Steady Flow through a Circular Vertical Pipe with Slip at the Permeable Boundaries with an Applied Magnetic Field

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1 Applied Mathematical Sciences, Vol. 4, 2010, no. 50, Steady Flow through a Circular Vertical Pipe with Slip at the Permeable Boundaries with an Applied Magnetic Field K. Elangovan and Nirmala P. Ratchagar Mathematics Section Faculty of Engineering and Technology Annamalai University Annamalainagar , India nirmalapasala@yahoo.co.in Abstract The hydromagnetic steady flow of a viscous incompressible conducting fluid in a circular vertical pipe with slip at the permeable boundaries is investigated. Analytical solutions are constructed for the governing boundary-value problem using differential geometry techniques and the important properties of the overall flow structure are discussed. Mathematics Subject Classification: 76V05 Keywords: Circular vetical pipe, permeable walls, magnetic field, slip coefficient 1 Introduction The study of flow of an electrically conducting fluid through a circular vertical pipe with permeable walls not only possesses a theoretical appeal but also model many biological and engineering problems such as magnetohydrodynamics (MHD) generators, plasma studies, blood flow problems, plasma studies, nuclear reactors, geothermal energy extraction, etc. A survey of MHD studies in the technological fields can be found in Moreau [1]. Furthermore, an extensive theoretical work has been carried out on the hydromagnetic fluid flow in a channel under various situation [2-4]. The flow of a conducting fluid in a

2 2446 K. Elangovan and N. P. Ratchagar circular pipe has been investigated by many authors (Gadiraju et al. (1992), Pube et al (1975), Ritler et al (1977) and Chamkha (1994)). Gadiraju et al (1992) investigated steady flow phase vertical flow in a pipe. Dube et al (1975) and Ritler et al (1977) reported solution for unsteady dusty gas flow in a circular pipe in the absence of magnetic field. Chamkha (1994) obtained extract solutions which generalise the results reported by Dube et al (1975) and Ritler et al (1977) by inclusion of the magnetic and viscous effects. Meanwhile, Beavers and Joseph [9] in their experimental work on boundary conditions at a naturally permeable wall confirmed the existence of slip at the interface separate the flow in the channel and the permeable boundaries. The importance of slip velocity on ultra filtration performance has been illustrated by Singh and Lawrence [10]. Pal et al [11] investigated the effect on slip on longitudinal dispersion of tracer particles in a channel bounded by porous media. The problem of laminar flow in channels of slowly varying width permeable boundaries was investigated in Makinde [12]. The main objective of the present paper is to study the combined effect of magnetic field and permeable walls slip velocity on the steady flow of a electrially conducting fluid in a circular vertical pipe of uniform width. The governing Navier-Stokes equations is reduced to a differential equations and analytical techniques are utilized for its solution. In the following sections, the problem is formulated, analysed and discussed. 2 Mathematical formulation Consider the steady flow of a viscous incompressible electrically conducting fluid through a circular vertical pipe of radius r driven by a constant pressure gradient. A uniform magnetic field is applied perpendicular to the flow direction. Then these assumptions the governing momentum equation can be written as O = p z ρg + 1 [ r z )] (μr u σb 2 r 0u μ k u (1) where p the fluid pressure, ρ the fluid density, σ the fluid conductivity, B 0 the magnetic field acceleration due to gravity g, viscosity μ and slip parameter β are all constant. We assume that the velocity u in the axial direction is a function of r only. To solve the equation (1), we use the following boundary conditions. The symmetric condition u r =0 at r = 0 (2)

