# Analysis of Heat and Mass Transfer on MHD Peristaltic Flow through a Tapered Asymmetric Channel

Authors: M. Kothandapani; J. Prakash; V. Pushparaj
Journal: Journal of Fluids
 (2015)

---

## Abstract

This paper describes the peristaltic flow of an incompressible viscous fluid in a tapered asymmetric channel with heat and mass transfer. The fluid is electrically conducting fluid in the presence of a uniform magnetic field. The propagation of waves on the nonuniform channel walls to have different amplitudes and phase but with the same speed is generated the tapered asymmetric channel. The assumptions of low Reynolds number and long wavelength approximations have been used to simplify the complicated problem 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 problem and interpreted.


---

## Body

## 1. Introduction

The study of peristaltic transport has enjoyed increased interest from investigators in several engineering disciplines. From a mechanical point of view, peristalsis offers the opportunity of constructing pumps in which the transported medium does not come in direct contact with any moving parts such as valves, plungers, and rotors. This could be of great benefit in cases where the medium is either highly abrasive or decomposable under stress. This has led to the development of fingers and roller pumps which work according to the principle of peristalsis. Applications include dialysis machines, open-heart bypass pump machines, and infusion pumps. After the first investigation reported by Latham [1], several theoretical and experimental investigations [2–5] about the peristaltic flow of Newtonian and non-Newtonian fluids have been made under different conditions with reference to physiological and mechanical situations.In view of the processes like hemodialysis and oxygenation, some progress is shown in the theory of peristalsis with heat transfer [6–8]. Such analysis of heat transfer is of great value in biological tissues, dilution technique in examining blood flow, destruction of undesirable cancer tissues, metabolic heat generation, and so forth. In addition, the mass transfer effect on the peristaltic flow of viscous fluid has been examined in the studies [9–11]. The effect of magnetic field on a Newtonian fluid has been reported for treatment of gastronomic pathologies, constipation, and hypertension.Recently, few attempts have been made in the peristaltic literature to study the combined effects of heat and mass transfer. Eldabe et al. [10] analyzed the mixed convective heat and mass transfer in a non-Newtonian fluid at a peristaltic surface with temperature-dependent viscosity. The influence of heat and mass transfer on MHD peristaltic flow through a porous space with compliant walls was studied by Srinivas and Kothandapani [11]. The effects of elasticity of the flexible walls on the peristaltic transport of viscous fluid with heat transfer in a two-dimensional uniform channel have been investigated by Srinivas and Kothandapani [12]. Ogulu [13] examined heat and mass transfer of blood under the influence of a uniform magnetic field.The problem of intrauterine fluid motion in a nonpregnant uterus caused by myometrial contractions is a peristaltic-type fluid motion and the myometrial contractions may occur in both symmetric and asymmetric directions. Further it is observed that the intrauterine fluid flow in a sagittal cross section of the uterus discloses a narrow channel enclosed by two fairly parallel walls with wave trains having different amplitudes and phase difference [14–16]. Keeping in view of the abovementioned reasons, a new theoretical investigation of peristaltic motion of a Newtonian fluid in the presence of heat and mass transfer in the most generalized form of the channel, namely, the tapered asymmetric channel, is carried out. The governing equations of motion, energy, and concentration are simplified by using the assumptions of long wavelength and low Reynolds number approximations. The exact solutions of velocity, temperature, and concentration of the fluid are generated. Also interesting flow quantities are analyzed by plotting various graphs.


