Effects of the hydromagnetic free convection on the oscillatory flow of a viscous incompressible fluid past a porous limiting surface

The unsteady two-dimensional free convection flow of a viscous incompressible and electrically conducting fluid past an infinite non-conducting and non-magnetic porous limiting surface (e.g. of a star) through which suction with uniform velocity occurs is considered when the free-stream velocity, the temperature of the limiting surface and the induced magnetic field are oscillating in the time about a constant mean value. Expressions, in closed form for the velocity, the skin-friction, the displacement thickness, the induced magnetic field and the electrical current density are obtained by the help of the two-sided Laplace transform technique, when the magnetic Prandtl numberP _m, and the Prandtl numberP are equal to one, and the magnetic parameterM is smaller to one. During the course of analysis the effects of magnetic parameterM, Grashof numberG and non-dimensional frequency are discussed.     

Hydromagnetic free-convection effects on the oscillatory flow in the Stokes' problem past a vertical limiting surface,I

This paper presents an approximate solution to a two-dimensional free-convection flow of a viscous, incompressible fluid past an infinite vertical, porous, limiting surface under the following conditions: (i) the fluid is electrically conducting; (ii) the limiting surface is electrically non-conducting; (iii) the free-stream velocity oscillates in time about a constant mean; (iv) suction velocity normal to the limiting surface is constant; (v) the limiting surface temperature is constant; (vi) the limiting surface is moved impulsively in its own plane with velocityU ₀; (vii) there exist free-convection currents due to the difference between the limiting surface temperature and the free-stream temperature; (viii) a uniform transverse magnetic field is applied. The mean velocity, mean temperature, mean induced magnetic field and related quantities are shown graphically, followed by a discussion.     

The effects on the flow field of retaining the Joule heating and viscous dissipation term in the energy equation for the hydromagnetic free-convective oscillatory flow past a porous limiting surface,I

With viscous dissipation and Joule heating taking into account the hydromagnetic two-dimensional oscillating free-convection flow, of a viscous, incompressible and electrically conducting fluid, past an infinite vertical porous limiting surface, is studied. For the solution of the problem it is considered that, the free-stream velocity, the plate temperature and the induced magnetic field are oscillating in the time about constant mean values. The flow is subjected to a constant suction velocity, through the porous surface, and a magnetic field of uniform strength is applied transversely to the direction of the flow. Analytical expressions for the flow field are obtained by solving the coupled non-linear system of equations which describe the flow. The influence of the various parameters entering into the problem is also extensively discussed signifying the importance of retaining the Joule heating and viscous dissipation term in the energy equation.     

Hydromagnetic free convection flow in the stokes problem for a porous vertical limiting surface with constant suction

This paper provides a comprehensive analysis of the effects of a uniform transverse magnetic field on the free-convection flow of a viscous incompressible and electrically conductive fluid (e.g., of a stellar atmosphere) past an impulsively started, infinite, porous, vertical limiting surface (e.g., of a star) with a constant suction. The magnetic Reynolds number is assumed small so that the induced magnetic field is considered negligible. Exact solution of the equations governing the flow is obtained in closed form with the help of the Laplace transform technique when the Prandtl numberP=1. Expressions are given for the velocity field, for the temperature field and for their related quantities. The results thus obtained are discussed quantitatively in the last section of this paper.     

Laminar hydromagnetic thermal boundary layer in fluctuating flow past a limiting surface with variable suction

An analysis of the temperature field in the case of the two-dimensional hydromagnetic flow of a viscous incompressible and electrically conducting fluid, (e.g., of a stellar atmosphere), past a porous, infinite, limiting surface in the presence of a transverse magnetic field, is considered when (i) the free stream velocity oscillates in time about a constant mean; (ii) the suction velocity normal to the limiting surface oscillates in magnitude but not in direction about a non-zero mean; and (iii) there is no heat transfer between the fluid and the wall. Approximate solution is obtained of the energy equation and are given expressions for the temperature field and for the temperature at the limiting surface, when the magnetic Prandtl numberP _m =1 and the magnetic parameterM<1. They are shown graphically followed by a discussion.Research supported by the Alexander S. Onassis Foundation.     

Effects of mass transfer on the hydromagnetic free convective flow in the Stokes' problem

The effects of the mass transfer on free convection flow of an electrically conducting viscous fluid (e.g., of a stellar atmosphere) past an impulsively started infinite vertical limiting surface (e.g., of a star) in presence of a transverse magnetic field is considered. Solutions for the velocity and skin-friction, in closed form are obtained with the help of the Laplace transform technique and the results obtained for various values of the parametersS _c (Schmidt number),P (Prandtl number) andM (Hartmann number) are given in graphical form. The paper is concluded with a discussion of the results obtained.     

