Hydromagnetic periodic flow in a circular pipe through porous medium with heat transfer in slip flow regime

Khem Chand1*, K. D. Singh2 and Sanjeev Kumar3

1Department of Mathematics & Statistics, H.P. University-Shimla-171 005, India.

2Department of Mathematics (ICDEOL). H. P. University-Shimla-171 005, India.

3Research Scholar, Department of Mathematics and Statistics, H.P. University-Shimla, India.

*Corresponding Author: khemthakur99@gmail.com, kdsinghshimla@gmail.com, sanju75sanju@gmail.com

ABSTRACT:

Heat transfer in hydromagnetic flow of electrically conducting, viscous, incompressible fluid through a circular pipe of uniform width filled with porous material under the influence of periodically varying pressure gradient and wall slip condition is investigated. The governing equation is solved by using the Bessel function. The solution for the velocity and the temperature profiles are obtained and evaluated numerically. The results have been expressed graphically to bring out the effects of various parameters enter in the governing equation.

KEY WORDS: Hydromagnetic, Periodic, Porous medium, Slip flow condition.

1. INTRODUCTION:

The study of flow of conducting fluid through circular pipe with permeable walls not only posses a theoretical appeal but also model many biological and engineering problems such as MHD generator, blood flow problem, plasma studies, chemical engineering, electronic, atomic power and geothermal energy extraction etc. Ostrach (1952, 1954) has studied the laminar natural convection flow between vertical heated plates when the walls are kept at constant temperature and also when the temperature varies linearly along the plates. Following Ostrach, Nanda and Sharma (1963) investigated the free convection flow with and without heat source in circular pipe. Also Agarwal and Upmanyu (1976) and Gupta et al (1979) have discussed laminar free convective flow with and without heat source in a circular pipe using the density variation as proposed by the Goren (1966). The analytical solution of the flow problem of a dusty viscous liquid through a circular pipe in the case of coaxial symmetry, when the pressure gradient varies harmonically with time has been obtained by Kishore and Pandey (1977). Free convective MHD flow between two coaxial circular pipes has been investigated by Krishan et al (1984). Thereafter, the flow of conducting fluid in a circular pipe has been investigated by many scholars (Gadiraju et al Dube et al Ritler et al and Chamkha). Gadiraju et al (1992) have studied the steady flow phase vertical flow in a pipe.

Dube et al (1975) Ritler et al (1997) obtained the solution for unsteady dusty gas flow in a circular pipe in the absence of magnetic field. Chamkha (1994) reported an exact solution which generalized the result obtained by Dube et al (1975) and Ritler (1997).

Meanwhile, Beavers and Joseph (1967) in their experimental work on boundary condition at a normal permeable wall confirmed the existence of slip at the interface separating the flow in the channel and the permeable boundaries. The importance of the slip velocity on the ultra filtration performance has been illustrated by Singh and Lawrence (1997). A survey of MHD studies in the technological fields can be found in Moreau (1990). Recently the combine effect of magnetic field and permeable wall slip velocity on the steady flow of an electrical conducting fluid in a circular vertical piped- uniform width has been analysed by Elangovan and Ratchagar (2010).

Motivated by these studies, we have investigated the combined effect of the magnetic field and the wall slip velocity on the heat transfer of an unsteady flow of a viscous incompressible and electrically conducting fluid in a circular pipe of uniform width filled with porous material under the influence of periodically varying pressure gradient. The governing Navier stokes equation is solved by utilizing the Bessel functions.

4.     RESULTS AND DISCUSSIONS:

Since, the fluid is viscous, incompressible and electrically conducting therefore the above mathematical analysis is suitable for liquid. Figure 1-5 shows variation in the velocity profile with the Hartmann number M, the permeability of the porous medium K, the amplitude of pressure gradient A, the slip parameter h and the frequency of oscillation  respectively. A parabolic axial velocity profile is observed with maximum value along the centre line. The effect of the slip parameter is clearly visible in these figures. However a general decease in the magnitude of the axial velocity profile are noticed with increasing magnetic field intensity and the wall slip parameter where as increasing porous medium permeability K and the amplitude of the pressure gradient A enhance the axial velocity profiles. With the variations in the frequency of oscillation   the velocity profiles first decreases up to  approximately thereafter the trend is reversed. The variation in the skin friction with frequency of oscillation is presented in Figure 6. It is observed that wall skin friction decreases with the increasing magnetic field parameter M and the amplitude of pressure gradient A, where as it increases with increasing velocity slip parameter h and the permeability of the porous medium.

