Melt Motion Due to Peltier Marking during Bridgman Crystal Growth with an Axial Magnetic Field

This paper treats a liquid-metal flow inside an electrically insulating cylinder with electrically conducting solids above and below the liquid region. There is a uniform axial magnetic field, and there is an electric current through the liquid and both solids. Since the lower liquid-solid interface is concave into the solid and since the liquid is a better electrical conductor than the adjacent solid, the electric current is locally concentrated near the centerline. The return to a uniform current distribution involves a radial electric current which interacts with the axial magnetic field to drive an azimuthal flow. The axial variation of the centrifugal force due to the azimuthal velocity drives a meridional circulation with radial and axial velocities. This problem models the effects of Peltier marking during the vertical Bridgman growth of semiconductor crystals with an externally applied magnetic field, where the meridional circulation due to the Peltier current may produce important mixing in the molten semiconductor. This revised version was published online in July 2006 with corrections to the Cover Date.     

Flow behind magnetogasdynamic exponential shock wave in self-gravitating gas

The propagation of a cylindrical (or spherical) shock wave driven out by a piston moving with time according to an exponential law, in a self-gravitating ideal gas with azimuthal magnetic field is investigated. The initial magnetic field is assumed to be varying according to an exponential law. Solutions are obtained for both the cases of isothermal and adiabatic flows. The effects of variation of ambient magnetic field, gravitational parameter and adiabatic exponent are worked out in detail. It is manifested that the increase in strength of ambient magnetic field has decaying effect on the shock wave however increase in the value of gravitational parameter has reverse effect on the shock strength. The compressibility of the medium is increased in the presence of gravitational field. Also, a comparison between the solutions obtained in the case of isothermal and adiabatic flows is done. Density, pressure, velocity and magnetic field increases, however mass decreases as we move inward from the shock front towards the piston.     

Formation and evolution of vortex rings induced by interactions between shock waves and a low-density bubble

The deformation and instability of a low-density spherical bubble induced by an incident and its reflected shock waves are studied by using the large eddy simulation method. The computational model is firstly validated by experimental results from the literature and is further used to examine the effect of incident shock wave strength on the formations and three-dimensional evolutions of the vortex rings. For the weak shock wave case (Ma?=?1.24), the baroclinic effect induced by the reflected shock wave is the key mechanism for the formation of new vortex rings. The vortex rings not only move due to the self-induced effect and the flow field velocity, but also generate azimuthal instability due to the pressure disturbance. For the strong shock wave case (Ma?=?2.2), a boundary layer is formed adjacent to the end wall owing to the approach of vortex ring, and unsteady separation of the boundary layer near the wall results in the ejection and formation of new vortex rings. These vortex rings interact in the vicinity of the end wall and finally collapse to a complicated vortex structure via azimuthal instability. For both shock wave strength cases, the evolutions of vortex rings due to the instability lead to the formation of the complicated structure dominated by the small-scale streamwise vortices.     

A self-similar solution of exponential shock waves in non-ideal magnetogasdynamics

A self similar method is used to analyze numerically the one-dimensional, unsteady flow of a strong cylindrical shock wave driven by a piston moving with time according to an exponential law in a plasma of constant density. The plasma is assumed to be a non-ideal gas with infinite electrical conductivity permeated by an axial magnetic field. Numerical solutions in the region between the shock and the piston are presented for the cases of adiabatic and isothermal flow. The general behaviour of density, velocity, and pressure profiles remains unaffected due to presence of magnetic field in non-ideal gas. However, there is a decrease in values of density, velocity and pressure in case of magnetogasdynamics as compared to non-magnetic case. It may be noted that the effect of magnetic field on the flow pattern is more significant in case of isothermal flow as compared to adiabatic flow. The effect of non-idealness, specific heat exponent and magnetic field strength on the variation of shock strength across the shock front is also investigated.     

Hydromagnetic capillary instability of a liquid jet with an axial flow

The hydromagnetic capillary instability of a jet of inviscid, impressible fluid of infinite electrical conductivity and subjected to a uniform axial magnetic field is studied, taking into account an axial flow in the jet. The results show that while the axial flow promotes instability due to capillary effects and the axial-flow effects can be completely suppressed by a magnetic field of sufficient strength.     

