπ·
24)π¦4.
(23)
The constant values appeared are listed out in Appendix.The nondimensional expression for the pressure rise per
wavelength Ξππ is
Ξππ = β«
1
0β«
1
0(ππ
ππ₯)
π¦=0
ππ₯ ππ‘. (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]
ππ₯π¦ =ππ’
ππ¦β
π
(1 β π)Ξ©, (25)
ππ₯π¦ = 3π¦2(Brπ· (π β 1)
6+π΄7π
3π2) β (
ππ
ππ₯)
β (π¦ (π β 1) + π΄8π cosh (ππ¦) + π΄11π sinh (ππ¦)
β2ππ¦ (π β 1)
2 (π β 2)) + πΊ(π΄10π cosh (ππ¦) β
π
π β 2
+ π΄13π sinh (ππ¦) + 1) + cosh (π2π¦)
β (π΅Gr (π β 1)
π2
+π΄5π
π2 β π22
βπ΅Brππ‘ (π β 11)
πππ2
)
β (π
π β 1)(
π΄5 cosh (π2π¦)π2 β π22
β πΉ sinh (ππ¦)
βπΊ
π β 2β πΈ cosh (ππ¦) +
π΄4 sinh (π1π¦)π2 β π21
+π΄7 (π
2π¦2+ 2)
π4+π΄6π¦
π2+(ππ/ππ₯) (π β 1)
π β 2)
+ π1 sinh (π1π¦)(π΄Gr (π β 1)
π21
+π΄4π
π1 (π2 β π21)
βπ΄Brππ‘ (π β 1)
πππ21
) β π΄9π cosh (ππ¦)
β π΄12π sinh (ππ¦) +2π΄7π
π4+π΄6ππ¦
π2,
ππ¦π₯ =1
(1 β π)
ππ’
ππ¦+
π
(1 β π)Ξ©,
ππ¦π₯ = (β1
π β 1)(3π¦
2(Brπ· (π β 1)
6+π΄7π
3π2)
β (ππ
ππ₯) (π¦ (π β 1) + π΄8π cosh (ππ¦)
+ π΄11π sinh (ππ¦) β2ππ¦ (π β 1)
2 (π β 2))
+ πΊ(π΄10π cosh (ππ¦) βπ
π β 2+ π΄13π sinh (ππ¦)
+ 1) + cosh (π2π¦)(π΅Gr (π β 1)
π2
+π΄5π
π2 β π22
βπ΅Brππ‘ (π β 11)
πππ2
) + π1 sinh (π1π¦)
β (π΄Gr (π β 1)
π21
+π΄4π
π1 (π2 β π21)
βπ΄Brππ‘ (π β 1)
πππ21
) +2π΄7π
π4+π΄6ππ¦
π2
β π΄9π cosh (ππ¦) β π΄12π sinh (ππ¦)) β (π
π β 1)
β (π΄5cosh (π2π¦)
π2 β π22
β πΉ sinh (ππ¦) β πΊπ β 2
β πΈ cosh (ππ¦) +π΄4 sinh (π1π¦)
π2 β π21
+π΄7 (π
2π¦2+ 2)
π4
+π΄6π¦
π2+(ππ/ππ₯) (π β 1)
π β 2) .
(26)
The numerical computation for the shear stress ππ₯π¦ 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 πΉ gives the pressure rise (Ξππ) always negative and
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Applied Bionics and Biomechanics 7
0 0.5 1 1.50
0.2
0.4
0.6
0.8
1
1.2
u
y
β1 β0.5
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; Ξ = 1.5; n = 0.1;; Nt = 0.3; Nb = 0.1;
; Rn = 0.3;π = π/6; π½ = 0.3; N = 0.4
Pr = 0.5
Gr = 0.7; Br = 0.9
y: β0.6781
Figure 2: Variation ofπ on the velocity π’ with respect to π¦.
hence pumping action cannot be perceived. To discuss thefound results quantitatively, we assume the form of theinstantaneous volume rate of the flow πΉ(π₯, π‘) [30, 33β37],
πΉ (π₯, π‘) = Ξ + π sin 2π (π₯ β π‘) + π sin [2π (π₯ β π‘) + π] (27)
in which πΉ = π/ππ, Ξ = π/ππ, πΉ = β«β2β1
π’ ππ¦ = π(β2) β π(β1),and Ξ 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 2β20 with the constant values π₯ = 0.5 andπ‘ = 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 2β7 express that the varia-tion of axial velocity π’with respect to π¦ for different values ofthe nonuniform parameter (π), amplitude of right wall (π),local temperature Grashof number (Gr), Brownian motionparameter (ππ), micropolar parameter (π), and couplingnumber (π). The dimensionless axial velocity profiles (π’)satisfy the boundary conditions and are varied by a smoothcurve for different (π) 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
β1 β0.5
b = 0
b = 0.1
b = 0.2b = 0.3
a = 0.4; m = 0.3; Ξ = 2; n = 0.1;; Nt = 0.3; Nb = 0.2;
; Rn = 0.1;π = π/2; π½ = 0.3; N = 0.8
Pr = 1.5
Gr = 0.4; Br = 0.2
Figure 3: Variation of π on the velocity π’ with respect to π¦.
