Magnetohydrodynamic flow in an infinite channel

The magnetohydrodynamic (MHD) flow of an incompressible, viscous, electrically conducting fluid in an infinite channel, under an applied magnetic field has been investigated. The MHD flow between two parallel walls is of considerable practical importance because of the utility of induction flowmeters. The walls of the channel are taken perpendicular to the magnetic field and one of them is insulated, the other is partly insulated, partly conducting. An analytical solution has been developed for the velocity field and magnetic field by reducing the problem to the solution of a Fredholm integral equation of the second kind, which has been solved numerically. Solutions have been obtained for Hartmann numbers M up to 200. All the infinite integrals obtained are transformed to finite integrals which contain modified Bessel functions of the second kind. So, the difficulties associated with the computation of infinite integrals with oscillating integrands which arise for large M have been avoided. It is found that, as M increases, boundary layers are formed near the nonconducting boundaries and in the interface region for both velocity and magnetic fields, and a stagnant region in front of the conducting boundary is developed for the velocity field. Selected graphs are given showing these behaviours.     

Magnetohydrodynamic flow on a half-plane

We investigate the magnetohydrodynamic flow (MHD) on the upper, half of a non-conducting plane for the case when the flow is driven by the current produced by an electrode placed in the middle of the plane. The applied magnetic field is perpendicular to the plane, the flow is laminar, uniform, steady and incompressible. An analytical solution has been developed for the velocity field and the induced magnetic field by reducing the problem to the solution of a Fredholm's integral equation of the second kind, which has been solved numerically. Infinite integrals occurring in the kernel of the integral equation and in the velocity and magnetic field were approximated for large Hartmann numbers by using Bessel functions. As the Hartmann number M increases, boundary layers are formed near the non-conducting boundaries and a parabolic boundary layer is developed in the interface region. Some graphs are given to show examples of this behaviour.     

BEM solution to magnetohydrodynamic flow in a semi‐infinite duct

We consider the magnetohydrodynamic flow that is laminar and steady of a viscous, incompressible, and electrically conducting fluid in a semi‐infinite duct under an externally applied magnetic field. The flow is driven by the current produced by a pressure gradient. The applied magnetic field is perpendicular to the semi‐infinite walls that are kept at the same magnetic field value in magnitude but opposite in sign. The wall that connects the two semi‐infinite walls is partly non‐conducting and partly conducting (in the middle). A BEM solution was obtained using a fundamental solution that enables to treat the magnetohydrodynamic equations in coupled form with general wall conductivities. The inhomogeneity in the equations due to the pressure gradient was tackled, obtaining a particular solution, and the BEM was applied with a fundamental solution of coupled homogeneous convection–diffusion type partial differential equations. Constant elements were used for the discretization of the boundaries (y = 0, ?a ? x ? a) and semi‐infinite walls at x = ±a, by keeping them as finite since the boundary integral equations are restricted to these boundaries due to the regularity conditions as y → ∞ . The solution is presented in terms of equivelocity and induced magnetic field contours for several values of Hartmann number (M), conducting length (l), and non‐conducting wall conditions (k). The effect of the parameters on the solution is studied. Flow rates are also calculated for these values of parameters. Copyright © 2011 John Wiley & Sons, Ltd.     

Boundary element method solution of MHD flow in a rectangular duct

The magnetohydrodynamic (MHD) flow of an incompressible, viscous, electrically conducting fluid in a rectangular duct with an external magnetic field applied transverse to the flow has been investigated. The walls parallel to the applied magnetic field are conducting while the other two walls which are perpendicular to the field are insulators. The boundary element method (BEM) with constant elements has been used to cast the problem into the form of an integral equation over the boundary and to obtain a system of algebraic equations for the boundary unknown values only. The solution of this integral equation presents no problem as encountered in the solution of the singular integral equations for interior methods. Computations have been carried out for several values of the Hartmann number (1 ? M ? 10). It is found that as M increases, boundary layers are formed close to the insulated boundaries for both the velocity and the induced magnetic field and in the central part their behaviours are uniform. Selected graphs are given showing the behaviours of the velocity and the induced magnetic field.     

