Analytical solution of the MHD free convective unsteady flow over a vertical plate with heat source

I. INTRODUCTION

Consider a two dimensional unsteady flow of a laminar, incompressible, viscous, electrically conducting and heat generation fluid past a semi-infinite vertical moving plate embedded in a uniform porous medium and subjected to a uniform transverse magnetic field in the presence of a pressure gradient has been considered with free convection, thermal diffusion and thermal radiation effects taking in to an account. According to the coordinate system the x* -axis is taken along the porous plate in the upward direction and y-* axis normal to it. The fluid is assumed to be gray, absorbing–emitting but not scattering medium. Now to solve the momentum, energy and concentration equations usual similarity transformations are introduced. We get a set of ordinary differential equation to obtain the solutions of the problem. The ordinary differential equations are solved by using perturbation technique. The expressions for velocity field, temperature distribution, concentration field, skin friction, Nusselt number (Nu) and Sherwood number (Sh) have been obtained. The results are discussed in detailed with the help of graphs to analyze the effect of different flow parameters. .

Several workers have studied the problem of free convection flow with mass transfer. Gupta et al ^[1] have studied heat and mass transfer on a stretching sheet with suction or blowing. Free convection and mass transfer flow through porous medium bounded by an infinite vertical limiting Surface with constant suction have been analyzed by Raptis et al ^[2]. The free convection and mass transfer flow through a porous medium past an infinite vertical porous plate with time dependent temperature and concentration medium have been discussed by Sattar ^[3]. Das et al ^[4] have studied numerical solution of mass transfer effects on unsteady flow past an accelerated vertical porous plate with suction. Viscous dissipation in external natural convection flows have been discussed by Gebhart B. ^[5]. Viscous dissipation effects on unsteady free convection and mass transfer flow past an accelerated vertical porous plate with suction have been discussed by Bala Siddulu Malga, Naikoti Kishan. ^[6].

Chandran et al. ^[7] have discussed the unsteady free convection flow with heat flux and accelerated motion. Soundalgekar et al. ^[8] have analyzed the transient free convection flow of a viscous dissipative fluid past a semi-infinite vertical plate.

Free-convection flow with thermal radiation and mass transfer past a moving vertical porous plate have analyzed by Makinde, O. D ^[9]. Unsteady MHD free convection flow of a compressible fluid past a moving vertical plate in the presence of radioactive heat transfer have been discussed by Mbeledogu, I. U, Amakiri, A.R.C and Ogulu, A, ^[10]. Numerical Study on MHD free convection and mass transfer flow past a vertical flat plate has been discussed by S. F. Ahmmed ^[11]

Recently, Das et al [4] have studied numerical solution of mass transfer effects on unsteady flow past an accelerated vertical porous plate with suction. The present study is extension of work; here we considered the effects of viscous dissipation on unsteady free convection and mass transfer boundary layer flow past an accelerated infinite vertical porous flat plate .In their paper they converted the governing equations which are in partial differential equations to ordinary differential equations by introducing similarity variables and then solved the governing equations by finite difference scheme. In the study we have solved the governing partial differential equations only by using the perturbation technique. The effects of the flow parameters on the velocity, temperature and the concentration distribution of the flow field have been studied with the help of graphs. This type of problem has some significant relevance to geophysical and astrophysical studies.

In our present work, we have studied about analytical study on unsteady free convection and mass transfer flow past an accelerated vertical porous plate. The governing equations for the unsteady case are also studied. Then these governing equations are transformed into dimensionless momentum, energy and concentration equations.. The obtained results of this problem have been discussed for the different values of well-known parameters with different time steps. The Wolfram mathematica students for 7 is used to draw graph of the flow.

I. The governing equation

Introducing a Cartesian co-ordinate system x* is chosen along the plate in the direction of flow and y*- axis normal to it. The fluid is assumed to be gray, absorbing–emitting but not scattering medium. Now to solve the momentum, energy and concentration equations usual similarity transformations are introduced.

Within the frame work of delete such assumptions the equations of continuity, momentum, energy and concentration are follows,

Abbildung in dieser Leseprobe nicht enthalten

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(1)

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(2)

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(3)

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(4)

where Abbildung in dieser Leseprobe nicht enthalten, Abbildung in dieser Leseprobe nicht enthaltenand Abbildung in dieser Leseprobe nicht enthaltenare the dimensional distances along the plate, perpendicular to the plate and dimensional time, respectively. Abbildung in dieser Leseprobe nicht enthaltenand Abbildung in dieser Leseprobe nicht enthaltenare the components of dimensional velocities along x* and y* directions, Abbildung in dieser Leseprobe nicht enthaltenis the fluid density, Abbildung in dieser Leseprobe nicht enthaltenis the velocity, Cp the specific heat at constant pressure, Abbildung in dieser Leseprobe nicht enthaltenis the fluid electrical conductivity, B0 is the magnetic induction, Abbildung in dieser Leseprobe nicht enthaltenis the permeability of the of the porous medium, Abbildung in dieser Leseprobe nicht enthaltenis the dimensional temperature, Abbildung in dieser Leseprobe nicht enthaltenis the coefficient of chemical molecular diffusivity, Abbildung in dieser Leseprobe nicht enthaltenis the coefficient of thermal diffusivity, Abbildung in dieser Leseprobe nicht enthaltenis the dimensional concentration is the thermal conductivity of the fluid, g is the acceleration due to gravity and Abbildung in dieser Leseprobe nicht enthalten and R are the local radioactive heat flux and the reaction rate constant respectively.

