Long-Time Stabilization of Solutions to the Ginzburg–Landau Equations of Superconductivity

 The long-time dynamical properties of solutions (φ,A) to the time-dependent Ginzburg–Landau (TDGL) equations of superconductivity are investigated. The applied magnetic field varies with time, but it is assumed to approach a long-time asymptotic limit. Sufficient conditions (in terms of the time rate of change of the applied magnetic field) are given which guarantee that the dynamical process defined by the TDGL equations is asymptotically autonomous, i.e., it approaches a dynamical system as time goes to infinity. Analyticity of an energy functional is used to show that every solution of the TDGL equations asymptotically approaches a (single) stationary solution of the (time-independent) Ginzburg–Landau equations. The standard “φ = − ∇ · A” gauge is chosen. (Received 30 June 2000; in revised form 30 December 2000)     

Long-Time Stabilization of Solutions to the Ginzburg–Landau Equations of Superconductivity

 The long-time dynamical properties of solutions (φ,A) to the time-dependent Ginzburg–Landau (TDGL) equations of superconductivity are investigated. The applied magnetic field varies with time, but it is assumed to approach a long-time asymptotic limit. Sufficient conditions (in terms of the time rate of change of the applied magnetic field) are given which guarantee that the dynamical process defined by the TDGL equations is asymptotically autonomous, i.e., it approaches a dynamical system as time goes to infinity. Analyticity of an energy functional is used to show that every solution of the TDGL equations asymptotically approaches a (single) stationary solution of the (time-independent) Ginzburg–Landau equations. The standard “φ = − ∇ · A” gauge is chosen.     

Electromagnetic fields induced by electric current in an infinite plate with an elliptical hole

Analytical solutions to the electromagnetic field in a thinconductive plate with an elliptical hole are derived by meansof complex potentials and conformal mapping techniques. Thesteady-state current field in a thin conductive plate is twodimensional (2D) and is explored by a standard complex variabletechnique. The current is disturbed around the elliptical hole,and produces a three dimensional magnetic field. In this case,using the complex variable method to solve the real magneticfield can be challenging. The magnetic boundary conditions takedifferent forms for the soft ferromagnetic and the para- ordiamagnetic materials under consideration. A simplified analysistaking account of the magnitude of the magnetic permeabilityof the magnetic material and air surrounding the material isproposed to reduce the magnetic field in a thin plate to 2Dcalculations. The magnetic field distributions are derived foreach material and the equations of the magnetic components atthe tip of elliptical hole are presented.     

Effects of partial slip,viscous dissipation and Joule heating on Von Kármán flow and heat transfer of an electrically conducting non-Newtonian fluid

The steady Von Kármán flow and heat transfer of an electrically conducting non-Newtonian fluid is extended to the case where the disk surface admits partial slip. The fluid is subjected to an external uniform magnetic field perpendicular to the plane of the disk. The constitutive equation of the non-Newtonian fluid is modeled by that for a Reiner–Rivlin fluid. The momentum equations give rise to highly non-linear boundary value problem. Numerical solutions for the governing non-linear equations are obtained over the entire range of the physical parameters. The effects of slip, magnetic parameter and non-Newtonian fluid characteristics on the velocity and temperature fields are discussed in detail and shown graphically. Emphasis has been laid to study the effects of viscous dissipation and Joule heating on the thermal boundary layer. It is interesting to find that the non-Newtonian cross-viscous parameter has an opposite effect to that of the slip and the magnetic parameter on the velocity and the temperature fields.     

Effect of Hall current on the unsteady MHD flow due to a rotating disk with uniform suction or injection

The unsteady flow of a viscous conducting fluid due to the rotation of an infinite, non-conducting, porous disk in the presence of an axial uniform steady magnetic field is studied without neglecting the Hall effect. The fluid is acted upon by a uniform injection or suction through the disk. The relevant equations are solved numerically with a special technique to resolve the conflict between the initial and boundary conditions. The solution shows that the inclusion of the injection or suction through the surface of the disk in addition to the Hall current gives some interesting results.     