3 Steady flow through a circular vertical pipe 2447 The slip condition β u r = u at r = R 0 (3) We make the above equations dimensionless using the following quantities p = P ; u = u ρv0 2 ; R 0 g = g V 2 0 k = k ρr 2 0 V 0 ; η = r R 0 ; ; Re = ρv 0R 0 μ In this model p z z = z R 0 β = β R 0 is constant. 2 u η + 1 u 2 η η (M + 1 )u = A. (4) k where M = σb2 0 R2 0 and A = Re. [ p + g] μ z The boundary conditions are u =0 at η = 0 (5) η β u = u at η = 1 (6) η Solving equation (4) with boundary conditions (5) and (6) using Hankel transform,we obtain The skin friction The mass flow rate u =2A i=1 τ = μ du dy r=r 0 = μρv 2 0 = 2AμρV 2 0 Q = 2πV 0 R 2 0 J 0 (ξ i η) ξ i (ξ 2 i + M + 1 k ).J 1(ξ i ) 1 du dη η=1 i=1 = 4πV 0 R 2 0 A 0 u.η.dη i=1 1 ξ 2 i + M + 1 k (7) (8) 1 ξ 2 i (ξ2 i + M + 1 k ) (9)

4 2448 K. Elangovan and N. P. Ratchagar 3 Results and Discussion Since the fluid is viscous and incompressible, the above mathematical analysis are suitable for liquid. Figures 1,2 and 3 show the fluid velocity profiles. A parabolic axial velocity profile is observed with maximum value at the channel centre line and minimum value at the walls. However a general decrease in the magnitude of the axial velocity profiles are notice with an increase in both wall slip (k) and Reynolds number (Re) and increases the magnetic field intesity (M) decreases the fluid velocity. The wall skin friction with respect to Reynolds number are shown in Figs. 4 and 5. A general decrease in wall friction is observed with an increase in wall slip and a decrease in magnetic field intensity. The volumetric flow rate Q for various values at the parameters M and k are shown in figures 5 and 6. It is clear that increasing M in increases Q for all values of k. u Fig. 1 : Velocity Profile for different M

5 Steady flow through a circular vertical pipe 2449 u Fig. 2 : Velocity Profile for different k u Fig 3 : Velocity Profile for different Re

6 2450 K. Elangovan and N. P. Ratchagar Fig 4. Skin friction for different M (μ =5;k =0.2) Fig 5. Skin friction for different k (μ =5;M =1) Fig 6. Mass flow rate for different M (k =0.2)

7 Steady flow through a circular vertical pipe 2451 References Fig 7. Mass flow rate for different k (M =1) 1. R. Moreau, Magnetohydrodynamics, Kluwer Academic Publishers, Dordrecht, (1990). 2. J. Hartmann, Hg-Dynamics -I, Maths-fys. Medd. 15 no. 6. (1937). 3. A.K. Barkakatic, I Paper. MHD heat transfer int the flow between two coaxial cylinders, Acta Mechanica, 97 (1984). 4. O.D. Makinde, Magneto hydromagnetic stablility of plane-poiseuille flow using Mult-Dect a symptotic technique, Mathematical and Computer Modelling, 37, no. 3-4, 251, (2003). 5. M. Gadiraju, J. Peddieason and S. Munukutia. Great solutions for two phase vertical pipe flow. Mechanics Research Comn. Vol. 19(1). pp (1992). 6. S.N. Dube and C.L. Sharma. A note on unsteady flow of a dusty viscous fluid in a circular pipe. J. Phy. Soc. Japan. Vol. 30(1), pp (1975). 7. J.H. Ritler and Peddieson. J. Transient two phase flows in channels and circular pipes. Proceedings 6th canandian congress of applied mechanics (1997). 8. J. Chamka. Unsteady flow of a dusty conducting fluid through a pipe. Mech. Research Comm. Vol. 21 (3) pp (1994). 9. G.S. Beavers and DD Joseph. Bondary conditions at a naturally permeable wall. J. Fluid Mech., 30(197)(1967).

8 2452 K. Elangovan and N. P. Ratchagar 10. R. Singh and R. L. Lawrence. Influence of slip velocity at a membrane surface on ultrafiltration performances - II (Tube flow system). Int. J. Mass Transfer, 12, 731 (1979). 11. D. Pal, R. Verabhadraih, P.N. Sivakumar, N. Rudraiah. Longitudinal dispersion of trocer particles in a channel bounded by porous media using slip condition. Int. J. Math. Sci. 7(755) (1984). 12. O.D. Makinde, Strongly exotherme explosions in a cylindrical pipe: a case study of series summation technique. Mech. Research Comm. 32(195) (2005). Received: November, 2009