0
0.2
0.4
0.6
0.8
1
1.2
u
0 0.5 1y
β1 β0.5
u: 1.2
a = 0.4; b = 0.2; Ξ = 1.6; m = 0.1;; Nt = 0.2; Nb = 0.7;
; Rn = 0.2;π = π/4; π½ = 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 π’ with respect to π¦.
π’withπ¦ for various values of the amplitude of right wall (π). Itis found that an increase of π results in increase of the velocityof the fluid near the right part of the channel. The effects ofGr andππ on the axial velocity pattern are shown in Figures4 and 5. When Gr and ππ values are larger and π’ 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 π on axialvelocity is depicted in Figure 6. It is observed that the flow
-
8 Applied Bionics and Biomechanics
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
β1 β0.5
u: 0.986
Nb = 0.01Nb = 0.02
Nb = 0.03
Nb = 0.1
a = 0.2; b = 0.3; Ξ = 1.4; m = 0.2;; Nt = 0.5; n = 0.2;
; Rn = 0.2;π = π/8; π½ = 0.4; N = 0.3
Pr = 0.75
Gr = 0.5; Br = 0.4
y: β0.1971
Figure 5: Variation ofππ on the velocity π’ with respect to π¦.
0
0.2
0.4
0.6
0.8
1
1.2
u
0 0.5 1 1.5y
β1 β0.5
u: 1.213
N NewtonianN = 0.4
N = 0.8
Nβ1
a = 0.4; b = 0.5; Ξ = 1.8; m = 0.4;; Nt = 0.5; Nb = 0.4;
; Rn = 0.3;π = π/5; π½ = 0.2; n = 0.2
Pr = 0.8
Gr = 1; Br = 0.5
y: 0.1202
Figure 6: Variation ofπ on the velocity π’ with respect to π¦.
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 πin Figure 7.
4.2. Heat Transfer and Nanoparticle Mass Transfer Distri-butions. The dimensionless nanoparticle temperature andconcentration profiles (π, π) satisfy the boundary conditionsand are varied by a smooth curve for different π½,ππ, and π πvalues. In the absence of amplitude of left wall π = 0, heat
0 0.5 10
0.2
0.4
0.6
0.8
1
1.2
u
1.5y
β1 β0.5
u: 1.146
n Newtoniann = 0.3
n = 0.6
n = 1
a = 0.3; b = 0.4; Ξ = 1.6; m = 0.2;; Nt = 0.5; Nb = 0.1;
; Rn = 0.2;π = π/3; π½ = 0.2; N = 0.9
Pr = 0.1
Gr = 0.7; Br = 0.3
y: 0.2471
Figure 7: Variation of π on the velocity π’ with respect to π¦.
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
β1 β0.5
π
π½ = 0π½ = 0.25
π½ = 0.5
π½ = 0.75
a = 0.3; b = 0.4; m = 0.4;Pr = 0.5; Nt = 0.2; Nb = 0.4;π = π/4; Rn = 0.1;
Figure 8: Temperature profile π(π¦) for π½.
source/sink π½ = 0, Prandtl number ππ = 1, thermal radiationparameter π π = 0, and nonuniform parameter π = 0 ournanoparticle temperature and concentration distributionsresults are in close agreement with earlier works of Tripathiand BeΜg [24]. The effects of heat source/sink parameter (π½)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
-
Applied Bionics and Biomechanics 9
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
β1 β0.5
π½ = 0π½ = 0.5
π½ = 1
π½ = 1.5
a = 0.3; b = 0.4; m = 0.4;Pr = 0.5; Nt = 0.2; Nb = 0.4;π = π/4; Rn = 0.1;
π
Figure 9: Nanoparticle volume fraction π(π¦) for π½.