Convergence,stability, and numerical solution of unsteady free convection magnetohydrodynamical flow between two slipping plates

In this study, the unsteady free convection magnetohydrodynamical flow of a viscous, incompressible, and electrically conducting fluid between two horizontally directed slipping plates is considered. The external magnetic filed is applied uniformly in the y-direction and the fluid is assumed to be of low conductivity so that the induced magnetic field is negligible. So the relevant variables, that is, the velocity and the temperature, depend only on one coordinate, the y-axis. The governing equations of velocity and temperature fields are obtained from the continuity, momentum, and energy equations. The boundary conditions for the velocity are taken in the most general form as Robins type which contain slipping parameter. Moreover, the upper plate is heated exponentially and the lower plate is adiabatic. Finite difference method (FDM) is used to simulate the numerical solutions of the problem in which the explicit forward difference in time variable t and central difference in space variable y is used. Hartmann number, Prandtl number, decay factor, and slipping parameter influences on the flow and temperature are shown graphically. It is seen that as the Hartmann number increases, the velocity magnitude drops, which is the well-known flattening tendency of the MHD flow. Also, the increase in decay factor causes an increase in both the velocity and temperature magnitudes at increasing time levels, but it does not change further close to the steady-state. Furthermore, the convergence and stability conditions of the considered scheme are obtained in terms of Hartmann number, Prandtl number, and the slip length.     

On the vanishing viscosity limit in a disk

We say that a solution of the Navier–Stokes equations converges in the vanishing viscosity limit to a solution of the Euler equations if their velocities converge in the energy (L ²) norm uniformly in time as the viscosity ν vanishes. We show that a necessary and sufficient condition for the vanishing viscosity limit to hold in a disk is that the space–time energy density of the solution to the Navier–Stokes equations in a boundary layer of width proportional to ν vanish with ν, and that one need only consider spatial variations whose frequencies in the radial or tangential direction lie in a band centered around 1/ν. The author was supported in part by NSF grant DMS-0705586 during the period of this work.     

Finite element analysis of thermoelastic field in a rotating FGM circular disk

This study focuses on the finite element analysis of thermoelastic field in a thin circular functionally graded material (FGM) disk subjected to a thermal load and an inertia force due to rotation of the disk. Due to symmetry, the FGM disk is assumed to have exponential variation of material properties in radial direction only. As a result of nonuniform coefficient of thermal expansion (CTE) and nonuniform temperature distribution, the disk experiences an incompatible eigenstrain which is taken into account. Based on the two dimensional thermoelastic theories, the axisymmetric problem is formulated in terms of a second order ordinary differential equation which is solved by finite element method. Some numerical results of thermoelastic field are presented and discussed for an Al₂O₃/Al FGM disk. The analysis of the numerical results reveals that the thermoelastic field in an FGM disk is significantly influenced by temperature distribution profile, radial thickness of the disk, angular speed of the disk, and the inner and outer surface temperature difference, and can be controlled by controlling these parameters.     

Polarization of vacuum by an external magnetic field in the (2+1)-dimensional quantum electrodynamics with a nonzero fermion density

The Green's function of the Dirac equation with an external stationary homogeneous magnetic field in the (2+1)-dimensional quantum electrodynamics (QED ₂₊₁) with a nonzero fermion density is constructed. An expression for the polarization operator in an external stationary homogenous magnetic field with a nonzero chemical potential is derived in the one-loopQED ₂₊₁ approximation. The contribution of the induced Chern—Simons term to the polarization operator and the effective Lagrangian for the fermion density corresponding to the occupation of n relativistic Landau levels in an external magnetic field are calculated. An expression of the induced Chern—Simons term in a magnetic field for the case of a finite temperature and a nonzero chemical potential is obtained. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 125, No. 1, pp. 132–151, October, 2000.     

Hydromagnetic forced flow against a porous rotating disk

This paper deals with the steady forced flow of a viscous, incompressible and electrically conducting fluid against a porous rotating disk when a uniform magnetic field acts perpendicular to the disk surface. For small suction the equations of motion are integrated numerically by Kármán-Pohlhausen method, but for large suction a series solution in the inverse powers of the suction parameter is obtained. The effects of disk porosity and magnetic field on the various flow parameters are discussed in detail.     

