Viscous lifting of a conducting fluid in the presence of a magnetic field

Summary This paper describes the effect of a magnetic field upon the viscous lifting of a conducting fluid for two types of lifting surfaces; conducting and non-conducting. It is shown that the magnetic field produces very small effects on the film thickness and mass flow rate for the case of the dielectric plate. For the conducting plate, the effects are more pronounced and increase with larger values of the ratio of plate conductivity to fluid conductivity. The analysis employed here is simplified to the extent that the effects of surface tension are not included.     

Magnetohydrodynamic Rayleigh problem for magnetic prandtl number close to one

Summary In this paper the solutions for indicial motion of an infinite flat plate are discussed. It is assumed that the fluid is incompressible and has constant viscosity and electric conductivity. It is also assumed that both the solid and fluid are semi-infinite or that the solid is thin with fluids on both sides. The conductivity of the wall is assumed in one case to be much greater than the conductivity of the fluid and in a second case to be much less than that of the fluid. In the limit the first case corresponds to a perfectly conducting wall; the second, to a perfectly insulated wall. The distributions of the velocity, magnetic field current, and vorticity are calculated. In the case where the magnetic diffusivity becomes larger than the viscous diffusivity, we show that there is a spreading of the layer in which the magnetic field changes and also a shrinking of the viscous layer. Both layers are very thick in comparison with the non-magneto-hydrodynamic case. This is due to diffusion of the Alfvén wave carrying the vorticity and the current.     

Combined heat and mass transfer by mixed convection MHD flow along a porous plate with chemical reaction in presence of heat source

An exact and a numerical solutions to the problem of a steady mixed convective MHD flow of an incompressible viscous electrically conducting fluid past an infinite vertical porous plate with combined heat and mass transfer are presented.A uniform magnetic field is assumed to be applied transversely to the direction of the flow with the consideration of the induced magnetic field with viscous and magnetic dissipations of energy.The porous plate is subjected to a constant suction velocity as well as a uniform mixed stream velocity.The governing equations are solved by the perturbation technique and a numerical method.The analytical expressions for the velocity field,the temperature field,the induced magnetic field,the skin-friction,and the rate of heat transfer at the plate are obtained.The numerical results are demonstrated graphically for various values of the parameters involved in the problem.The effects of the Hartmann number,the chemical reaction parameter,the magnetic Prandtl number,and the other parameters involved in the velocity field,the temperature field,the concentration field,and the induced magnetic field from the plate to the fluid are discussed.An increase in the heat source/sink or the Eckert number is found to strongly enhance the fluid velocity values.The induced magnetic field along the x-direction increases with the increase in the Hartmann number,the magnetic Prandtl number,the heat source/sink,and the viscous dissipation.It is found that the flow velocity,the fluid temperature,and the induced magnetic field decrease with the increase in the destructive chemical reaction.Applications of the study arise in the thermal plasma reactor modelling,the electromagnetic induction,the magnetohydrodynamic transport phenomena in chromatographic systems,and the magnetic field control of materials processing.     

Hydromagnetic flow near an accelerated plate in the presence of a magnetic field

Summary Hydromagnetic flow of a viscous incompressible fluid due to uniformly accelerated motion of an infinite flat plate in the presence of a magnetic field fixed relative to the plate, is discussed. It is assumed that the induced magnetic field is negligible compared to the imposed magnetic field. It is observed under these conditions that the velocity at any point and at any instant decreases when the strength of the magnetic field is increased.     

Self-similar solution to the problem of the expansion of a cylindrical column of conducting gas in a longitudinal magnetic field

The nonstationary radial motion of a long cylindrical column of conducting gas in a time-varying longitudinal magnetic field is considered. Exact solutions are found by the method of separating the variables for the system of equations of magnetohydrodynamics on the assumption that the statistical pressure of the plasma at the boundary of the column is proportional to the external magnetic pressure. Some numerical computations are performed and the energetic characteristics of the interaction process are calculated. The ratio of the useful work done by the gas over an infinite time interval to the initial energy of the column is given as a function of the magnetic Reynolds number. We note that a similar method was applied in [1], where not only was the average temperature taken over the cross section, but the inertia of the medium was also neglected. When the inertia is taken into account, we have the additional requirement that the statistical pressure be proportional to the magnetic pressure at the boundary of the column.A physically similar model may be interpreted, for example, as the expansion of a compressible conducting gas column in a nonconducting incompressible fluid situated in a permeable cylinder of some radius R infinite along the axis of symmetry. The requirement that the statistical pressure be proportional to the magnetic pressure reduces to the condition that the external pressure on the boundary of the permeable cylinder of radius R should vary according to a specific law, which may easily be determined.     