Stationary cylindrical vortex in a viscous electrically conducting fluid

An exact solution of the magnetohydrodynamic equations is constructed which describes steady vortex flow in a stationary cylinder on the axis of which a conductor carrying a known current is located. The solution is obtained under the assumption that the fluid is viscous and has finite electrical conductivity and that the magnetic field has only the axial and azimuthal components in a cylindrical coordinate system. It is found that the action of the Lorentz force is compensated by changing the pressure. Fluid flow occurs from the periphery to the axis of the cylinder under a pressure gradient, with flow rotation and swirling. The fluid flow causes a concentration of the magnetic lines near the axis of the cylinder, providing an exponential decrease in the magnetic field strength with distance from the axis. This flow can be considered as a model of a local increase in the magnetic field strength due to the transfer of its force lines by the flow of the electrically conducting fluid.     

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.     

Three-dimensional magnetohydrodynamic description of the process of collision of a solar wind shock wave and the Earth’s bow shock

The propagation of a solar wind shock wave along the surface of the Earth’s bow shock is investigated within the framework of an ideal magnetohydrodynamic model in the three-dimensional non-plane-polarized formulation. The most characteristic values of the solar wind parameters and the interplanetary magnetic field strength are considered for the plane front of a solar wind shock wave moving at various velocities along the Sun-Earth radius. The global three-dimensional pattern of the interaction is constructed as a function of the angle of inclination of the surface of the bow shock to the solar wind velocity and the azimuthal angle along the curve of intersection of the fronts of the interacting shock waves. The evolution of the flow developed in the neighborhood of the bow shock is investigated and the parameters of the medium and magnetic field are calculated.     

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.     

The effect of rotation on the nonlinear magnetoelastic response of a circular cylindrical tube

In this paper we use a recently developed concise general theory of nonlinear magnetoelasticity to analyze the mechanical response of (a) a circular cylindrical tube under steady rotation about its axis in an azimuthal magnetic field, and (b) a solid circular cylinder also under steady rotation about its axis in an axial magnetic field. It is found that for problem (a) the magnetic field can either enhance or counteract the effect of rotation, while for problem (b) the magnetic field reinforces the effect of rotation.     

Theoretical and numerical study of Couette-Taylor flow with an axial flow using lattice Boltzmann method

In this study, the numerical models for swirling flows developed by Li et al and Zhou for lattice Boltzmann method (LBM) are chosen. These models were firstly validated using the Couette-Taylor flow between two concentric cylinders simulations. Numerical results showed the efficiency of the Zhou's model. Numerical simulation results using LBM are in good agreement for the steady and unsteady regimes compared to the literature review. In a second step, the Zhou model was then adopted to our study to determine the Couette-Taylor instabilities with an axial flow. Two protocols are tested. The first one (direct protocol) starts with an azimuthal flow without any axial flow (Re = 0). Once the regime is established, an axial flow is then superposed to the Couette-Taylor flow (with a sudden or a progressive manner). The second protocol (inverse protocol) starts with an axial flow at a given Reynolds number (Poiseuille flow). Once the regime is established, an azimuthal flow is the executed (with a sudden or a progressive manner). The effect of various parameters controlling the physical situation is also discussed. The increase of the azimuthal velocity mainly led to the emergence and development of Taylor vortices. Its influence decreases when the axial Reynolds number increases. The relevant result for this study is the change of the critical axial Reynolds number Re_c (total disappearance of instabilities) with both protocols and both manners.     

Magnetohydrodynamic stability of heterogeneous incompressible nondissipative conducting liquids

Summary This discussion which is restricted to the flow of heterogeneous, incompressible, inviscid, perfect conducting liquids between two rotating or nonrotating coaxial cylinders is divided into three parts. In the first part the stability of the liquid in question between two coaxial nonrotating cylinders with an applied magnetic field perpendicular to the flow is investigated. A sufficient condition for stability is found and if the motion is unstable an upper bound for the amplification factor is given. As a particular case the stability of the liquid with uniform steady velocity, and density varying as a function of a distance from the axis of the cylinders is discussed and it is found that the effect of the magnetic field makes the flow more stable.In part two the stability of the liquid in question between two rotating coaxial cylinders with an applied magnetic field in the tangential direction is discussed. A necessary and sufficient condition for stability is derived.In part three the stability of the same liquid between two rotating coaxial with an applied magnetic field in the axial direction is treated. A sufficient condition for stability is derived. In general, we infer that in the case of parallel flow the magnetic field plays the same role in the liquid as gravitational field in Synge's) discussion.     

