Oblique collision of a plane-polarized Alfvén discontinuity and a slow shock wave propagating in opposite directions in magnetohydrodynamics

Collision of plane fronts of a plane-polarized Alfvén discontinuity and a slow shock wave propagating in opposite directions at a certain angle is considered within the framework of an ideal magnetohydrodynamic model. The initial state of an infinitely conducting medium at rest with a frozen-in magnetic field is assumed to be given. Calculations are carried out for various values of the shock wave Mach number and the magnetic field strength using a special software which makes it possible to find an exact solution of the Riemann problem of breakdown of a discontinuity between the states downstream of the interacting waves by means of a computer. The wave flow structure is investigated and a bifurcation map of flow restructuring is constructed. Domains of the initial parameters for which the interaction differs qualitatively are distinguished. The parameters of the medium and magnetic field are found as functions of the angle between the colliding discontinuities and the inclination of the magnetic field. The results obtained may be used in investigations of magnetic reconnection.     

Steady magnetic waves

Steady simple waves are investigated in an incompressible conducting ideal inhomogeneously and isotropically magnetizable fluid moving along the lines of force of a magnetic field. The integration of the system of equations describing such waves is reduced to the calculation of quadrature expressions in the case of an arbitrary magnetization law. It is shown that, depending on the magnetic properties of the medium, different types of steady waves are possible: magnetizing waves in a diamagnetic fluid and demagnetizing waves in a paramagnetic fluid. The results are given of calculations of demagnetizing waves in a conducting ferromagnetic fluid. An analysis is made of the various possible flow regimes of a conducting magnetizable fluid at the point of a perfectly conducting corner.     

Plane waves in a semi-infinite fluid saturated porous medium

The field equations governing the propagation of waves in an incompressible liquid-saturated porous medium are investigated and a general solution is presented. It has been revealed that coupled longitudinal and transverse waves propagate in the porous medium. The propagation of transverse waves in the fluid phase is completely due to the interaction between the solid and fluid phases. The dispersion relationship and attenuation features are discussed. Unlike other investigations, all explicit forms of the arguments are derived. The reflection of the plane harmonic waves at the plane, traction-free boundary, which shows the influence of the dissipation on the velocity, and the attenuation coefficients of the reflected waves is studied. It is of interest that pore pressure is produced in the process of reflection, even in the case of the incidence of transverse waves.     

Some geometric acoustic problems of one-dimensional unsteady and two-dimensional steady flows

The behavior of discontinuities (weak shocks) of the parameters of a disturbed flow and their interaction with the discontinuities of the basic flow in the geometric acoustics approximation, when the variation of the intensity of such shocks along the characteristics or the bicharacteristics is described by ordinary differential equations, has been investigated by many authors. Thus, Keller [1] considered the case when the undisturbed flow is three-dimensional and steady, and the external inputs do not depend on the flow parameters. An analogous study was made by Bazer and Fleischman for the MGD isentropic flow of an ideal conducting medium [2], while Lugovtsov [3] studied the three-dimensional steady flow of a gas of finite conductivity for small magnetic Reynolds numbers and no electric field. Several studies (for example, [4]) have considered the behavior of discontinuities of the solutions from the general positions of the theory of hyperbolic systems of quasilinear equations. Finally, the interaction of weak shocks (or the equivalent continuous disturbances) with shock waves was studied in [5–11].In what follows we consider one-dimensional (with plane, cylindrical, and spherical waves) and quasi-one-dimensional unsteady flows, and also plane and axisymmetric steady flows. Two problems are investigated: the variation of the intensity of weak shocks in the presence of inputs which depend on the stream parameters, and the interaction of weak shocks with strong discontinuities which differ from contact (tangential) discontinuities.The thermodynamic properties of the gas are considered arbitrary. We note that the resulting formulas for the interaction coefficients of the weak and strong discontinuities are also valid for nonequilibrium flow.     

Generation of a magnetic field in the shear deformation region of a conducting material upon high–velocity penetration

It is shown that when a high–velocity impactor penetrates into a conducting target with a transverse magnetic field, conditions for considerable field amplification are produced in the shear deformation region on the lateral surface of the impactor. Field generation in a conducting medium deformed in shear is considered within the framework of a plane one–dimensional problem of magnetohydrodynamics. The results obtained indicate that along the boundary of the cavity produced by the impactor in the target with a magnetic field, a thin layer with a very high field intensity (about 100 T) is formed. The possibility of explosion of this layer due to the magnetic pressure acting in it is analyzed.     

Stationary simple waves in an electrically conducting magnetizable fluid

A study is made of the analog of Prandt1—Meyer flow in an incompressible electrically conducting ideal fluid that is magnetizable in accordance with an arbitrary isotropic law. It is shown that inhomogeneity of the magnetization in a conducting fluid makes possible the existence of stationary simple waves with varying magnetic permeability. For a paramagnetic fluid magnetized to saturation, the equations of these waves are integrated completely in the case of a magnetic field parallel to the velocity. Some regions of such flows of magnetizable fluids are discussed in the present paper for the example of the problem of flow around a perfectly conducting profile.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 136–143, September–October, 1980.I thank I.E. Tarapov for his interest in the work and valuable comments made in a discussion.     