Effects of slip on steady Bödewadt flow and heat transfer of an electrically conducting non-Newtonian fluid

The steady flow and heat transfer arising due to the rotation of a non-Newtonian fluid at a larger distance from a stationary disk is extended to the case where the disk surface admits partial slip. The constitutive equation of the non-Newtonian fluid is modeled by that for a Reiner–Rivlin fluid. The fluid is subjected to an external uniform magnetic field perpendicular to the plane of the disk. The momentum equation gives rise to a highly nonlinear boundary value problem. Numerical solution of the governing nonlinear equations are obtained over the entire range of the physical parameters. The effects of slip, non-Newtonian fluid characteristics and the magnetic interaction parameter on the momentum boundary layer and thermal boundary layer are discussed in detail and shown graphically. It is observed that slip has prominent effects on the velocity and temperature fields.     

The limits of Riemann solutions to the isentropic magnetogasdynamics

The objective of this note is to prove that the Riemann solutions of the isentropic magnetogasdynamics equations converge to the corresponding Riemann solutions of the transport equations by letting both the pressure and the magnetic field vanish. The delta shock wave can be obtained as the limit of two shock waves and the vacuum state can be obtained as the limit of two rarefaction waves. Moreover the relation between the speed of formation of singular density and those of the vanishing pressure and the vanishing magnetic field is discussed in detail.     

One-dimensional analysis for magneto-thermo-mechanical response in a functionally graded annular variable-thickness rotating disk

In this paper, the magneto-thermo-mechanical response of a functionally graded magneto-elastic material (FGMM) annular variable-thickness rotating disk is investigated. The material properties namely material stiffness, heat conduction coefficient, thermal expansion coefficient, mass density and magnetic permeability are assumed to vary continuously along the radial direction according to a power law. The thickness profile of the disk placed in a uniform magnetic field and subjected to the thermal load is assumed to be hyperbolic in nature. The effects of the magnetic field, grading index and geometric nonlinearity on the mechanical and thermal stresses of the disk are investigated. For a specific value of the grading index the maximum radial stress due to magneto-mechanical load in a mounted FGMM disk with hyperbolic convergent profile is found away from the center. This result is different from other thickness profile disks where the radial stresses are always at the center. It is observed that unlike radial stress in a mounted FGM disk subjected to mechanical load only where it is always tensile, the radial stress due to magneto-thermal load in a mounted FGMM disk can be both tensile and compressive type. It is seen that a decrease in the value of grading index invokes shifting of the location of the maximum temperature in FGMM disk with hyperbolic convergent profile towards the outer surface of the disk.     

Non-blow-up of the 3D ideal magnetohydrodynamics equations for a class of three-dimensional initial data in cylindrical domains

The non blow-up of the 3D ideal incompressible magnetohydrodynamics (MHD) equations is proved for a class of three-dimensional initial data characterized by uniformly large vorticity and magnetic field in bounded cylindrical domains. There are no conditional assumptions on properties of solutions at later times, nor are the global solutions close to some 2D manifold. The approach of proving regularity is based on investigation of fast, singular, oscillating limits and nonlinear averaging methods in the context of almost periodic functions. We establish the global regularity of the 3D limit resonant MHD equations without any restrictions on the size of the 3D initial data. After establishing the strong convergence to the limit resonant equations, we bootstrap this into the regularity on arbitrarily large time intervals for solutions of the 3D MHD equations with weakly-aligned uniformly large vorticity and magnetic field at t = 0. Bibliography: 36 titles. Dedicated to the memory of O. A. Ladyzhenskaya Published in Zapiski Nauchnykh Seminarov POMI, Vol. 318, 2004, pp. 203–219.     

Sur les équations de la magnétothermoélasticité

Summary Writing the MTE equations in non-dimensional variables, the author points out the essential parameters and the structure of the equations. The equation of the MTE system characteristics is determined, as well as the displacement velocities of the waves of weak discontinuity. Furthermore, the equation which is satisfied by the current density vectorJ is determined.The paper considers the Cauchy problem for the infinite perfectly conductive plane, in the presence of a temperature and of a magnetic field. The general solution for the current density and the magnetic field are determined from Maxwell's equation. The solution obtained permits one to determine the displacement and the temperature fields by means of the classical thermoelasticity.The method of solving the Cauchy problem is based upon the representation of the general solution by a continous superposition of plane waves.     