Electromagnetic twisting of a penny-shaped crack in an elastic conducting cylinder

This paper deals with the electromagnetoelastic problem of an elastic, conducting circular cylinder with a penny-shaped crack under a uniform axial current flow and a constant axial magnetic field. The current flow is disturbed by the presence of the crack and the torsional stresses are caused by the interactions between the magnetic field and the disturbed current. Two problems concerning the electric current density field and the electromagnetoelastic field are formulated by means of integral transform techniques and reduced to two Fredholm integral equations of the second kind. Numerical calculations are carried out and stress intensity factors are obtained for several values of the geometric parameters.     

Influence of a homogeneous axial magnetic field on Taylor–Couette flow of ferrofluids with low particle–particle interaction

Experimental results concerning the stability of Couette flow of ferrofluids under magnetic field influence are presented. The fluid cell of the Taylor–Couette system is subject to a homogeneous axial magnetic field and the axial flow profiles are measured by ultrasound Doppler velocimetry. It has been found that an axial magnetic field stabilizes the Couette flow. This effect decreases with a rotating outer cylinder. Moreover, it could be observed that lower axial wave numbers are more stable at a higher axial magnetic field strength. Since the used ferrofluid shows a negligible particle–particle interaction, the observed effects are considered to be solely based on the hindrance of free particle rotation.     

Self-similar solutions for unsteady flow behind an exponential shock in an axisymmetric rotating dusty gas

Similarity solutions are obtained for one-dimensional unsteady isothermal and adiabatic flows behind a strong exponential cylindrical shock wave propagating in a rotational axisymmetric dusty gas, which has variable azimuthal and axial fluid velocities. The shock wave is driven by a piston moving with time according to an exponential law. Similarity solutions exist only when the surrounding medium is of constant density. The azimuthal and axial components of the fluid velocity in the ambient medium are assumed to obey exponential laws. The dusty gas is assumed to be a mixture of small solid particles and a perfect gas. To obtain some essential features of the shock propagation, small solid particles are considered as a pseudo-fluid; it is assumed that the equilibrium flow conditions are maintained in the flow field, and that the viscous stresses and heat conduction in the mixture are negligible. Solutions are obtained for the cases when the flow between the shock and the piston is either isothermal or adiabatic, by taking into account the components of the vorticity vector. It is found that the assumption of zero temperature gradient results in a profound change in the density distribution as compared to that for the adiabatic case. The effects of the variation of the mass concentration of solid particles in the mixture \(K_p\) , and the ratio of the density of solid particles to the initial density of the gas \(G_a\) are investigated. A comparison between the solutions for the isothermal and adiabatic cases is also made.     

Self-similar solution of cylindrical shock wave propagation in a rotational axisymmetric mixture of a non-ideal gas and small solid particles

Similarity solutions are obtained for one- dimensional isothermal and adiabatic unsteady flow behind a strong cylindrical shock wave propagating in a rotational axisymmetric dusty gas, which has a variable azimuthal fluid velocity together with a variable axial fluid velocity. The shock is assumed to be driven out by a moving piston and the dusty gas to be a mixture of non-ideal (or perfect) gas and small solid particles, in which solid particles are continuously distributed. It is assumed that the equilibrium flow-condition is maintained and variable energy input is continuously supplied by the piston. The shock Mach number is not infinite, but has a finite value. The azimuthal and axial component of the fluid velocity in the ambient medium are assumed to be vary and obey power laws, and the density of the ambient medium is taken to be constant. In order to obtain the similarity solutions the angular velocity of the ambient medium is assumed to be decreasing as the distance from the axis increases. Effects of the variation of the parameter of non-idealness of the gas in the mixture, the mass concentration of solid particles and the ratio of the density of solid particles to the initial density of the gas are investigated.     