Regular reflection of a magnetohydrodynamic shock wave from a conducting wall

In two-dimensional supersonic gasdynamics, one of the classical steady-state problems, which include shock waves and other discontinuities, is the problem concerning the oblique reflection of a shock wave from a plane wall. It is well known [1–3] that two types of reflection are possible: regular and Mach. The problem concerning the regular reflection of a magnetohydrodynamic shock wave from an infinitely conducting plane wall is considered here within the scope of ideal magnetohydrodynamics [4]. It is supposed that the magnetic field, normal to the wall, is not equal to zero. The solution of the problem is constructed for incident waves of different types (fast and slow). It is found that, depending on the initial data, the solution can have a qualitatively different nature. In contrast from gasdynamics, the incident wave is reflected in the form of two waves, which can be centered rarefaction waves. A similar problem for the special case of the magnetic field parallel to the flow was considered earlier in [5, 6]. The normal component of the magnetic field at the wall was equated to zero, the solution was constructed only for the case of incidence of a fast shock wave, and the flow pattern is similar in form to that of gasdynamics. The solution of the problem concerning the reflection of a shock wave constructed in this paper is necessary for the interpretation of experiments in shock tubes [7–10].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 102–109, May–June, 1977.The author thanks A. A. Barmin, A. G. Kulikovskii, and G. A. Lyubimov for useful discussion of the results obtained.     

Coupled magnetoacoustic waves in a conducting paramagnetic fluid

We consider the propagation of small disturbances in a paramagnetic conducting fluid in a uniform constant magnetic field. Because of coupling of the mechanical and magnetic effects, coupled magnetoacoustic oscillations of a wave nature develop in a certain (resonant) frequency region. The usual MHD waves and uniform magnetization oscillations occur far from resonance. Dissipative processes are accounted for.The equations of motion for a conducting paramagnetic fluid in which interaction of the hydrodynamic velocity with the magnetization and the magnetic field was taken into account phenomenologically were obtained in [1], One of the consequences of this interaction is the propagation of coupled magnetoelastic waves in the fluid; this phenomenon is examined in the present paper.     

Investigation of surfaces of discontinuity in perfectly conducting magnetizing media

Relationships on discontinuities in magnetizing perfectly conducting media in a magnetic field are investigated. The magnetic permeabilities before and after the discontinuity are assumed to be constant, but unequal, quantities. It is shown that shocks of two kinds, fast and slow, are possible in the formulation under consideration in the hydrodynamics of magnetizing media, as in magnetic hydrodynamics: It is shown that the entropy decreases on the rarefaction shocks diminishing the magnetic permeability, but can grow on the rarefaction shocks increasing the magnetic permeability, but such waves are not evolutionary. The relationships on discontinuities in the mechanics of a continuous medium are written down in general form in [1] with the electromagnetic field, polarization, and magnetization effects taken into account. Relationships on discontinuities in the ferrohydrodynamic and elec trohydrodynamic approximations were written down in [2] and [3–5], respectively, for the cases when the magnetic permeability and dielectric permittivity of the medium ahead of and behind the discontinuity are arbitrary functions of their arguments and are identical. A system of relationships on discontinuities propagated into a magnetizing perfectly conducting medium is investigated in this paper. The method proposed in [6] is used in the investigation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 104–110, January–February, 1976.We are grateful to A. A. Barmin for discussing the paper and for valuable remarks.     

On non-linear magnetohydrodynamic problems of an Oldroyd 6-constant fluid

In this work we develop a mathematical model to predict the velocity profile for an unidirectional, incompressible and steady flow of an Oldroyd 6-constant fluid. The fluid is electrically conducting by a transverse magnetic field. The developed governing equation is non-linear. This equation is solved analytically to obtain the general solution. The governing non-linear equation is also solved numerically subject to appropriate boundary conditions (three cases of typical plane shearing flows) by an iterative technique with the finite-difference discretizations. A parametric study of the physical parameters involved in the problems such as the applied magnetic field and the material constants is conducted. The obtained results are illustrated graphically to show salient features of the solutions. Numerical results show that the applied magnetic field tends to reduce the flow velocity. Depending on the choice of the material parameters, the fluid exhibits shear-thickening or shear-thinning behaviours.     

Oscillatory Motion of a Viscous Fluid in Contact with a Flat Layer of a Porous Medium

Analytical solutions are obtained for two problems of transverse internal waves in a viscous fluid contacting with a flat layer of a fixed porous medium. In the first problem, the waves are considered which are caused by the motion of an infinite flat plate located on the fluid surface and performing harmonic oscillations in its plane. In the second problem, the waves are caused by periodic shear stresses applied to the free surface of the fluid. To describe the fluid motion in the porous medium, the unsteady Brinkman equation is used, and the motion of the fluid outside the porous medium is described by the Navier–Stokes equation. Examples of numerical calculations of the fluid velocity and filtration velocity profiles are presented. The existence of fluid layers with counter-directed velocities is revealed.     