Magnetohydrodynamic flow in electrodynamically coupled rectangular ducts

In Sezgin^1,2 the problems considered are the magnetohydrodynamic (MHD) flows in an electrodynamically conducting infinite channel and in a rectangular duct respectively, in the presence of an applied magnetic field. In the present paper we extend the solution procedure of these papers to two rectangular channels connected by a barrier which is partially conductor and partially insulator. The problem has been reduced to the solution of a pair of dual series equations and then to the solution of a Fredholm's integral equation of the second kind. The infinite series obtained were transformed to finite integrals containing Bessel Junctions of the second kind to avoid the computations of slowly converging infinite series and infinite integrals with oscillating integrands. The results obtained compared well with those of Butsenieks and Shcherbinin3 which were obtained for the perfectly conducting barrier separating the flows.     

A singular perturbation problem of laminar flow in a uniformly porous channel with large injection and with an applied transverse magnetic field

Summary The problem of the steady flow of an electrically conducting viscous fluid through porous walls of a channel in the presence of an applied transverse magnetic field is considered. A solution for the case of small M ²/R (where M = Hartmann number, R = suction Reynolds number) with large blowing at the walls has been given by Terrill and Shrestha [3]. Their solution, on differentiating three times, is found to become infinite at the centre of the channel. Physically this means that there must be a viscous layer at the centre of the channel and Terrill and Shrestha are neglecting the shear layer. In this paper the solution given by Terrill and Shrestha is extended by obtaining an extra term of the series of expansion and the method of inner and outer expansion is used to obtain the complete solution which includes the viscous layer. The resulting series solutions are confirmed by numerical results.     

Flow and heat transfer over an unsteady stretching surface with Hall effect

The present investigation is concerned with the effect of Hall currents on the flow and heat transfer of an electrically conducting fluid over an unsteady stretching surface in presence of a strong magnetic field. The induced magnetic field is neglected while the electron-atom collision frequency is assumed to be relatively high, so that the Hall effect is assumed to exist. The incorrect similarity transformation of Elbashbeshy and Bazid (Heat Mass Transfer 41:1–4, 2004). is corrected and a physically realistic distribution of the velocity and temperature is obtained. Using a similarity transformation the governing time dependent boundary layer equations for momentum and thermal energy are reduced to a set of coupled ordinary differential equations which are then solved numerically by the shooting method. Effects of the magnetic field, M , Hall parameter, m, and the unsteadiness parameter, S, on the velocity and temperature profiles as well as the local skin friction coefficients and the heat transfer rate are shown graphically.     

Magnetohydrodynamic unsteady free convection past a hot vertical plate

The free convection flow of an electrically conducting liquid from an infinite plate has been studied in the presence of a uniform magnetic field. General expressions for the velocity field, induced magnetic field, skin-friction and temperature distribution have been obtained when the plate is a perfect conductor and its temperature varies with the law t ⁿ e ^at . The results have been presented through some graphs and tables with the magnetic Prandtl number unity as its value.     

Hydromagnetic free convection flow with induced magnetic field effects

An exact solution is presented for the hydromagnetic natural convection boundary layer flow past an infinite vertical flat plate under the influence of a transverse magnetic field with magnetic induction effects included. The transformed ordinary differential equations are solved exactly, under physically appropriate boundary conditions. Closed-form expressions are obtained for the non-dimensional velocity (u), non-dimensional induced magnetic field component (B _x ) and wall frictional shearing stress i.e. skin friction function (τ _x ) as functions of dimensionless transverse coordinate (η), Grashof free convection number (G _r ) and the Hartmann number (M). The bulk temperature in the boundary layer (Θ) is also evaluated and shown to be purely a function of M. The Rayleigh flow distribution (R) is derived and found to be a function of both Hartmann number (M) and the buoyant diffusivity parameter (ϑ ^*). The influence of Grashof number on velocity, induced magnetic field and wall shear stress profiles is computed. The response of Rayleigh flow distribution to Grashof numbers ranging from 2 to 200 is also discussed as is the influence of Hartmann number on the bulk temperature. Rayleigh flow is demonstrated to become stable with respect to the width of the boundary layer region and intensifies with greater magnetic field i.e. larger Hartman number M, for constant buoyant diffusivity parameter ϑ ^*. The induced magnetic field (B _x ), is elevated in the vicinity of the plate surface with a rise in free convection (buoyancy) parameter G _r , but is reduced over the central zone of the boundary layer regime. Applications of the study include laminar magneto-aerodynamics, materials processing and MHD propulsion thermo-fluid dynamics.     