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
β1 β0.5
π
a = 0.2; b = 0.4; m = 0.3;Pr = 0.8; Nt = 0.7; π½ = 0.8;
π = π/3; Rn = 1
Nb β 0
Nb = 1
Nb = 2
Nb = 3
Figure 10: Temperature profile π(π¦) forππ.
and 11. When ππ is larger and π curve is lower, so the masstransfer effect is higher for a largerππ 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 ofππ, 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
β1 β0.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
π
Nb = 1
Nb = 2
Nb = 3
a = 0.2; b = 0.4; m = 0.3;Pr = 0.8; Nt = 0.7; π½ = 0.8;
π = π/3; Rn = 1
Nb = 0.5
Figure 11: Nanoparticle volume fraction π(π¦) forππ.
0
0.2
0.4
0.6
0.8
1
0 0.5 1y
β1 β0.5
π
a = 0.2; b = 0.3; m = 0.2;Pr = 0.6; Nt = 0.5; Nb = 0.5;
π = π/4; π½ = 1
Rn = 0Rn = 0.3
Rn = 0.6
Rn = 1
Figure 12: Temperature profile π(π¦) for π π.
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 (Ξ©) with respect to π¦ for various values of cou-pling number (π), Brownian motion (ππ), and nonuniformparameter (π) are shown in Figures 14β16. 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
-
10 Applied Bionics and Biomechanics
0 0.5 1y
β1 β0.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
π
a = 0.2; b = 0.3; m = 0.2;Pr = 0.6; Nt = 0.5; Nb = 0.5;
π = π/4; π½ = 1
Rn = 0Rn = 0.3
Rn = 0.6
Rn = 1
Figure 13: Nanoparticle volume fraction π(π¦) for π π.
0 0.5 1 1.50
0.2
0.4
0.6
0.8
1
1.2
u
y
β1 β0.5
y: 0.1202u: 1.213
N newtonianN = 0.4
N = 0.8
Nβ 1
a = 0.4; b = 0.5; Ξ = 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 Ξ©(π¦) for different valuesofπ.
velocity is illustrated in Figure 14. It is seen that in the micro-rotation velocity increases by increasing the coupling numberπ (i.e., microrotation velocity for the micropolar nanofluidis wider than that of the region for Newtonian nanofluid).In the case of π = 0, there is no appreciable differencebetween Newtonian nanofluids and micropolar nanofluids.The microrotation velocity for the Brownian motion param-eter (ππ) is plotted in Figure 15. It is seen that, with theincrease in the Brownian motion parameter, microrotationvelocity profile decreases andmaximummicrorotation veloc-ity occurs at ππ β 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
β1 β0.5
Nb = 0.02
Nb = 0.03
Nb = 0.04Nb = 0.01
Ξ©
β0.02
a = 0.2; b = 0.3; Ξ = 1.4;m = 0.2;n = 0.2; Gr = 0.5; Br = 0.4; Rn = 0.2; π = π/8;
Pr = 0.75;Nt = 0.5; π½ = 0.4;N = 0.3
Figure 15:Microrotation velocity profileΞ©(π¦) for different values ofππ.
0
1
2
3
4
5
6
7
8
9
10
0 0.5 1 1.5y
β1 β0.5
m = 0.0m = 0.2
m = 0.4
m = 0.6
a = 0.1; b = 0.3; Ξ = 1.5;
n = 0.1; π = π/6; π½ = 0.3;
Pr = 0.5;Nt = 0.3;Nb = 0.1; Br = 0.9, Rn = 0.3;
N = 0.4
Γ10β3
Ξ©
Gr = 0.7
Figure 16: Microrotation velocity profile Ξ©(π¦) for different valuesofπ.
is also viewed that the microrotation velocity for a divergentchannel (π > 0) is higher compared to its value for a uniformchannel (π = 0).
4.4. Shear Stress Distribution. It is well known that the stresstensor is not symmetric in micropolar nanofluid. In Figures17β19, we have plotted the shear stresses ππ₯π¦ at the left wall forvalues of the nonuniform parameter, micropolar parameter,and coupling number. One can observe from Figure 17 that
-
Applied Bionics and Biomechanics 11
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
πxy
a = b = 0.2;N = 0.8,Ξ = 2.5; n = 0.1;Pr = 1.5;Nt = 0.3;Nb = 0.2; Gr = 0.4;Br = 0.2; Rn = 0.1;π = π/4; π½ = 0.3
y = β1
Figure 17: Shear stress profile ππ₯π¦(π₯) for different values ofπ.
