The unsteady hydromagnetic thermal boundary layer with suction

The unsteady two-dimensional laminar flow of a viscous incompressible and electrically conducting fluid near an oscillating porous plate in the presence of uniform suction, is investigated. The solutions for the velocity, magnetic field, electric current density, temperature and Nusselt number are given in a closed form for the case of the magnetic Prandtl number being equal to unity. The other significant constants are the Eckert number, the fluid Prandtl number and the frequency of oscillation. The influence of these parametres on the solutions is given in both tabulated and graphical forms.     

MHD forced convection boundary layer flow with a flat plate and porous substrate

The flow and heat transfer for an electrically conducting fluid with a porous substrate and a flat plate under the influence of magnetic field is considered. The magnetic field is assumed to be uniform and also along normal to the surface. The momentum and energy equations are transformed to ordinary differential equations by using suitable similarity transformation and are solved by standard techniques. But the energy equation is solved by considering two boundary layers, one in the porous substrate and the other above the porous substrate. Numerical results are presented through graphs with various values of magnetic parameter for both velocity and thermal boundary layers along with Nusselt number and for various values of Prandtl number and Eckert number in thermal boundary layer.     

Flow and heat transfer of an electrically conducting fluid of second grade in a porous medium over a stretching sheet subject to a transverse magnetic field

An analysis is performed for flow and heat transfer of a steady laminar boundary layer flow of an electrically conducting fluid of second grade in a porous medium subject to a transverse uniform magnetic field past a semi-infinite stretching sheet with power-law surface temperature or power-law surface heat flux. The effects of viscous dissipation, internal heat generation of absorption and work done due to deformation are considered in the energy equation. The variations of surface temperature gradient for the prescribed surface temperature case (PST) and surface temperature for the prescribed heat flux case (PHF) with various parameters are tabulated. The asymptotic expansions of the solutions for large Prandtl number are also given for the two heating conditions. It is shown that, when the Eckert number is large enough, the heat flow may transfer from the fluid to the wall rather than from the wall to the fluid when Eckert number is small. A physical explanation is given for this phenomenon.     

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.     

Entropy generation during fluid flow in a channel under the effect of transverse magnetic field

Entropy generation due to fluid flow and heat transfer inside a horizontal channel made of two parallel plates under the effect of transverse magnetic field is numerically investigated. The flow is assumed to be steady, laminar, hydro-dynamically and thermally fully developed of electrically conducting fluid. Both horizontal walls are maintained at constant temperatures higher than that of the fluid. The governing equations in Cartesian coordinate are solved by an implicit finite difference technique. After the flow field and the temperature distributions are obtained, the entropy generation profiles are computed and presented graphically. The factors, which were found to affect the problem under consideration are the magnetic parameter, Eckert number, Prandtl number, and the temperature parameter (θ_∞). It was found that, entropy generation increased as all parameters involved in the present problem increased.     

Magnetic effect on the formation of longitudinal vortices in a rotating laminar boundary layer

This paper presents a numerical study of magnetic effect on the formation of longitudinal vortices in a rotating laminar boundary layer. The criterion for the position marking the onset of longitudinal vortices is defined in this paper. The onset position characterized by the rotational Goertler number G_δ,rot, depends on the local rotation number, Reynolds number, the magnetic field parameter, the Prandtl number and the wave number. The results show that positive rotation destabilizes the flow. The flow is found to become more unstable to the vortex mode of instability as the value of magnetic field parameter M increases. The numerical data shows good agreement with the experimental results. Copyright © 2005 John Wiley & Sons, Ltd.     

Separation of species of a binary fluid mixture confined between two concentric rotating circular cylinders in presence of a strong radial magnetic field

The effect of a radial magnetic field on separation of a binary mixture of incompressible viscous thermally and electrically conducting fluids confined between two concentric rotating circular cylinders with different angular velocity is examined. The equations governing the motion, temperature and concentration in cylindrical polar coordinate are solved analytically. The solution obtained in closed form for concentration distribution is plotted against the radial distances from the surface of the inner circular cylinder for various values of non-dimensional parameters. It is found that the non-dimensional parameters viz. the Hartmann number, thermal diffusion number, baro diffusion number, rotational Reynolds number, the product of Prandtl number and Eckert number, magnetic Prandtl number and the ratio of the angular velocities of inner and outer cylinders affects the species separation of rarer and lighter component significantly. The problem discussed here derives its application in the basic fluid dynamics separation processes to separate the rarer component of the different isotopes of heavier molecules where electromagnetic method of separation does not work.     

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.     