The variation of the temperature profile is shown in the figure 7. It is observed that temperature profile decrease with the increase of the frequency of oscillation and Prandtl number.

The amplitude of heat transfer in term of non-dimensional Nusselt number  is presented in figure 8. It is clear from this figure that it increases with the increase of Prandtl number .

5.     CONCLUDING REMARKS:

1.     A parabolic axial velocity profile is observed with maximum value along the centre line.

2.     The permeability and the amplitude of the pressure gradient enhance the axial velocity whereas the effect of the magnetic field and the wall slip is to reduce its magnitude.

3.     The wall skin friction decreases with the increasing magnetic field and the amplitude of pressure gradient.

4.    The rate of heat transfer increases with the Prandtl number.

6.     REFERENCES:

1.        Agarwal, R. S. and Upmanyu, K.G., (1976).  Laminar free convection flow with and without heat source in circular pipes, Bull. Calcutta Math Soc., 68, pp.  285-292.

2.        Beavers, G. S. and Joseph D. D, (1967). Boundary conditions at a naturally permeable wall, J. Fluid Mech., 30, pp. 197-207.

3.        Chamkha, J, (1994). Unsteady flow of a dusty conducting fluid through a pipe, Mech. Research Comm., 21(3), pp.281-286.

4.        Dube, S. N. and Sharma, C. L., (1975). A note on unsteady flow of a dusty viscous fluid in a circular pipe, J. Phy. Soc. Japan. 13(10), pp.  298-310.

5.        Elangovan K. and Ratchagar, N. P., (2010).  Steady flow through a circular vertical pipe with slip at the permeable boundary with an applied magnetic field, Applied Mathematical Sciences, 4(50), pp. 2445-2452.

6.        Gadiraju, M., Peddieason, J. and Munukutia, S., (1992) Great solution for two phase vertical pipe flow, Mechanics Research Comm. 19(1), pp. 7-13.

7.        Goren, S.L, (1966).On free convection in water at C, Chem. Engng., 21, pp. 515-518.

8.        Gupta, M., Dubey, G.K. and Sharma, H.S., (1979).  Laminar free convection flow with and without heat source through coaxial circular pipes, Indian J. pure appl. Math., 10(7), pp.  792-799.

9.        Kishore, N. and Pandey, R.D., (1977). On the flow of a dusty viscous liquid through a circular pipe, Proc. Indian Acad. Sci. 85 A, (5), pp. 299-302.

10.     Krishan, B., Gupta, G.D. and Sharma, G.C., (1984). Free convective MHD flow between two coaxial circular pipes, Bull. Cal. Math. Soc. 76, pp. 315-320.

11.     Moreau, R., (1990). Magnetohydrodynamics, Kluwer Academic Publisher, Dordrecht.

12.     Nanda, R.S., and Sharma, V.P., (1963). Free convective flow with and without heat source in circular pipes, Appl. scientific Res., A 11(3), pp. 279-291.

13.      Ostrach, S., (1952). Laminar natural convection flow and heat transfer of liquid with and without heat source in channels with constant wall temperature, NACA TN, 2863.

14.     Ostrach, S., (1954). Combine natural and forced convection laminar flow and heat transfer of fluids with and without heat source on channels with linearly varying wall temperature, NACA TN, 3141.

15.     Ritler, J.H. and Peddieson, J., (1997).  Transient two phase flows in channels and circular pipes, Proceeding 6th Canadian congress of applied Mechanics.

16.     Singh, R and Lawrence, R. L., (1979). Influence of a slip velocity at a membrane surface on ultra filtration performances-11(tube flow system), Int. J. Heat and Mass Transfer, 22(5), pp. 731-737.

Received on 25.02.2013                                    Accepted on 14.02.2013

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Research J. Science and Tech 5(1): Jan.-Mar.2013 page 148-152