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
πxy
nβ 0n = 0.4
n = 0.8
n = 1
a = 0.5; b = 0.3; Ξ = 1.8;m = 0.5;Pr = 0.8;Nt = 0.2;Nb = Rn = 0.1;Gr = 3; Br = 2; π = π/2; π½ = 0.4;
N = 0.9
Figure 18: Shear stress profile ππ₯π¦(π₯) for different values of π.
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 parameterπ, while it increases as the coupling number π 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
πxy
N = 0
N = 0.4
N = 0.8
Nβ 1
a = 0.2; b = 0.3; Ξ = 1.2;n = 0.3; Pr = 1;Nt = 0.4;Nb = 0.2; Gr = 0.5; Br = 0.5;
Rn = 0.2; π = π/4; π½ = 0.1;m = 0.2
Figure 19: Shear stress profile ππ₯π¦(π₯) for different values ofπ.
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 ofπ on trappingare presented in Figures 20(b) and 20(c). This figure revealsthat the size of lower bolus decreases with an increase in π.FromFigures 20(c) and 20(d), it is clear that the trapped bolusincreases in size as π 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 whenπ 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 π has reverse behav-ior when compared to heat transfer.
(iv) Coupling number and Brownian motion parametershave opposite effects on the microrotation velocity.
-
12 Applied Bionics and Biomechanics
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
π(x, y)2.0
1.5
1.0
0.5
0.0
x
β2 β1 0 1 2
y
β0.2
β0.1
β0.1
β0.6
β0.9
β0.7
β0.7
β0.8
β0.3
β0.6
β0.2β0.8β0.9
(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
π(x, y)2.0
1.5
1.0
0.5
0.0
β2 β1β1.4
0 1 2
y
x
β0.8
β0.8
β0.6
β0.6
β0.6
β0.9
β0.1
β0.1
β0.2
β0.2β0.3
0.6
0.6
β0.7
β0.7
(b)
0
0
π(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
β2
β0.5
β0.8
β0.8
β0.7
β0.7
β0.9
β0.9
β0.6
β0.6
β0.2
β0.3
β0.1
β0.1
β1 0 1 2
y
x
(c)
0π(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
β2
β0.2
β0.5
β0.8
β0.8
β0.9
β0.9
β0.1β0.7
β0.7
β0.6
β0.6
β0.6
β1 0 1 2
y
x
(d)
Figure 20: Streamlines for π = 0, π = 0.01, π = 0.1 (a), π = 0.3, π = 0.01, π = 0.1 (b), π = 0.3, π β 1, π = 0.1 (c), and π = 0.3,π β 1, π = 0.9 (d). The other parameters chosen are π = 0.2, π = 0.3, π = 0.3, Ξ = 1.5, π = π/2, ππ‘ = 0.2, ππ = 0.3, π π = 0.9, Pr = 0.5,Gr = 0.6, Br = 0.2, π½ = 0.2, and π‘ = 0.2.
(v) The walls shear stress ππ₯π¦ decreases with the increaseof nonuniform parameter π 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:
π1 =
βπ΄2 + βπ΄22 β 4π΄3
2,
π2 =
βπ΄2 β βπ΄22 β 4π΄3
2,
-
Applied Bionics and Biomechanics 13
π΄ =βsinh (π1β1)
sinh (π2β2) cosh (π1β1) β sinh (π1β1) cosh (π1β2),
π΅ =cosh (π1β1)
sinh (π2β2) cosh (π1β1) β sinh (π1β1) cosh (π1β2),
πΆ = βπ·β1 +ππ‘
ππ