Magnetohydrodynamic flow past a drop at low Reynolds and Hartmann numbers

The flow of an electrically conductive liquid past a solid spherical particle at low Reynolds and Hartmann numbers in longitudinal and transverse magnetic fields was first investigated in [1,2]. The effect of a weak magnetic field on the strength of the resistance of a conductive drop in a dielectric medium was considered in [3]. In the present paper we consider the motion of a conductive liquid drop in an electrically conductive medium and calculate the strength of the resistance in the Stokes approximation for an arbitrary orientation of the uniform magnetic field and in the Oseen approximation for the case in which the direction of the magnetic field coincides with the direction of the oncoming stream. As in the previous studies, we do not consider the possibility of the formation of a double layer on the interface between the phases.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 19–25, November–December, 1978.The authors are grateful to G. I. Petrov and the participants in the seminar they conducted for their comments on the work.     

On the structure of MHD shock waves in a viscous non-ideal gas

The structure of one-dimensional magnetohydrodynamics (MHD) shock waves is studied using the Navier–Stokes equations for the non-ideal gas phase. The exact solutions are obtained for the flow variables (i.e. particle velocity, temperature, pressure and change-in-entropy) within the shock transition region. The equation of state for a non-ideal gas is considered as given by Landau and Lifshitz. The effects of the non-idealness parameter and coefficient of viscosity of the gas are analysed on the flow variables assuming the magnetic field having only constant axial component. The findings confirm that the thickness of MHD shock front increases with decreasing values of the non-idealness parameter.     

A linear analysis of instabilities in Couette flow of nematic liquid crystals

This paper presents a new method for determining the critical threshold values at which both homogeneous and roll-type instabilities may appear when thin layers of nematic liquid crystal are subjected to Couette flow. The anisotropic axis of the material is initially uniformly aligned parallel to the axial direction and the effect of a stabilising, axially oriented, uniform magnetic field is allowed for in calculations. Carrying out a linear stability analysis of the continuum equations, we employ a Fourier series method to derive an expression for determining the critical shear rate at which instability occurs. Thresholds for both homogeneous and roll instabilities are obtained and their variation with rotation rate and applied magnetic field strength is examined for the material MBBA.     

MHD simulation of the distribution of the gasdynamic parameters and magnetic field behind the Earth’s bow shock under sharp variations in the solar wind dynamic pressure

The distributions of the gasdynamic parameters (density, pressure, and velocity) and the magnetic field behind the Earth’s bow shock (on the outer boundary of the magnetosheath) generated under sharp variations in the solar wind dynamic pressure are found in the three-dimensional non-planepolarized formulation with allowance for the interplanetary magnetic field within the framework of the ideal magnetohydrodynamic model using the solution to the MHD Riemann problem of breakdown of an arbitrary discontinuity. Such a discontinuity which depends on the inclination of an element of the bow shock surface arises when a contact discontinuity traveling together with the solar wind and on which the solar wind density and, consequently, the dynamic pressure, increases or decreases suddenly impinges on the Earth’s bow shock and propagates along its surface initiating the development of to six waves or discontinuities (shocks). The general interaction pattern is constructed for the entire bow shock surface as a mosaic of exact solutions to the MHD Riemann problem obtained on computer using an original software (MHD Riemann solver) so that the flow pattern is a function of the angular surface coordinates (latitude and longitude). The calculations are carried out for various jumps in density on the contact discontinuity and characteristics parameters of the solar wind and interplanetary magnetic field at the Earth’s orbit. It is found that there exist horseshoe zones on the bow shock in which the increase in the density and the magnetic field strength in the fast shock waves or their reduced decrease in the fast rarefaction waves penetrating into the magnetosheath and arising as a result of sharp variation in the solar wind dynamic pressure is superposed on significant drop in the density and growth in the magnetic field strength in slow rarefaction waves. The distributions of the hydrodynamic parameters and the magnetic field can be used to interpret measurements carried out on spacecraft in the solar wind at the libration point and orbiters in the neighborhood of the Earth’s magnetosphere.