## 2. Mathematical Formulation

Consider the unsteady, combined convective heat and mass transfer, MHD flow of an electrically conducting viscous fluid in a two-dimensional tapered asymmetric channel. LetY¯=H¯1 and Y¯=H¯2, respectively, the lower and upper wall boundaries of the tapered asymmetric channel. The flow is generated by sinusoidal wave trains propagating with constant speed c along the tapered asymmetric channel (Figure 1). The geometry of the wall surface [15, 16] is defined as(1a)H2¯X¯,t¯=d+m′X¯+b2sin2πλX¯-ct¯.iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiupperwall,(1b)H1¯X¯,t¯=-d-m′X¯-b1sin2πλX¯-ct¯+ϕ.iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiilowerwall,where d is the half width of the channel at the inlet, b1 and b2 are the amplitudes of lower and upper walls, respectively, c is the phase speed of the wave, m′  (m′≪1) is the nonuniform parameter, λ is the wavelength, the phase difference ϕ varies in the range 0≤ϕ≤π, ϕ=0 corresponds to symmetric channel with waves out of phase (i.e., both walls move towards outward or inward simultaneously), and further b1, b2, and ϕ satisfy the following conditions for the divergent channel at the inlet:
(2)b12+b22+2b1b2cos⁡(ϕ)≤2d2.Figure 1
Schematic diagram of the tapered asymmetric channel.It is assumed that the temperature and concentration at lower wall areT0 and C0, respectively, while the temperature and concentration at the upper wall are T1 and C1, respectively (T0<T1&C0<C1). The equations of continuity, momentum, energy, and concentration are described as follows [11, 12]:
(3)∂U¯∂X¯+∂V¯∂Y¯=0,ρ∂U¯∂t¯+U¯∂U¯∂X¯+V¯∂U¯∂Y¯=-∂P¯∂X¯+μ∂2U¯∂X¯2+∂2U¯∂Y¯2-σB02U¯,ρ∂V¯∂t¯+U¯∂V¯∂X¯+V¯∂V¯∂Y¯=-∂P¯∂Y¯+μ∂2V¯∂X¯2+∂2V¯∂Y¯2,ρζ∂T¯∂t¯+U¯∂T¯∂X¯+V¯∂T¯∂Y¯=κ∂2T¯∂X¯2+∂2T¯∂Y¯2+μ2∂U¯∂X¯2+∂V¯∂Y¯2+∂U¯∂Y¯+∂V¯∂X¯2,∂C¯∂t¯+U¯∂C¯∂X¯+V¯∂C¯∂Y¯=Dm∂2C¯∂X¯2+∂2C¯∂Y¯2+DmKTTm∂2T¯∂X¯2+∂2T¯∂Y¯2,
where U¯, V¯ are the components of velocity along X¯ and Y¯ directions, respectively, t¯ is the dimensional time, μ is the coefficient of viscosity, σ is the electrical conductivity of the fluid, B0 is the uniform applied magnetic field, ρ is the fluid density, ζ is the specific heat at constant volume, P- is the pressure, T¯ is the temperature, C¯ is the concentration of the fluid, Tm is the mean temperature, κ is the thermal conductivity, Dm is the coefficient of mass diffusivity, and KT is the thermal diffusion ratio.The corresponding boundary conditions are(4a)U¯=0,T¯=T0,C¯=C0atY¯=H¯1,(4b)U¯=0,T¯=T1,C¯=C¯1,atY¯=H¯2.By introducing the following set of nondimensional variables in (3),
(5)x′=X¯λ,y′=Y¯d,t′=ct¯λ,u′=U¯c,v′=V¯c,δ=dλ,h1=H1¯d,h2=H2¯d,p′=d2P¯cλμ,θ=T¯-T0T1-T0,Θ=C¯-C0C1-C0,
we obtain
(6)Rδ∂u′∂t′+u′∂u′∂x′+v′∂u′∂y′=-∂p′∂x′+δ2∂2u′∂x′2+∂2u′∂y′2-M2u′,Rδ3∂v′∂t′+u′∂v′∂x′+v′∂v′∂y′=-∂p′∂y′+δ2∂2v′∂x′2+∂2v′∂y′2,Rδ∂θ∂t′+u′∂θ∂x′+v′∂θ∂y′=1Pr⁡δ2∂2θ∂x′2+∂2θ∂y′2+E2δ2∂u′∂x′2+∂v′∂y′2llllllllllllllll+∂u′∂y′+δ∂v′∂x′2,Rδ∂Θ∂t′+u′∂Θ∂x′+v′∂Θ∂y′=1Scδ2∂2Θ∂x′2+∂2Θ∂y′2+Srδ2∂2θ∂x′2+∂2θ∂y′2,
where a  (=b1/d) and b  (=b2/d) are nondimensional amplitudes of lower and upper walls, respectively, k1=(λm′/d) is nonuniform parameter, R  (=cdρ/μ) is the Reynolds number, Sc=μ/Dmρ is the Schmidt number, Sr=ρDmKT(T1-T0)/μ(C1-C0)Tm is the Soret number, M=σ/μB0d is the Hartmann number, Pr⁡=ρνζ/κ is the Prandtl number, and E=c2/ζ(T1-T0) is the Eckert number.In order to discuss the results quantitatively, we assume the instantaneous volume rate of the flowF(x,t), periodic in (x-t) [17–19], as
(7)Fx,t=Q+asin2πx-t+ϕ+bsin2πx-t,
where Q is the time-average of the flow over one period of the wave and
(8)F=∫h1h2udy.


