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

**Khem**** Chand ^{1}*,
K. D. Singh^{2} and Sanjeev Kumar^{3}**

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

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

^{3}Research 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

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.

**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 *Chem. Engng*.,
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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
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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), I*nt. J. Heat and Mass Transfer,* 22(5),
pp. 731-737.

Received on 25.02.2013 Accepted
on 14.02.2013

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