Free convection effects on the oscillatory flow in the Stokes's problem past an infinite porous vertical limiting surface with constant suction — I

An analysis of Rayleigh's problem (also Stokes's problem) for the flow of a viscous fluid (e.g. of a stellar atmosphere) past an impulsively started infinite, vertical porous limiting surface (e.g. of a star) with constant suction, when the free stream velocity oscillates in time about a constant mean, has been carried out. On solving the coupled non-linear equations in approximate way, expressions for the mean velocity, the mean temperature, the mean skin-friction and the mean rate of heat transfer, expressed in terms of Nusselt number, are obtained. The effects of Grashof numberG, Eckert numberE and Prandtl numberP, on these quantities, is discussed for the cases of an externally heating and cooling of the limiting surface, by the free convection currents, and the variations of them are shown graphically.     

Effects of free convection on the hydromagnetic accelerated flow past a vertical porous limiting surface

The effects of free convection on the accelerated flow of a viscous, incompressible and electrically conducting fluid (e.g. of a stellar atmosphere) past a vertical, infinite, porous limiting surface (e.g. of a star) in the presence of a transverse magnetic field, is considered. The magnetic Reynolds number of the flow is taken to be small enough, so that the induced magnetic field is negligible. Expressions for velocity and skin-friction are obtained by using Laplace transform, when the Prandtl number is equal to one (P=1). Graphs showing variations of velocity and skin-friction, for different values ofG (Grashof number) andM (magnetic parameter) are plotted, and the results of them are discussed.     

Oscillatory hydromagnetic flow through a porous medium in the presence of free convection and mass transfer flow

Unsteady two-dimensional hydromagnetic free convection and mass transfer flow of an electrically-conducting viscous-incompressible fluid, through a highly porous medium bounded by a vertical plane surface of constant temperature is considered. The free-stream velocity of the fluid vibrates about a mean constant value and the surface absorbs the fluid with constant velocity. Expressions for the velocity, temperature, concentration are obtained. Effects of Gr (Grashof number), Gm (modified Grashof number),K (permeability of the porous medium), (frequency parameter), andM (magnetic parameter) upon the velocity field are discussed.     

Unsteady hydromagnetic boundary-layer flow on a semi-infinite wall

Unsteady two-dimensional flow of a viscous, incompressible and electrically conducting fluid which is confined on one side of a semi-infinite wall, in presence of a transverse magnetic field is investigated. The wall is initially at rest and then it is suddenly accelerated in its own plane with a velocity which is an arbitrary function of the time. Solution of the problem is obtained, for two particular cases, i.e., corresponding to two free-stream velocities, with the help of the finite difference approximation technique using the explicit method for uniformly accelerated motion of the wall.     

Magnetohydrodynamics effects on mass transfer flow through porous medium

Magnetohydrodynamic unsteady flow through a porous medium with the presence mass transfer is considered. The porous medium is bounded by a vertical surface and this surface absorbs the fluid with a constant velocity. Also the free-stream velocity vibrates about a mean constant value. The influences of the permeability parameter and magnetic parameter on the velocity field are discussed.     

Hydromagnetic flow near an oscillating porous limiting surface

Unsteady hydromagnetic flow near a harmonically oscillating limiting surface (e.g., of a star) is considered in presence of a transverse magnetic field. Exact solutions, for a periodic boundary layer without a mean steady flow,are obtained when the magnetic Prandtl number is unity and there is a normal velocity of injection imposed at the wall. The results are also presented for the case when the wall is subjected to a normal velocity of suction instead of injection. It is observed that two distinct boundary (or hydromagnetic boundary) layers exist and tend to coalesce into a single layer when the magnetic field parameter approaches zero. The thicknesses of these boundary layers are significantly affected by the injection/suction velocity and the applied magnetic field.     

Hydromagnetic thermal boundary-layer flow near an oscillating limiting surface

Unsteady hydromagnetic thermal boundary layer flow of an electrically conducting, viscous, incompressible fluid near a harmonically oscillating limiting surface (e.g. in stellar atmospheres) is considered when the viscous dissipation and the Joule heating terms are retained in the energy equation. Exact solution of the problem, with the aid of the Laplace transform technique, is obtained when the magnetic Prandtl number is unity. For the purpose of application and discussion of the results, the variations of the temperature and the rate of heat transfer (expressed in the form of the Nusselt number) are presented for different sets of values of the magnetic parameter for the case of two particular fluids — air and water — when the limiting surface is under the condition of either cooling or heating.     