Thermal instability in natural convection flow over a boundary layer subject to external electrical and magnetic fields

This paper presents a numerical study of external electrical and magnetic effects on the formation of longitudinal vortices in natural convection flow over a heated horizontal plate. The criterion on the position marking on the onset of longitudinal vortices is defined in the present paper. The onset position characterized by the Grashof number depends on the Prandtl number, the wave number, the electric field parameter, and the magnetic field parameter. The flow is found more stable as the value of the magnetic field parameter increases. Moreover, the stabilizing effect is also found on the flow when the positive electric field parameter Ec is applied. The results of the present numerical prediction show reasonable agreement with the experimental data with zero magnetic field and electric field parameters in literature.     

Regime diagram and three-dimensional effects of convective interactions in the hydrodynamic Czochralski model

The generalized diagram of the critical Grashof numbers as functions of the Prandtl number is presented. The diagram shows the zones of occurrence of flow field and temperature fluctuations in the axisymmetric and three-dimensional formulations of the crystal growth model using the pulling from a melt. The structure of thermals at high Prandtl numbers and the distinctive features of the three-dimensional convection structure in the zones of stabilization and hazardous mode changeover at different Prandtl numbers are discussed. The effect of crystal rotation on the flow and temperature field patterns is estimated.     

外加磁场对熔融半导体CZ法拉晶过程中流动场影响的数值计算

数值分析结果表明,外加磁场可以改变熔融半导体中的流型,几千高斯的磁场可以显著地减小熔体的流动,但对温度场影响不大。     

外加磁场对熔融半导体CZ法拉晶过程中流动场影响的数值计算

数值分析结果表明,外加磁场可以改变熔融半导体中的流型,几千高斯的磁场可以显著地减小熔体的流动,但对温度场影响不大。     

MHD boundary layer flow and heat transfer over a stretching sheet with induced magnetic field

In this paper, the problem of steady magnetohydrodynamic boundary layer flow and heat transfer of a viscous and electrically conducting fluid over a stretching sheet is studied. The effect of the induced magnetic field is taken into account. The transformed ordinary differential equations are solved numerically using the finite-difference scheme known as the Keller-box method. Numerical results are obtained for various values of the magnetic parameter, the reciprocal magnetic Prandtl number and the Prandtl number. The effects of these parameters on the flow and heat transfer characteristics are determined and discussed in detail. When the magnetic field is absent, the closed analytical results for the skin friction are compared with the exact numerical results. Also the numerical results for the heat flux from the stretching surface are compared with the results reported by other authors when the magnetic field is absent. It is found that very good agreement exists.     

Effect of viscous dissipation and heat source on flow and heat transfer of dusty fluid over unsteady stretching sheet

This paper investigates the problem of hydrodynamic boundary layer flow and heat transfer of a dusty fluid over an unsteady stretching surface.The study considers the effects of frictional heating(viscous dissipation) and internal heat generation or absorption.The basic equations governing the flow and heat transfer are reduced to a set of non-linear ordinary differential equations by applying suitable similarity transformations.The transformed equations are numerically solved by the Runge-Kutta-Fehlberg-45 order method.An analysis is carried out for two different cases of heating processes,namely,variable wall temperature(VWT) and variable heat flux(VHF).The effects of various physical parameters such as the magnetic parameter,the fluid-particle interaction parameter,the unsteady parameter,the Prandtl number,the Eckert number,the number density of dust particles,and the heat source/sink parameter on velocity and temperature profiles are shown in several plots.The effects of the wall temperature gradient function and the wall temperature function are tabulated and discussed.     

MHD natural convection flow and heat transfer in a laterally heated partitioned enclosure

This study looks at MHD natural convection flow and heat transfer in a laterally heated enclosure with an off-centred partition. Governing equations in the form of vorticity–stream function formulation are solved using the polynomial differential quadrature (PDQ) method. Numerical results are obtained for various values of the partition location, Rayleigh, Prandtl and Hartmann numbers. The results indicate that magnetic field significantly suppresses flow, and thus heat transfer, especially for high Rayleigh number values. The results also show that the x-directional magnetic field is more effective in damping convection than the y-directional magnetic field, and the average heat transfer rate decreases with an increase in the distance of the partition from the hot wall. The average heat transfer rate decreases up to 80% if the partition is placed at the midpoint and an x-directional magnetic field is applied. The results also show that flow and heat transfer have little dependence on the Prandtl number.     