Thermal radiation effects on MHD flow past a semi-infinite inclined plate in the presence of mass diffusion

MHD mixed free-forced heat and mass convective steady incompressible laminar boundary layer flow of a gray optically thick electrically conducting viscous fluid past a semi-infinite inclined plate for high temperature and concentration differences is studied. A uniform magnetic field is applied perpendicular to the plate. The density of the fluid is assumed to reduce exponentially with temperature and concentration. The usual Boussinesq approximation is neglected due to the high temperature and concentration differences between the plate and the ambient fluid. The Rosseland approximation is used to describe the radiative heat flux in the energy equation. The boundary layer equations governing the flow are reduced to ordinary differential equations, which are numerically solved by applying an efficient technique. The effects of the density/temperature parameter n, the density/concentration parameter m, the local magnetic parameter M_x and the radiation parameter R are examined on the velocity, temperature and concentration distributions as well as the coefficients of skin-friction, heat flux and mass flux.     

Boundary element solution of unsteady magnetohydrodynamic duct flow with differential quadrature time integration scheme

A numerical scheme which is a combination of the dual reciprocity boundary element method (DRBEM) and the differential quadrature method (DQM), is proposed for the solution of unsteady magnetohydrodynamic (MHD) flow problem in a rectangular duct with insulating walls. The coupled MHD equations in velocity and induced magnetic field are transformed first into the decoupled time‐dependent convection–diffusion‐type equations. These equations are solved by using DRBEM which treats the time and the space derivatives as nonhomogeneity and then by using DQM for the resulting system of initial value problems. The resulting linear system of equations is overdetermined due to the imposition of both boundary and initial conditions. Employing the least square method to this system the solution is obtained directly at any time level without the need of step‐by‐step computation with respect to time. Computations have been carried out for moderate values of Hartmann number (M?50) at transient and the steady‐state levels. As M increases boundary layers are formed for both the velocity and the induced magnetic field and the velocity becomes uniform at the centre of the duct. Also, the higher the value of M is the smaller the value of time for reaching steady‐state solution. Copyright © 2005 John Wiley & Sons, Ltd.     

MHD flow of a power-law fluid over a rotating disk

Magnetohydrodynamic flow of an electrically conducting power-law fluid in the vicinity of a constantly rotating infinite disk in the presence of a uniform magnetic field is considered. The steady, laminar and axi-symmetric flow is driven solely by the rotating disk, and the incompressible fluid obeys the inelastic Ostwald de Waele power-law model. The three-dimensional boundary layer equations transform exactly into a set of ordinary differential equations in a generalized similarity variable. These ODEs are solved numerically for values of the magnetic parameter m up to 4.0. The effect of the magnetic field is to reduce, and eventually suppress, the radially directed outflow. An accompanying reduction of the axial flow towards the disk is observed, together with a thinning of the boundary layer adjacent to the disk, thereby increasing the torque required to maintain rotation of the disk at the prescribed angular velocity. The influence of the magnetic field is more pronounced for shear-thinning than for shear-thickening fluids.     

Buoyant MHD flows in a vertical channel: the levitation regime

Buoyant magnetohydrodynamic (MHD) flows with Joulean and viscous heating effects are considered in a vertical parallel plate channel. The applied magnetic field is uniform and perpendicular to the plates which are subject to adiabatic and isothermal boundary conditions, respectively. The main issue of the paper is the levitation regime, i.e., the fully developed flow regime for large values of the Hartmann number M, when the hydrodynamic pressure gradient evaluated at the temperature of the adiabatic wall is vanishing. The problem is solved analytically by Taylor series method and the solution is validated numerically. It is found that the fluid velocity points everywhere and for all values of M downward. For small M’s, the velocity field extends nearly symmetrically (with respect to the mid-plane) over the whole section of the channel between the adiabatic and the isothermal walls. For large values of M, by contrast, the fluid levitates over a broad transversal range of the channel, while the motion becomes concentrated in a narrow boundary layer in the neighborhood of the isothermal wall. Accordingly, the fluid temperature is nearly uniform in the levitation range and decreases rapidly within the boundary layer in front of the isothermal wall. It also turns out that not only the volumetric heat generation by the Joule effect, but also that by viscous friction increases rapidly with increasing values of M, the latter effect being even larger than the former one for all M.     