## 3. Exact Analytical Solution

Using the long wavelength approximation and neglecting the wave number along with low Reynolds number and omitting prime, one can find from (6) that
(9)0=-∂p∂x+∂2u∂y2-M2u,(10)0=-∂p∂y,(11)0=1Pr⁡∂2θ∂y2+E∂u∂y2,(12)0=∂2Θ∂y2+ScSr∂2θ∂y2.
Equation (10) shows that p is not function of y.The corresponding boundary conditions are(13a)iiiiiiiiiiiiiiiiiiiiu=0,θ=1,andΘ=1aty=h2=1+k1x+bsin2πx-t,(13b)iiiiiiiiiiiiiiiiiiiiiiiiiiu=0,θ=0,andΘ=0aty=h1=-1-k1x-asin2π(x-t)+ϕwhich satisfy, at the inlet of channel,
(14)a2+b2+2abcos⁡(ϕ)≤4.
The set of (9)–(12), subject to the conditions (13a) and (13b), are solved exactly for u, θ, and Θ, and we have
(15)u=∂p/∂xM2×1-cosh⁡⁡Mh2-cosh⁡⁡Mh1sinh⁡⁡Mh1M2sinh⁡⁡Mh1-h2×cosh⁡⁡Mycosh⁡⁡Mh1+∂p/∂xM2×cosh⁡⁡Mh2-cosh⁡⁡Mh1sinh⁡⁡MyM2sinh⁡⁡Mh1-h2-1,θ=A1+B1y-Pr⁡EA28cosh⁡⁡2My-2y2M2-Pr⁡EB28cosh⁡⁡2My+2y2M2-Pr⁡EAB8sinh⁡⁡(2My),Θ=4D3ScSrM2h12-D4h1-D1ScSre2Mh1-D2ScSre-2Mh1+D4y+D1ScSre2My+D2ScSre-2My-4D3ScSrM2y2,∂p∂x=FM3cosh⁡Mh1sinh⁡Mh1-h2×sinh⁡⁡Mh2-sinh⁡⁡Mh1lllll×sinh⁡⁡Mh1-h2lllllllll-cosh⁡⁡Mh2-cosh⁡⁡Mh1sinh⁡⁡Mh1-1+FM3sinh⁡⁡Mh1-h2cosh⁡Mh2-cosh⁡Mh12-FM2h2-h1,
where
(16)A=∂p/∂x-BM2sinh⁡(Mh1)M2cosh⁡(Mh1),B=∂p/∂xcosh⁡(Mh2)-cosh⁡(Mh1)M2sinh⁡(M(h1-h2)),A1=Pr⁡EA28cosh⁡⁡2Mh1-2h12M2+Pr⁡EB28cosh⁡⁡2Mh1+2h12M2+Pr⁡EAB8sinh⁡⁡[2Mh1]-Dh1,B1=1h2-h11+Pr⁡EA28cosh⁡(2Mh2)-2h22M2lllllllllllllllllllllllllllllllllll-cosh⁡(2Mh1)+2h12M2llllllllllllll+Pr⁡EB28cosh⁡(2Mh2)+2h22M2lllllllllllllllllllllllllllllll-cosh⁡(2Mh1)-2h12M2llllllllllllll+Pr⁡EAB8sinh⁡(2Mh2)-sinh⁡(2Mh1)Pr⁡EAB28,D1=Pr⁡EA216+Pr⁡EB216+Pr⁡EAB8,D2=Pr⁡EA216+Pr⁡EB216-Pr⁡EAB8,D3=Pr⁡EA216-Pr⁡EB216,D4=1h1-h24D3ScSrM2h12-h22lllllllllllllll-D1ScSre2Mh1-e2Mh2llllllllllllll-D2ScSre-2Mh1-e-2Mh2-1.
The coefficient of heat transfer at the lower wall is given by
(17)Z=h1xθy.