Steady mixed convection flow of Maxwell fluid over an exponentially stretching vertical surface with magnetic field and viscous dissipation

The steady mixed convection flow and heat transfer from an exponentially stretching vertical surface in a quiescent Maxwell fluid in the presence of magnetic field, viscous dissipation and Joule heating have been studied. The stretching velocity, surface temperature and magnetic field are assumed to have specific exponential function forms for the existence of the local similarity solution. The coupled nonlinear ordinary differential equations governing the local similarity flow and heat transfer have been solved numerically by Chebyshev finite difference method. The influence of the buoyancy parameter, viscous dissipation, relaxation parameter of Maxwell fluid, magnetic field and Prandtl number on the flow and heat transfer has been considered in detail. The Nusselt number increases significantly with the Prandtl number, but the skin friction coefficient decreases. The Nusselt number slightly decreases with increasing viscous dissipation parameter, but the skin friction coefficient slightly increases. Maxwell fluid reduces both skin friction coefficient and Nusselt number, whereas buoyancy force enhances them.     

Thermal instability of a boundary layer subjected to external electric and magnetic fields

This paper presents a numerical study of the electrical and magnetic fields on thermal instability in a boundary layer. The criterion on the position marking on the onset of longitudinal vortices is defined in the present paper. The results show that the onset position characterized by the Grashof number depends on the Prandtl number, the Reynolds number, the wave number, the electric field parameter, and the Hartmann number. The flow becomes more stable as the magnetic field increases. However, the destabilizing effect is found on the flow when the negative electric field parameter is applied. The results of the present numerical prediction show reasonable agreement with the experimental data in the case of zero Hartmann number and zero electric field parameter in the literature. Copyright © 2010 John Wiley & Sons, Ltd.     

Numerical study of Prandtl effect on MHD flow at a lid‐driven porous cavity

In this paper, the lattice Boltzmann method is used to study the Prandtl number effect on flow structure and heat transfer rates in a magnetohydrodynamic flow mixed convection in a lid‐driven cavity filled with a porous medium. The right and left walls are at constant but different temperatures (θ_h and θ_c), while the other walls are adiabatic. Gallium and salt water (0.02 < Pr < 13.4) are used as samples of the electroconducting fluids in the cavity. Typical sets of streamlines and isotherms are presented to analyze the flow patterns set up by the competition among the forced flow created by the lid‐driven wall, the buoyancy force of the fluid and the magnetic force of the applied magnetic field. Mathematical formulations in the porous media were constructed based on the Brinkman–Forchheimer model, while the multidistribution‐function model was used for the magnetic field effect. Numerical results were obtained and the effects of the Prandtl number and the other effective parameters such as Richardson, Hartman, and Darcy numbers were investigated. It was found that the fluid fluctuations within the cavity were reduced by increasing the Hartman number. A similar pattern was observed for the Darcy number reduction. Heat transfer was essentially dominated by the conduction for the low Prandtl number and forced convection dominated as the Prandtl number increased. Also, the average Nusselt number was raised by increasing the Prandtl number. It was discovered that a remarkable heat transfer enhancement of up to 28% could be reached by increasing the Prandtl number (from 0.02 to 13.4) at constant Richardson and Darcy numbers. Copyright © 2011 John Wiley & Sons, Ltd.     

Heat convection from a cylinder performing steady rotation or rotary oscillation – Part I: Steady rotation

In this paper, the problem of laminar, two dimensional heat convection from a circular cylinder performing steady rotation is investigated. The cylinder is␣placed with its axis horizontal in a quiescent fluid of infinite extent. Because of viscous dissipation, the flow process is confined to the region adjacent to the cylinder and is mainly driven by shear and buoyancy forces. The study is based on the solution of the full conservation equations of mass, momentum and energy for Rayleigh numbers up to 10⁴ and Reynolds numbers (based on surface velocity) up to 400 while Prandtl number ranges between 0.7 and 7.0. For the range of parameters considered, the study revealed that the rate of heat transfer increases with the increase of Rayleigh number and decreases with the increase of speed of rotation. The increase of Prandtl number resulted in an appreciable increase in the average Nusselt number only at low Reynolds numbers. The effect of Prandtl number at high Reynolds number is negligibly small. The resulting flow field in all cases is steady with no vortex shedding. The streamlines and isotherms are plotted for a number of cases to show the details of the velocity and thermal fields. Received on 15 December 1997     

Slip effects on a mixed convection flow of a third-grade fluid near the orthogonal stagnation point on a vertical surface

A mixed convection flow of a third-grade fluid near the orthogonal stagnation point on a vertical surface with slip and viscous dissipation effects is investigated. The governing partial differential equations for the third-grade fluid are converted into a system of nonlinear ordinary differential equations by using a similarity transformation. The effects of various parameters, including the Weissenberg number, third-grade parameter, local Reynolds number, Prandtl number, Eckert number, mixed convection parameter, velocity slip, and thermal slip on the velocity and temperature profiles, local skin friction coefficient, and local Nusselt number are discussed.