Research Article | Open Access

M. Kothandapani, J. Prakash, V. Pushparaj, "Analysis of Heat and Mass Transfer on MHD Peristaltic Flow through a Tapered Asymmetric Channel", *Journal of Fluids*, vol. 2015, Article ID 561263, 9 pages, 2015. https://doi.org/10.1155/2015/561263

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

**Academic Editor:**Miguel Onorato

#### 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.

#### 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. Let and , respectively, the lower and upper wall boundaries of the tapered asymmetric channel. The flow is generated by sinusoidal wave trains propagating with constant speed along the tapered asymmetric channel (Figure 1). The geometry of the wall surface [15, 16] is defined aswhere is the half width of the channel at the inlet, and are the amplitudes of lower and upper walls, respectively, is the phase speed of the wave, is the nonuniform parameter, is the wavelength, the phase difference varies in the range , corresponds to symmetric channel with waves out of phase (i.e., both walls move towards outward or inward simultaneously), and further , , and satisfy the following conditions for the divergent channel at the inlet:

It is assumed that the temperature and concentration at lower wall are and , respectively, while the temperature and concentration at the upper wall are and , respectively . The equations of continuity, momentum, energy, and concentration are described as follows [11, 12]: where , are the components of velocity along and directions, respectively, is the dimensional time, is the coefficient of viscosity, is the electrical conductivity of the fluid, is the uniform applied magnetic field, is the fluid density, is the specific heat at constant volume, is the pressure, is the temperature, is the concentration of the fluid, is the mean temperature, is the thermal conductivity, is the coefficient of mass diffusivity, and is the thermal diffusion ratio.

The corresponding boundary conditions areBy introducing the following set of nondimensional variables in (3), we obtain where and are nondimensional amplitudes of lower and upper walls, respectively, is nonuniform parameter, is the Reynolds number, is the Schmidt number, is the Soret number, is the Hartmann number, is the Prandtl number, and is the Eckert number.

In order to discuss the results quantitatively, we assume the instantaneous volume rate of the flow , periodic in [17–19], as where is the time-average of the flow over one period of the wave and

#### 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 Equation (10) shows that is not function of .

The corresponding boundary conditions arewhich satisfy, at the inlet of channel, The set of (9)–(12), subject to the conditions (13a) and (13b), are solved exactly for , , and , and we have where The coefficient of heat transfer at the lower wall is given by

#### 4. Numerical Results and Discussion

The effect of various flow parameters on temperature is plotted in Figure 2 for the fixed values of and . The influence of the nonuniform parameter () on is depicted in Figure 2(a). It is noticed that the temperature increases nearer to the lower wall of the tapered channel while the situation is reversed as nonuniform parameter () 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 () on 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 . It is observed that the temperature increases with an increase in the time-average flow rate in the entire tapered channel. The variations of the Hartmann number against 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 , , , , , and on the heat transfer coefficient (). 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 , , , , , and .

**(a)**

**(b)**

**(c)**

**(d)**

**(e)**

**(f)**

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**(c)**

**(d)**

**(e)**

**(f)**

The effect of various physical parameters on the concentration of the fluid is shown in Figure 4. Figure 4(a) exposes that the fluid concentration decreases with an increase of . It is noticed from Figure 4(b) that the concentration of the fluid increases as increases. Figure 4(c) shows that the absolute value of concentration distribution increases at the central part of channel when is increased. Figure 4(d) is plotted to see the influence of on the concentration. We notice that an increase in increases . The concentration for the phase difference is shown in Figure 4(e). It is observed that an increase in causes increase in . In Figure 4(f), the cause of Prandtl number on is captured. It is detected that with an increase in the concentration of the fluid decreases. Figure 4(g) depicts the concentration profile corresponding to various values of . It is seen that the concentration distribution decreases with an increase in the mean flow rate . From Figure 4(h), one can view that the concentration profile decreases with increase of .

**(a)**

**(b)**

**(c)**

**(d)**

**(e)**

**(f)**

**(g)**

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#### 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 when the Prandtl number , time-average flow rate , occlusion parameter , and Eckert number are increased while it decreases when is increased.(2)Mass transfer increases with the increase of , , and while it decreases with the increase of , , , and .(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 the , , , , , and are increased.