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.     

Hydromagnetic flow in a rotating channel

The steady flow in a parallel plate channel rotating with an angular velocity Ω and subjected to a constant transverse magnetic field is analysed. An exact solution of the governing equations is obtained. The solution in the dimensionless form contains two parameters: the Hartmann number, M ², and K ² which is the reciprocal of the Ekman number. The effects of these parameters on the velocity and magnetic field distributions are studied. For large values of the parameters, there arise thin boundary layers on the walls of the channel.     

Pulsatile magneto-biofluid flow and mass transfer in a non-Darcian porous medium channel

A numerical study of pulsatile flow and mass transfer of an electrically conducting Newtonian biofluid via a channel containing porous medium is considered. The conservation equations are transformed and solved under boundary conditions prescribed at both walls of the channel, using a finite element method with two-noded line elements. The influence of magnetic field on the flow is studied using the dimensionless hydromagnetic number, Nm, which defines the ratio of magnetic (Lorentz) retarding force to the viscous hydrodynamic force. A Darcian linear impedance for low Reynolds numbers is incorporated in the transformed momentum equation and a second order drag force term for inertial (Forchheimer) effects. Velocity and concentration profiles across the channel width are plotted for various values of the Reynolds number (Re), Darcy parameter (λ), Forchheimer parameter (Nf), hydro-magnetic number (Nm), Schmidt number (Sc) and also with dimensionless time (T). Profiles of velocity varying in space and time are also provided. The conduit considered is rigid with a pulsatile pressure applied via an appropriate pressure gradient term. Increasing the hydromagnetic number (Nm) from 1 to 15 considerably depresses biofluid velocity (U) indicating that a magnetic field can be used as a flow control mechanism in, for example, medical applications. A rise in Nf from 1 to 20 strongly retards the flow development and decreases the velocity, U, across the width of the channel. The effects of other parameters on the flowfield are also discussed at length. The flow model also has applications in the analysis of electrically conducting haemotological fluids flowing through filtration media, diffusion of drug species in pharmaceutical hydromechanics, and also in general fluid dynamics of pulsatile systems.     

The stability of the modified plane Poiseuille flow in the presence of a transverse magnetic field

The stability against small disturbances of the pressure-driven plane laminar motion of an electrically conducting fluid under a transverse magnetic field is investigated. Assuming that the outer regions adjacent to the fluid layer are electrically non-conducting and not ferromagnetic, the appropriate boundary conditions on the magnetic field perturbations are presented. The Chebyshev collocation method is adopted to obtain the eigenvalue equation, which is then solved numerically. The critical Reynolds number R_c, the critical wave number α_c, and the critical wave speed c_c are obtained for wide ranges of the magnetic Prandtl number P_m and the Hartmann number M. It is found that except for the case when P_m is sufficiently small, the magnetic field has both stabilizing and destabilizing effects on the fluid flow, and that for a fixed value of M the fluid flow becomes more unstable as P_m increases.     

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.     

Stochastic methods in dispersion theory

In this study we show how methods from the theory of stochastic processes can be applied to problems in dispersion theory.First, we show that Taylor dispersion with adsorbing boundaries is easily transformed into a new Taylor dispersion problem without adsorbing boundaries. The transformed problem can then be solved using any of the traditional methods used for Taylor dispersion.Secondly, we consider the dispersion of particles in a channel (between parallel plates) with one partially adsorbing surface and one perfectly reflecting boundary. We determine the exact law of the position of adsorption for an arbitrary channel flow in terms of an infinite series of iterated integrals of the flow field, which is assumed to be a function of the cross-channel coordinate only. We also consider the case of shear flow over an adsorbing plane, by taking the limit where one of the boundaries is taken to infinity     

Invariants lagrangiens et integrales premieres en magnetodynamique des fluides

This paper deals with the study of the flow of an electrically conducting but perfect, compressible fluid. Starting from the general representation of the velocity and magnetic fields, one obtains the system of lagrangian invariants corresponding to the magnetohydrodynamic equations; this system describes the evolution of certain caracteristic scalar fields along the fluid trajectories of the magnetohydrodynamic flow. Lastly, the system of first integrals corresponding to the case of stationary magnetohydrodynamic flow, with aligned magnetic field is derived.