## 4. Numerical Results and Discussion

The effect of various flow parameters on temperature is plotted in Figure2 for the fixed values of x=0.6 and t=0.4. The influence of the nonuniform parameter (k1) on θ is depicted in Figure 2(a). It is noticed that the temperature (θ) increases nearer to the lower wall of the tapered channel while the situation is reversed as nonuniform parameter (k1) increases. Figure 2(b) reveals that the temperature profile increases with increase of Eckert number. It is considered from Figure 2(c) that the temperature profile increases as the amplitude of lower tapered channel increases. The variation of the Prandtl number (Pr⁡) on θ is shown in Figure 2(d). This figure indicates that an increase in Prandtl number results in increase in the temperature of the fluid. Figure 2(e) depicts the effect of temperature for the various values of Q. It is observed that the temperature θ increases with an increase in the time-average flow rate Q in the entire tapered channel. The variations of the Hartmann number M against θ are shown in Figure 2(f). This figure indicates that by increasing 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). These figures display the typical oscillatory behavior of heat transfer which may be due to the phenomenon of peristalsis. Figures 3(a)–3(f) reveal that the absolute value of the heat transfer coefficient increases by increasing E, a, Q, M, ϕ, and Pr⁡.Temperature distribution (for fixed value oft=0.4 and x=0.6): (a) Q=1.7, ϕ=π/3, M=1, E=0.5, Pr⁡=1.7, a=0.3, and b=0.4; (b) E=0.5, k1=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, k1=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, k1=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.
(a)
(b)
(c)
(d)
(e)
(f)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, k1=0.25, M=2.5, Pr⁡=1.5, ϕ=π/3, k1=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, and M=1.6; (f) Q=1.72, x=0.42, E=1, a=0.2, k1=0.23, M=1.2, ϕ=π/4, and b=0.3.
(a)
(b)
(c)
(d)
(e)
(f)The effect of various physical parameters on the concentration of the fluid(Θ) is shown in Figure 4. Figure 4(a) exposes that the fluid concentration decreases with an increase of E. It is noticed from Figure 4(b) that the concentration of the fluid increases as a increases. Figure 4(c) shows that the absolute value of concentration distribution increases at the central part of channel when k1 is increased. Figure 4(d) is plotted to see the influence of M on the concentration. We notice that an increase in M increases Θ. The concentration for the phase difference is shown in Figure 4(e). It is observed that an increase in ϕ causes increase in Θ. In Figure 4(f), the cause of Prandtl number Pr⁡ on Θ is captured. It is detected that with an increase in Pr⁡ the concentration of the fluid decreases. Figure 4(g) depicts the concentration profile corresponding to various values of Q. It is seen that the concentration distribution decreases with an increase in the mean flow rate Q. From Figure 4(h), one can view that the concentration profile decreases with increase of Sc.Mass transfer distribution (fixed value ofx=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, k1=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.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)


## 5. Conclusion

The present analysis can serve as a model which may help in understanding the mechanism of physiological flow of a Newtonian fluid in tapered asymmetric channel. The analytical solutions for the temperature, concentration, and coefficient of heat transfer have been obtained under long wavelength and low Reynolds number approximations. The features of the flow characteristics are analyzed by plotting graphs and discussed in detail. The main observations found from the present study are given as follows.(1)
There is an increase in the temperature(θ) when the Prandtl number Pr⁡, time-average flow rate Q, occlusion parameter a, and Eckert number are increased while it decreases when M 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 at the core part of channel when nonuniform parameter is increased and the reverse situation is observed in respect of the concentration of the fluid.(4)
The absolute value of the heat transfer coefficient increases when theE, a, Q, M, ϕ, and Pr⁡ are increased.



---

*Source: 561263-2015-01-26.xml*