Effects of magnetohydrodynamic flow past a vertical plate with variable surface temperature

An analysis is performed to study the magnetohydrodynamic flow of an electrically conducting, viscous incompressible fluid past a semi-infinite vertical plate with variable surface temperature under the action of transversely applied magnetic field. The heat due to viscous dissipation and the induced magnetic field are assumed to be negligible. The dimensionless governing equations are unsteady, two-dimensional, coupled and non-linear governing equations. It is found that the magnetic field parameter has a retarding effect on the velocities of air and water.     

The effect of radiation on transient natural convection past a doubly infinite plate

The present research note is concerned with the transient (short time) simultaneous free convection and radiation analysis of a viscous fluid along a doubly infinite vertical isothermal flat plate. To simplify a very complicated problem, an incompressible flow field is used in the analysis. Generally, the exact numerical solution of this problem is quite lengthy. However, by considering an optically thick radiating gas, expressed by the Rosseland diffusion approximation, the solution is much simpler. Moreover, this case leads to a complete similarity transformation of the governing partial differential equations into a set of ordinary differential equations. An exact numerical solution is obtained of the resulting ordinary differential equations for a Prandtl number equal to 0.733 and for a wide range of involved parameters.     

Nonstationary conducting free flow in a transverse magnetic field

In magnetohydrodynamic flow the viscous friction at the walls can be substantial. The role of viscous friction can be considerably reduced by using a free or a semirestricted flow of the conducting fluid. Nonstationary phenomena in one-dimensional motion of a free plane incompressible fluid flow in a transverse magnetic field are examined. The narrow sides of the flow come into contact with the sectional electrodes connected through external circuits with an active-inductive load. The magnetic Reynolds number and the magnetody-dynamic interaction parameter are assumed to be large. When the electric field due to electromagnetic induction in the channel is much smaller than the field due to the external circuits, the problem can be reduced to the characteristic Cauchy problem for a quasilinear hyperbolic system of first-order equations which can be solved by the method of characteristics using a computer.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 34–39, July–August, 1970.     

Effects of radiation in an optically thin gray gas flowing past a vertical infinite plate in the presence of a magnetic field

This paper studies the two dimensional flow of an electrically conducting fluid which is an optically thin gray gas past a stationary vertical infinite plate in the presence of radiation. It is assumed that the temperature of the plate and the suction at the plate are constant. The presence of the induced magnetic field is also taken into account. Numerical solutions for the velocity and the induced magnetic field are derived and the effects of the radiation parameter are discussed.     

Spontaneous rotation in the exact solution of magnetohydrodynamic equations for flow between two stationary impermeable disks

Magnetohydrodynamic (MHD) flow of a viscous electrically conducting incompressible fluid between two stationary impermeable disks is considered. A homogeneous electric current density vector normal to the surface is specified on the upper disk, and the lower disk is nonconducting. The exact von Karman solution of the complete system of MHD equations is studied in which the axial velocity and the magnetic field depend only on the axial coordinate. The problem contains two dimensionless parameters: the electric current density on the upper plate Y and the Batchelor number (magnetic Prandtl number). It is assumed that there is no external source that produces an axial magnetic field. The problem is solved for a Batchelor number of 0–2. Fluid flow is caused by the electric current. It is shown that for small values of Y, the fluid velocity vector has only axial and radial components. The velocity of motion increases with increasing Y, and at a critical value of Y, there is a bifurcation of the new steady flow regime with fluid rotation, while the flow without rotation becomes unstable. A feature of the obtained new exact solution is the absence of an axial magnetic field necessary for the occurrence of an azimuthal component of the ponderomotive force, as is the case in the MHD dynamo. A new mechanism for the bifurcation of rotation in MHD flow is found.     