Unsteady hydromagnetic thermal boundary layer flow

Unsteady hydromagnetic thermal boundary layer flow past a non-conducting infinite porous wall in presence of a transverse magnetic field is considered. The magnetic Reynolds number of the flow is taken to be small enough so that the induced magnetic field is negligible. It is assumed that the normal velocity of suction/injection at the wall varies att ^?/12. Solution of the problem, in the form of power series, is obtained for two cases:

1. When the wall temperature is the same as that of the free-stream, and
2. When the difference in the temperatures of the wall and that of the free-stream varies as some power of time. The variations of the skin-friction, the temperature and the rate of heat transfer are shown graphically followed by a quantitative discussion.

     

Free convection and mass transfer effects on the unsteady laminar boundary-layer accelerated flow past a vertical porous limiting surface with uniform suction

An analysis of the mass transfer and free convection effects on the unsteady laminar accelerated flow of a viscous incompressible fluid past an infinite vertical porous limiting surface is presented when the free stream is accelerated and the limiting surface temperature and concentration changes with step-wise variations. Expressions for velocity and skin-friction are obtained by using Laplace transform, when the Prandtl number and the Schmidt number are equal to one. Graphs showing variations of velocity and skin-friction, for different values of Gr (Grashof number) and Gc (modified Grashof number) are plotted, and the results of them are discussed.     

Free convection effects on the oscillatory flow in the Stokes's problem past an infinite porous vertical limiting surface with heat sources

In the present work we study the free convection flow past an impulsively started infinite, vertical porous limiting surface (e.g. of a star) which is subjected to a constant suction velocity, with heat sources when the free stream velocity oscillates in time about a constant mean. A perturbation solution is sought for obtaining the expressions of mean velocity and mean temperature. The results are compared with the corresponding ones without heat sources.     

Unsteady hydromagnetic free-convection flow with radiative heat transfer in a rotating fluid of arbitrary optical thickness,III

This paper investigates transient effects on the flow of a thermally-radiating and electrically-conducting compressible gas in a rotating medium bounded by a vertical flat plate. The transience is provoked by a time-dependent perturbation on a constant plate temperature. The problem particularly focusses on an optically thick gas and a gas of arbitrary optical thickness when the difference between the wall and free-stream temperatures is small. Analytical results are possible only for limiting values of time and these results are discussed quatitatively. Indeed the assumption of small temperature difference is more appropriate for plates which are opaque than transparent.     

Effects of mass transfer on an oscillating free-convection flow in the Stokes's problem past an infinite porous vertical limiting surface with heat sources and variable suction

An analysis of the effects of the mass transfer on the unsteady free-convection flow of a viscous incompressible fluid, past an impulsively started infinite porous vertical limiting surface with heat sources is presented, when the free-stream velocity and the suction velocity, are oscillating in the time about constant mean values. Approximate solutions for the coupled nonlinear equations are derived for the mean velocity, the mean temperature, the mean skin-friction, and the mean rate of heat transfer. All the above quantities are shown graphically followed, by a discussion.     

Magnetohydrodynamic free convection effects on the stokes problem for an incompressible viscous fluid past an infinite vertical limiting surface

An analysis of the transverse magnetic field effects on the free convective flow of an incompressible, electrically conducting viscous fluid past an infinite non-conducting and non-magnetic, vertical limiting surface (e.g., of a star), has been carried out. The limiting surface is assumed to move after receiving an initial impulse. Exact solutions to equations governing the flow are derived with the help of the Laplace transform technique. The velocity, the induced magnetic field, the skin-friction and the electric current density are shown graphically. The effects of the Grashof numberG, the Prandtl numberP, and the magnetic parameterM are described during the course of discussion.     

Free convection and mass transfer effects on the control of the unsteady boundary layer on an infinite vertical porous limiting surface

The present study presents an analytical solution to the flow field of the unsteady laminar accelerated flow of a viscous incompressible fluid past an infinite vertical porous limiting surface, when the freestream is accelerated and the limiting surface temperature and concentration are given functions of time. The expressions for the velocity, temperature and skin friction are obtained by using Laplace transform, when the Prandtl and Schmidt numbers are given. Graphs showing variations of the velocity and the skin friction, for different values ofG _r andG _c (modified Grashof number), as well as of the temperature are plotted and the results are discussed.