The boundary conditions for the velocity, temperature and concentration fields are given as follows

where Abbildung in dieser Leseprobe nicht enthaltenand Abbildung in dieser Leseprobe nicht enthaltenare the wall dimensional temperature and concentration respectively. Abbildung in dieser Leseprobe nicht enthaltenis the free stream dimensional concentration. U0 and Abbildung in dieser Leseprobe nicht enthaltenare constants.

From the equation (1), we consider the velocity as the exponential formAbbildung in dieser Leseprobe nicht enthalten

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(5)

where, A is the real positive constant, Abbildung in dieser Leseprobe nicht enthaltenand Abbildung in dieser Leseprobe nicht enthaltenare small less than unity and v0 is a scale of suction velocity which has non-zero positive constant.

In the free stream, from equation (2) we get

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(6)

Eliminate Abbildung in dieser Leseprobe nicht enthaltenusing equation (2) and equations (6), we obtain

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(7)

Abbildung in dieser Leseprobe nicht enthalten = Abbildung in dieser Leseprobe nicht enthaltenAbbildung in dieser Leseprobe nicht enthalten + Abbildung in dieser Leseprobe nicht enthalten Abbildung in dieser Leseprobe nicht enthalten

(8)

Substituting equation (4.2.10) into equation (4.2.9), we have

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(9)

where, Abbildung in dieser Leseprobe nicht enthaltenis the coefficient of the kinematic viscosity.

The radioactive heat flux term by using the Roseland approximation is given by

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(10)

where, Abbildung in dieser Leseprobe nicht enthalten and Abbildung in dieser Leseprobe nicht enthalten are respectively the Stefan-Boltzmann constant and the mean absorption coefficient. We assume that the temperature difference within the flow are sufficiently small such that Abbildung in dieser Leseprobe nicht enthalten may be expressed as a linear function of the temperature. This is accomplished by expanding in a Taylor series about Abbildung in dieser Leseprobe nicht enthalten and neglecting higher order terms, thus

Abbildung in dieser Leseprobe nicht enthalten

(11)

By using equations (10) and (11), into equation (3) is reduced to

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(12)

Introducing the non-dimensional quantities and parameters

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[...]

^[1] P.S. Gupta and A.S. Gupta, “Heat and mass transfer on a stretching sheet with suction or blowing”. Can J Chem Eng., 55, 744–746, 1977.

^[2] A.Raptis. ”Free convection and mass transfer flow through porous medium bounded by an infinite vertical limiting surface with constant suction.” G.T Zivnidis, N.Kafousis, Letters in heat and mass transfer, 8, 5,417-424, 1981.

^[3] M. A.Satter, “Free convection and mass transfer flow through a porous medium past an infinite vertical porous plate dependent temperature and concentration.” Int.J. Pure Appl.Math. 23, 759-766, 1994.

^[4] S.S. Das, S.K. Sahoo and G.C. Dash, Bull, “Numerical solution of mass transfer effects on unsteady flow past an accelerated vertical porous plate with suction.”Math. Sci. Soc., 2, 29,(1), 33–42, 2006.

^[5] B.Gebhart andMollendorf, “Viscous dissipation in external natural convection flows.” Journal of Fluid Mechanics, 38, 97-107, 1969.

^[6] Bala Siddulu Malga, Naikoti Kishan. ” Viscous dissipation effects on unsteady free convection and mass transfer flow past an accelerated vertical porous plate with suction”. Pelagia research library advances in applied science research, 2011, 2 (6):460-469.

^[7] P.Chandran, N.C.Sacheti and A.K.Singh, “Unsteady free convection flow with heat flux and accelerated motion.” J. Phys. Soc. Japan 67,124-129, 1998.

^[8] V. M. Soundalgekar, B. S. Jaisawal, A. G. Uplekar and H. S. Takhar, “The transient free convection flow of a viscous dissipative fluid past a semi-infinite vertical plate.” J. Appl. Mech. Engng.4, 203-218, 1999.

^[9] O. D Makinde “Free-convection flow with thermal radiation and mass transfer past a moving vertical porous plate”, Int. Comm. Heat Mass Transfer, 32, pp. 1411– 1419, 2005

^[10] Mbeledogu, I. U, Amakiri, A.R.C and Ogulu “Unsteady MHD free convection flow of a compressible fluid past a moving vertical plate in the presence of radioactive heat transfer”, Int. J. of Heat and Mass Transfer, 50, pp. 1668– 1674, 2007.

^[11] S. F. Ahmmed, S. Mondal and A. Ray.” Numerical studies on MHD free convection and mass transfer flow past a vertical flat plate”. IOSR Journal of Engineering (IOSRJEN). ISSN 2250-3021, vol. 3, Issue 5, pp 41-47, May. 2013.