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.
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
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
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.
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Received on 25.02.2013 Accepted on 14.02.2013
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