Numerical simulation of flow of a micropolar fluid between a porous disk and a non-porous disk

Two dimensional steady, laminar and incompressible motion of a micropolar fluid between an impermeable disk and a permeable disk is considered to investigate the influence of the Reynolds number and the micropolar structure on the flow characteristics. The main flow stream is superimposed by constant injection velocity at the porous disk. An extension of Von Karman’s similarity transformations is applied to reduce governing partial differential equations (PDEs) to a set of non-linear coupled ordinary differential equations (ODEs) in dimensionless form. An algorithm based on finite difference method is employed to solve these ODEs and Richardson’s extrapolation is used to obtain higher order accuracy. The numerical results reflect the expected physical behavior of the flow phenomenon under consideration. The study indicates that the magnitude of shear stress increases strictly and indefinitely at the impermeable disk while it decreases steadily at the permeable disk, by increasing the injection velocity. Moreover, the micropolar fluids reduce the skin friction as compared to the Newtonian fluids. The magnitude of microrotation increases with increasing the magnitude of R and the micropolar parameters. The present results are in excellent comparison with the available literature results.     

The microwave heating of three-dimensional blocks: semi-analytical solutions

The microwave heating of three-dimensional blocks, by the transversemagnetic waveguide mode TM₁₁, is considered in a long rectangularwaveguide. The governing equations are the forced heat equationand a steady-state version of Maxwell's equations, while theboundary conditions take into account both convective and radiativeheat loss. Semi-analytical solutions, valid for small thermalabsorptivity, are found using the Galerkin method. The electricalconductivity and the thermal absorptivity are assumed to betemperature dependent, while both the electrical permittivityand magnetic permeability are taken to be constant. Both a quadraticrelation and an Arrhenius-type law are used for the temperaturedependency. As the Arrhenius-type law is not amenable analytically,it is approximated by a rational–cubic function. A multivaluedsteady-state temperature versus power relationship is foundto be possible for both types of temperature dependency. Atthe critical power level thermal runaway occurs when the temperaturejumps from the lower (cool) temperature branch to the upper(hot) temperature branch of the solution. The semi-analyticalsolutions are compared with numerical solutions of the governingequations for various special cases such as the limits of smalland large heat loss at the edges of the block. An excellentcomparison is obtained between the semi-analytical and numericalsolutions, on both temperature branches for the Arrhenius-typelaw. For the quadratic temperature dependency the comparisonis excellent on the low branch but the semi-analytical theorysignificantly underpredicts the temperature on the upper solutionbranch.     

Radiation of short waves by a pair of plates backed by a semi-infinite set of ribbed stiffeners

The following acoustic diffraction problem is considered. The upper half-space is filled by an acoustic medium. Two semi-infinite thin plates are situated on the boundary of the upper half-space. One of the plates is backed by a semi-infinite periodical set of ribbed stiffeners. The source of an acoustic field is positioned on the other plate. The problem is reduced to an infinite system of linear algebraic equations. Such a system can be solved in a shortwave approximation. An expression for the acoustic field at a large distance from the junction of the plates is obtained. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1994, pp. 38–46. Translated by A. V. Badanin.     

Layer potentials and boundary value problems for the helmholtz equation in the complement of a thin obstacle

Our purpose is to show in a precise manner the mathematical approach of the problem of the acoustic diffraction by an infinitely thin screen. The classical equations of acoustics are transformed into integral equations. The sound field diffracted by the obstacle is described by a double layer potential, the density of which is equal to the step of the potential across the screen. To avoid the mathematical difficulties, the infinitely thin screen will be considered as the limit of a sequence of obstacles with finite thickness. On such obstacles, the operators can be defined, thanks to the theory of Pseudo-differential operators and Pseudo-Poisson kernels. Then a limiting process is used and gives existence and unicity of the desired solutions in Sobolev spaces.     

A problem of generalized magneto-thermoelastic thin slim strip subjected to a moving heat source

The generalized thermoelastic theory with thermal relaxation, in the context of Lord and Shulman theory, is used to investigate the magneto-thermoelastic problem of a thin slim strip placed in a magnetic field and subjected to a moving plane of heat source. The generalized magneto-thermoelastic coupled governing equations are formulated. By means of the Laplace transform and numerical Laplace inversion, the governing equations are solved. Numerical calculations for the considered variables are performed and the obtained results are presented graphically. The effects of moving heat source speed and applied magnetic field on temperature, stress and displacement are studied. It is found from the graphs that the temperature, thermally induced displacement and stress in the strip are found to decrease at large heat source speed, and the magnetic field significantly influences the variations of non-dimensional displacement and stress. However, it has no effect on the non-dimensional temperature.