Effects of variable thermal conductivity,viscous dissipation on steady MHD natural convection flow of low Prandtl fluid on an inclined porous plate with Ohmic heating

Abstract Aim of the paper is to investigate the effects of linearly varying thermal conductivity, viscous dissipation and Ohmic heating on steady free convection flow of a viscous incompressible electrically conducting liquid having low Prandtl number along an inclined isothermal non-conducting porous plate in the presence of transverse magnetic field. The governing equations of continuity, momentum and energy are transformed into ordinary differential equations using similarity transformation. The resulting coupled and non-linear ordinary differential equations are solved using Runge-Kutta fourth order method and shooting technique. The velocity and temperature distributions are discussed numerically and presented through graphs. Skin-friction coefficient and Nusselt number at the plate are derived, discussed and their numerical values for various values of physical parameters are presented through tables.     

Viscous lifting and drainage of a conducting fluid in the presence of a transverse magnetic field

A theoretical investigation of the effects of a transverse magnetic field on the combined problem of viscous lifting and drainage of a conducting fluid on a plate is presented. The effects of inertia and transverse magnetic field on the liquid film thickness is studied for two cases namely a plate withdrawn with a constant velocity and one withdrawn with a constant acceleration. The expressions for the flow rate and the free surface profiles are obtained for the above two cases. It is found that the free surface profiles are convex in nature as in the non-magnetic case thus showing that the inertia does not effect the general pattern of flow, and the effect of the magnetic field is to retard both the lifting and drainage of the fluid.     

Hall effect on MHD free convection boundary layer flow past a vertical flat plate

Effects of Hall current on MHD free convection boundary layer flow of a viscous incompressible electrically conducting fluid past a heated vertical flat plate of finite dimension in the presence of a uniform transverse magnetic field have been studied. An exact solution of the governing equations describing the flow has been obtained. The velocity field, induced magnetic field and bulk temperature distributions in the boundary layer flow have been discussed. It is found that the velocity components increase with an increase in Hall parameter. It is noticed that the induced magnetic field components are radically influenced by the Hall parameter. It is also found that the magnitude of bulk temperature in the x-direction decreases with an increase in either Hall parameter or magnetic parameter. On the other hand, the magnitude of the bulk temperature in the z-direction increases with an increase in Hall parameter whereas it decreases with an increase in magnetic parameter.     

Exact solutions for the incompressible viscous magnetohydrodynamic fluid of a rotating-disk flow with Hall current

The focus of the present study is to obtain exact solutions for the flow of a viscous hydromagnetic fluid due to the rotation of an infinite disk in the presence of an axial uniform steady magnetic field with the inclusion of Hall current effect. In place of the traditional von Karman's axisymmetric evolution of the flow, the rotational non-axisymmetric stationary conducting flow is taken into consideration here, whose governing equations allow an exact solution to develop bounded everywhere in the normal direction to the wall.The three-dimensional equations of motion are treated analytically yielding derivation of exact solutions, which differ from those of corresponding to the classical von Karman's conducting flow. Making use of this solution, analytical formulas for the angular velocity components, for the current density field as well as for the wall shear stresses are extracted. The critical peripheral locations at which extrema of the local skin friction occur are also determined. It is proved from the analytical results that for the specific flow the properly defined thicknesses decay as the magnetic field strength increases in magnitude, approaching their hydrodynamic value in the limit of large Hall numbers.Interaction of the resolved flow field with the surrounding temperature is further analyzed via the energy equation. The temperature field is shown to accord with the dissipation function. According to the Fourier's heat law, a constant heat transfer from the disk to the fluid occurs, though it increases by the presence of magnetic field, the increase is slowed down by the Hall effect eventually reaching its hydrodynamic limit.     

Exact solutions for the incompressible viscous magnetohydrodynamic fluid of a rotating disk flow

The main interest of the present investigation is to generate exact solutions to the steady Navier-Stokes equations for the incompressible Newtonian viscous electrically conducting fluid flow motion due to a disk rotating with a constant angular speed. For an external uniform magnetic field applied perpendicular to the plane of the disk, the governing equations allow an exact solution to develop taking into account of the rotational non-axisymmetric stationary conducting flow.Making use of the analytic solution, exact formulas for the angular velocity components as well as for the wall shear stresses are extracted. It is proved analytically that for the specific flow the properly defined thicknesses decay as the magnetic field strength increases in magnitude. Interaction of the resolved flow field with the surrounding temperature is further analyzed via the energy equation. The temperature field is shown to accord with the dissipation and the Joule heating. According to Fourier's heat law, a constant heat transfer from the disk to the fluid occurs, though decreases for small magnetic fields because of the dominance of Joule heating, it eventually increases for growing magnetic field parameters.     