π΄ cosh (π1β1) +ππ‘
ππ
π΅ sinh (π2β1) ,
π· =1
β2 β β1
+ππ‘
ππ (β2 β β1)π΄ (cosh (π1β2)
β cosh (π1β1)) +ππ‘
ππ (β2 β β1)π΅ (sinh (π2β2)
β sinh (π2β1)) ,
πΈ = π΄8 (ππ
ππ₯) + π΄9 + πΊπ΄10,
πΉ = π΄11 (ππ
ππ₯) + π΄12 + πΊπ΄13,
πΊ = (π΄17 β π΄14
π΄16 β π΄19
)(ππ
ππ₯) + (
π΄18 β π΄15
π΄16 β π΄19
) ,
π» = β(π΄14 +π΄16 (π΄17 β π΄14)
π΄16 β π΄19
)(ππ
ππ₯) β π΄15
β (π΄16 (π΄18 β π΄15)
π΄16 β π΄19
) ,
π΄1 =ππ‘ + ππ
ππ (β2 β β1),
π΄2 =πππππ΄1
1 + π πππ
,
π΄3 =πππ½
1 + π πππ
,
π΄4 =π2(1 β π)π΄
(2 β π) π1
(Brππ‘ππ
β Gr) ,
π΄5 =π2(1 β π)π΅
(2 β π) π2
(Brππ‘ππ
β Gr) ,
π΄6 = β(1 β π)BrπΆπ2
2 β π,
π΄7 = β(1 β π)Brπ·π2
2 β π,
π΄8 =(1 β π) β1
(2 β π) cosh (πβ1)
β(1 β π) sinh (πβ1) (β1 cosh (πβ2) β β2 cosh (πβ1))
sinh (π (β1 β β2)) cosh (πβ1),
π΄9 = β(π΄33
cosh (πβ1)
+(cosh (πβ1) β cosh (πβ2)) sinh (πβ1)
sinh (π (β1 β β2)) cosh (πβ1)) ,
π΄10 =1
(2 β π) cosh (πβ1)
βsinh (πβ1) (cosh (πβ2) β cosh (πβ1))(2 β π) sinh (π (β1 β β2)) cosh (πβ1)
,
π΄11 =(1 β π) (β1 cosh (πβ2) β β2 cosh (πβ1))
sinh (π (β1 β β2)),
π΄12 =(π΄34 cosh (πβ1) β π΄33 cosh (πβ2))
sinh (π (β1 β β2)),
π΄13 =(cosh (πβ2) β cosh (πβ1))(2 β π) sinh (π (β1 β β2))
,
π΄14 =β21
2 β πβπ΄8 sinh (πβ1) + π΄11 cosh (πβ1)
π,
π΄15 = βπ (π΄9 sinh πβ1 + π΄12 cosh (πβ1))
π
+ cosh (π1β1) ((1 β π)π΄ (Brππ‘ β Grππ)
πππ21
βππ΄4
π1 (π21 β π2)) + sinh (π2β1)
β ((1 β π)π΅ (Brππ‘ β Grππ)
πππ22
βππ΄5
π2 (π22 β π2))
+ππ΄6β
21
2π2+ (
ππ΄7
3π2β(1 β π)Brπ·
6) β31 +
2ππ΄7β1
π4,
π΄16 =2β1
2 β π+π(π΄13 cosh (πβ1) + π΄10 sinh (πβ1))
π,
π΄17 =β22
2 β πβπ΄8 sinh (πβ2) + π΄11 cosh (πβ2)
π,
π΄18 = βπ (π΄9 sinh πβ2 + π΄12 cosh (πβ2))
π
+ cosh (π1β2) ((1 β π)π΄ (Brππ‘ β Grππ)
πππ21
βππ΄4
π1 (π21 β π2)) + sinh (π2β2)
β ((1 β π)π΅ (Brππ‘ β Grππ)
πππ22
βππ΄5
π2 (π22 β π2))
+ππ΄6β
22
2π2+ (
ππ΄7
3π2β(1 β π)Brπ·
6) β32 +
2ππ΄7β2
π4,
π΄19 =2β2
2 β π+π(π΄13 cosh (πβ2) + π΄10 sinh (πβ2))
π,
π΄20 =π΄17 β π΄14
π΄16 β π΄19
,
π΄21 =(1 β π)Brππ‘π΄
πππ21
β(1 β π)Grπ΄
π21
βππ΄4
π1 (π21 β π2),
π΄22 =(1 β π)Brππ‘π΅
πππ22
β(1 β π)Grπ΅
π22
βππ΄5
π2 (π22 β π2),
-
14 Applied Bionics and Biomechanics
π΄23 =ππ΄7
3π2β(1 β π)Brπ·
6,
π΄24 =π΄18 β π΄15
π΄16 β π΄19
,
π΄25 = βπ΄15 β π΄16π΄24,
π΄26 = βπ΄4 β π΄16π΄20,
π΄27 =π (π΄10 β π΄8)
π,
π΄28 =π (π΄13 β π΄11)
π,
π΄29 =π (π΄24π΄13 β π΄12)
π,
π΄30 =π (π΄24π΄10 β π΄9)
π,
π΄31 =2π΄7π
π4+2π΄24
2 β π,
π΄32 =ππ΄6
2π2,
π΄33 =π΄4 sinh (π1β1)
π21 β π2
+π΄5 cosh (π1β1)
π22 β π2
βπ΄6β1
π2
β π΄7 (π2β21 + 2
π4) ,
π΄34 =π΄4 sinh (π1β2)
π21 β π2
+π΄5 cosh (π1β2)
π22 β π2
βπ΄6β2
π2
β π΄7 (π2β22 + 2
π4) .
(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.
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