#### Conflict of Interests

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

#### References

- T. W. Latham,
*Fluid Motion in a Peristaltic Pump*, MIT Press, Cambridge, Mass, USA, 1966. - A. H. Shapiro, M. Y. Jaffrin, and S. L. Weinberg, “Peristaltic pumping with long wave lengths at low Reynolds number,”
*Journal of Fluid Mechanics*, vol. 37, no. 4, pp. 799–825, 1969. View at: Publisher Site | Google Scholar - G. Radhakrishnamacharya and V. Radhakrishna Murty, “Heat transfer to peristaltic transport in a non-uniform channel,”
*Defence Science Journal*, vol. 43, no. 3, pp. 275–280, 1993. View at: Google Scholar - T. Hayat, Y. Wang, K. Hutter, S. Asghar, and A. M. Siddiqui, “Peristaltic transport of an Oldroyd-B fluid in a planar channel,”
*Mathematical Problems in Engineering*, vol. 2004, no. 4, pp. 347–376, 2004. View at: Publisher Site | Google Scholar | MathSciNet - K. Vajravelu, S. Sreenadh, and V. R. Babu, “Peristaltic transport of a Herschel-Bulkley fluid in an inclined tube,”
*International Journal of Non-Linear Mechanics*, vol. 40, no. 1, pp. 83–90, 2005. View at: Publisher Site | Google Scholar - S. Nadeem, T. Hayat, N. S. Akbar, and M. Y. Malik, “On the influence of heat transfer in peristalsis with variable viscosity,”
*International Journal of Heat and Mass Transfer*, vol. 52, no. 21-22, pp. 4722–4730, 2009. View at: Publisher Site | Google Scholar - T. Hayat, M. U. Qureshi, and Q. Hussain, “Effect of heat transfer on the peristaltic flow of an electrically conducting fluid in a porous space,”
*Applied Mathematical Modelling*, vol. 33, no. 4, pp. 1862–1873, 2009. View at: Publisher Site | Google Scholar - K. S. Mekheimer, S. Z. A. Husseny, and Y. A. Elmaboud, “Effects of heat transfer and space porosity on peristaltic flow in a vertical asymmetric channel,”
*Numerical Methods for Partial Differential Equations*, vol. 26, no. 4, pp. 747–770, 2010. View at: Publisher Site | Google Scholar | MathSciNet - A. Ogulu, “Effect of heat generation on low Reynolds number fluid and mass transport in a single lymphatic blood vessel with uniform magnetic field,”
*International Communications in Heat and Mass Transfer*, vol. 33, no. 6, pp. 790–799, 2006. View at: Publisher Site | Google Scholar - N. T. M. Eldabe, M. F. El-Sayed, A. Y. Ghaly, and H. M. Sayed, “Mixed convective heat and mass transfer in a non-Newtonian fluid at a peristaltic surface with temperature-dependent viscosity,”
*Archive of Applied Mechanics*, vol. 78, no. 8, pp. 599–624, 2008. View at: Publisher Site | Google Scholar - S. Srinivas and M. Kothandapani, “The influence of heat and mass transfer on MHD peristaltic flow through a porous space with compliant walls,”
*Applied Mathematics and Computation*, vol. 213, no. 1, pp. 197–208, 2009. View at: Publisher Site | Google Scholar | MathSciNet - S. Srinivas and M. Kothandapani, “Peristaltic transport in an asymmetric channel with heat transfer—a note,”
*International Communications in Heat and Mass Transfer*, vol. 35, no. 4, pp. 514–522, 2008. View at: Publisher Site | Google Scholar - T. A. Ogulu, “Effect of heat generation on low Reynolds number fluid and mass transport in a single lymphatic blood vessel with uniform magnetic field,”
*International Communications in Heat and Mass Transfer*, vol. 33, no. 6, pp. 790–799, 2006. View at: Publisher Site | Google Scholar - O. Eytan, A. J. Jaffa, and D. Elad, “Peristaltic flow in a tapered channel: application to embryo transport within the uterine cavity,”
*Medical Engineering and Physics*, vol. 23, no. 7, pp. 473–482, 2001. View at: Google Scholar - M. Kothandapani and J. Prakash, “The peristaltic transport of Carreau nanofluids under effect of a magnetic field in a tapered asymmetric channel: application of the cancer therapy,”
*Journal of Mechanics in Medicine and Biology*, vol. 15, Article ID 1550030, 2014. View at: Publisher Site | Google Scholar - M. Kothandapani and J. Prakash, “Effect of radiation and magnetic field on peristaltic transport of nanofluids through a porous space in a tapered asymmetric channel,”
*Journal of Magnetism and Magnetic Materials*, vol. 378, pp. 152–163, 2015. View at: Publisher Site | Google Scholar - L. M. Srivastava, V. P. Srivastava, and S. N. Sinha, “Peristaltic transport of a physiological fluid. Part-I. Flow in non-uniform geometry,”
*Biorheology*, vol. 20, no. 2, pp. 153–166, 1983. View at: Google Scholar - M. Kothandapani and J. Prakash, “Influence of heat source, thermal radiation and inclined magnetic field on peristaltic flow of a Hyperbolic tangent nanofluid in a tapered asymmetric channel,”
*IEEE Transactions on NanoBioscience*, 2014. View at: Publisher Site | Google Scholar - M. Kothandapani and J. Prakash, “Effects of thermal radiation parameter and magnetic field on the peristaltic motion of Williamson nanofluids in a tapered asymmetric channel,”
*International Journal of Heat and Mass Transfer*, vol. 81, pp. 234–245, 2015. View at: Publisher Site | Google Scholar

#### Copyright

Copyright © 2015 M. Kothandapani 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.