Natural convection in unsteady MHD couette flow

 The combined effect of natural convection and uniform transverse magnetic field on the couette flow of an electrically conducting fluid between two parallel plates for impulsive motion of one of the plates in discussed. Under the assumption of negligible induced magnetic field and applied magnetic field being fixed relative to the fluid or plate, the governing equations have been solved exactly, and the expressions for velocity and temperature field have been presented for two different cases. A comparative study is made between the velocity field for magnetic field fixed with respect to plate and magnetic field fixed with respect to fluid. Received on 12 July 1999     

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 laminar and turbulent free jet

The flow of a laminar and turbulent plane free jet of a viscous, incompressible, electrically conducting fluid in the presence of a transverse magnetic field is studied. Using the Prandtlvon Mises transformation, in both the cases, closed form solutions are obtained. A remarkable similarity with the nonmagnetic solutions is noticed. Critical values of the magnetic interaction parameter are determined when the dragging action of the initial momentum is exactly balanced by the retarding effect of the magnetic field resulting in the zero axial velocity field.On leave from V.V. (P.G.) College, Shamli-247776, INDIA.     

Wave propagation in relativistic extended thermodynamics of a charged fluid

An analysis of discontinuity wave propagation across a constant state for a relativistic charged fluid with finite electric conductivity is performed. Beside thecontact wave usual in fluid dynamics, acharge wave arises which propagates with the drift speed of the ions. The remaining eight waves are all coupled together and reduce to magnetoacustic and Alfven waves when the electric conductivity tends to infinity, while they become purely electromagnetic waves when the electric permutivity and the magnetic permeability of the medium are the same as in vacuo.     

Fluctuating flow of an elastico-viscous fluid along an infinite porous plate with an applied magnetic field

The problem of oscillating free stream flow of an elastico-viscous, incompressible, and electrically conducting fluid along an infinite plate with suction varying periodically with time, is considered in the presence of a transverse magnetic field. The effect of the elasticity of the fluid, the magnetic fluid, and the fluctuation of suction velocity on the velocity and the skin friction is examined.     

MHD free-convective flow of micropolar and Newtonian fluids through porous medium in a vertical channel

We have studied the fully-developed free-convective flow of an electrically conducting fluid in a vertical channel occupied by porous medium under the influence of transverse magnetic field. The internal prefecture of the channel is divided into two regions; one region filled with micropolar fluid and the other region with a Newtonian fluid or both the regions filled by Newtonian fluids. Analytical solutions of the governing equations of fluid flow are found to be in excellent agreement with analytical prediction. Analytical results for the details of the velocity, micro-rotation velocity and temperature fields are shown through graphs for various values of physical parameters. It is noticed that Newtonian fluids prop up the linear velocity of the fluid in contrast to micropolar fluid. Also the skin friction coefficient at both the walls is derived and its numerical values are offered through tables.     

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.     

The stokes problems for a conducting fluid with a suspension of particles

The Stokes problems of an incompressible, viscous, conducting fluid with embedded small spherical particles over an infinite plate, set into motion in its plane by impulse and by oscillation, in the presence of a transverse magnetic field, are studied. The velocities of the fluid and of the particles and the wall shear stress are obtained. The stress is found to increase due to the particles and the magnetic field, with the effect of the particles diminishing as the field strength is increased.Nomenclature H ₀ strength of the imposed magnetic field - k density ratio of particles to fluid (per unit volume of flow field) - m _e ² H ₀ ² / - t time - y co-ordinate normal to the plate - u fluid velocity - v particle velocity - _e magnetic permeability of the fluid - kinematic viscosity of the fluid - electric conductivity of the fluid - fluid density - particle relaxation time - frequency of oscillation of the plate     

Two-dimensional magnetic fields in magnetohydrodynamic channels with steel walls at finite magnetic reynolds numbers

Plane problems on the distribution of a two-dimensional magnetic field in magnetohydrodynamic channels with ferromagnetic walls at appreciable magnetic Reynolds numbers and prescribed flow hydrodynamics are studied. An integral representation for the total magnetic induction is constructed with the use of a complex influence function describing the field resulting from a unit current. This makes it possible to obtain arbitrarily close approximations to exact solutions of the problems on a digital computer. Influence functions for various channels can be determined by mirror reflections and conformai mappings. The method is illustrated by numerical calculations of the distribution of the magnetic field for the flow of a conducting fluid along a plane ferromagnetic wall and the flow of a fluid in the space between ferromagnetic walls. Calculations are carried out on the effect of an external circuit and an inhomogeneous transverse velocity profile on the distribution of the magnetic field.Translated from Zhumal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 3–11, September–October, 1971.In conclusion the authors thank G. A. Lyubimov, A. B. Vatazhin, V. V. Gogosov, and A. E. Yakuberiko for useful discussion of the formulation of the problem and of results of the study.