Effect of Hall current on MHD mixed convection boundary layer flow over a stretched vertical flat plate

In this paper, the steady magnetohydrodynamic (MHD) mixed convection boundary layer flow of an incompressible, viscous and electrically conducting fluid over a stretching vertical flat plate is theoretically investigated with Hall effects taken into account. The governing equations are solved numerically using an implicit finite-difference scheme known as the Keller-box method. The effects of the magnetic parameter, the Hall parameter and the buoyancy parameter on the velocity profiles, the cross flow velocity profiles and the temperature profiles are presented graphically and discussed. Investigated results indicate that the Hall effect on the temperature is small, and the magnetic field and Hall currents produce opposite effects on the shear stress and the heat transfer at the stretching surface.     

The influence of magnetic field upon the collapse of a cylindrical shock wave

In the present paper, the problem of propagation of collapsing cylindrical shock wave in an ideal gas permeated by a transverse magnetic field with infinite electrical conductivity is investigated. Here it is assumed that the medium ahead of the shock front is uniform and at rest. Also, its counter pressure concerning the motion of the wave front is neglected. This problem admits a self similar solution of second kind. The similarity exponent has been computed by solving a nonlinear eigenvalue problem and integrating numerically the self-similar equations for various values of adiabatic heat exponent and Cowling number. Numerical computations have been performed to determine the flow field behind the shock wave. The influence of magnetic field strength and adiabatic heat exponent on the flow parameters for various cases is presented.     

Numerical study of magnetohydrodynamic laminar flow separation in a channel with smooth expansion

The flow of an electrically conducting incompressible viscous fluid in a plane channel with smooth expansion in the presence of a uniform transverse magnetic field has been analysed. A solution technique for the governing magnetohydrodynamic equations in primitive variable formulation has been developed. A co‐ordinate transformation has been employed to map the infinite irregular domain into a finite regular computational domain. The governing equations are discretized using finite‐difference approximations in staggered grid. Pressure Poisson equation and pressure correction formulae are derived and solved numerically. It is found that with increase in the magnetic field, the size of the flow separation zone diminishes and for sufficiently large magnetic field, the separation zone disappears completely. The peak u‐velocity decreases with increase in the magnetic field. It is also found that the asymmetric flow in a symmetric geometry, which occurs at moderate Reynolds numbers, becomes symmetric with sufficient increase in the transverse magnetic field. Thus, a transverse magnetic field of suitable strength has a stabilizing effect in controlling flow separation, as also in delaying the transition to turbulence. Copyright © 2008 John Wiley & Sons, Ltd.     

Scaling group transformation for the effect of temperature-dependent nanofluid viscosity on an mhd boundary layer past a porous stretching surface

This paper deals with a steady two-dimensional flow of an electrically conducting incompressible fluid over a porous vertical stretching sheet. The flow is permeated by a uniform transverse magnetic field. The fluid viscosity is assumed to vary as a linear function of temperature. The partial differential equations governing the problem under consideration are transformed by a special form of Lie group transformations, namely, scaling group of transformations, into a system of ordinary differential equations, which are solved numerically using the Runge-Kutta-Gill algorithm and the shooting method. The conclusion is drawn that the flow field and temperature profiles are significantly influenced by the Lewis number, Brownian motion number, and thermophoresis number.     

Transient-state flow of a conducting liquid in an MHD generator at constant flow rate in the presence of side walls

It is usual in studies of transient [nonsteady] flow for a viscous incompressible conducting fluid in an MHD channel to take the distance between the side walls as infinite, which allows the initial equations to be simplified, these reducing to a single equation for the velocity if the magnetic Reynolds number is small [1–3]. A real system has a finiteratio of the sides, so it is desirable to establish the effects of the side walls.