Dynamics of heat conducting fluids inside heat conducting domains

The existence of a solution is proved for a model of compressible, heat conducting fluid inside a heat conducting domain. Boundary conditions for the temperature involving a radiation term are assumed. Compared to other approaches, the scheme used for the proof requires lower regularity of the domain boundary in the case of its application to barotropic compressible fluids.     

Asymptotic analysis of heat conducting elastic materials

In this paper, an asymptotic one dimensional model for a clamped curved rod is rigorously derived as a limit of respective three dimensional models. The rod is made of isotropic elastic, heat conductive linearly responding material. Asymptotic analysis is used with respect to the thickness of domains.     

Calculation of moving phase boundaries with latent heat

An integrodifferential equation is derived which describes the motion of a phase transition of nonvanishing latent heat in a homogeneous body from a knowledge of only the Green's function and heat source, if present. As the temperature distribution no longer appears in the final equation for the surface, a reduction of one dimension over and above those afforded by symmetries is achieved. This is practically imperative for computer solutions involving realistic geometries. The method is applicable to problems in such diverse fields as metallurgy, vulcanology, and laser interaction with materials.     

Scattering of cylindrical waves by an elastic inclusion in an electrically conducting compressible medium

We study the problems of diffraction of cylindrical waves at an elastic homogeneous weakly conducting cylinder in a liquid weakly conducting medium and at a magnetohydrodynamic liquid cylinder in a magnetoelastic medium.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 35, 1992, pp. 127–131.     

Asymptotic behaviour in a conducting electromagnetic medium

We investigate the initial-boundary value problem for Maxwell'sequations in linear conducting materials together with dissipativeboundary conditions. We show that it is possible to introducethe free energy and derive from it a domain of dependence. Weprove the existence, uniqueness and asymptotic stability ofthe solution when one of Maxwell's equations is considered asa constraint for the electromagnetic fields.     

A theorem of instability by linear approximation for a one-dimensional non-linearly elastic body

On the basis of and in a development of the ideas and results of A.A. Movchan (Sr.), that extend to continuous bodies the definitions and main fundamental theorems of Lyapunov on stability and instability, a criterion for instability of the equilibrium position of a one-dimensional non-linearly elastic body subject to potential external forces is established. For the specified simplest type of continuous elastic system (which possesses, however, a number of fundamental properties of continuous elastic systems including unboundedness of the operator of linear approximation and discreteness of its spectrum) a theorem of instability by linear approximation is stated and proved. The method of proof is a version of Persidskii's sector method.     

A two-dimensional model of the dynamics of sharp bending of a non-linearly elastic plate

To describe the dynamics of the bending of a thin non-linearly elastic plate, a version of perturbation theory is proposed which correctly takes into account the non-linearity of the medium, the non-uniformity of the deformations along the plate thickness and the boundary conditions on its surface. An effective (2 + 1)-dimensional model is constructed which generalizes the static non-linearly geometrical Föppl-Karman equations. Two-dimensional solitons of the longitudinal deformation are obtained. The conditions for their existence and stability are investigated.     

A non-equilibrium phase transition in a dissipative forest model

The shape of the biostress force for a stressed Lotka–Volterra network is for the first time derived from Lindblad’s dissipative dynamics. Numerical solutions for stressed prey–predator systems with limited resources show a threshold. A non-equilibrium phase transition to a phase with ecosystem dying after a few enforced oscillations (waldsterben phase) occurs.     

The forms of the relation between the stress and strain tensors in a non-linearly elastic material

New relations for the stress and strain tensors, which comprise energy pairs, are obtained for a non-linearly elastic material using a similar method to that employed by Novozhilov, based on a trigonometric representation of the tensors. Shear strain and stress tensors, not used previously, are introduced in a natural way. It is established that the unit tensor, the deviator and the shear tensor form an orthogonal tensor basis. The stress tensor can be expanded in a strain-tensor basis and vice versa. By using this expansion, the non-linear law of elasticity can be written in a compact and physically clear form. It is shown that in the frame of the principal axes the stresses are expressed in terms of the strains and vice versa using linear relations, while the non-linearity is contained in the coefficients, which are functions of mixed invariants of the tensors, introduced by Novozhilov, the generalized moduli of bulk compression and shear and the phase of similitude of the deviators. Relations for different energy pairs of tensors are considered, including for tensors of the true stresses and strains, where the generalized moduli of elasticity have a physical meaning for large strains.     

A non-equilibrium model of a porous medium saturated with immiscible fluids

A model of a thermoelastic porous medium, saturated with two immiscible fluids, is considered. It is assumed that there are no phase transitions, the contribution of pulsations to the stress and kinetic energy is small, and that the components of the medium are in thermodynamic equilibrium. The non-equilibrium of the state, related to the finite time of redistribution of the fluids among the pores of the channels due to the presence of surface forces, is taken into account. A general form of the governing relations, necessary and sufficient to satisfy the principles of thermodynamic compatibility and independence of the choice of system of coordinates, is obtained. It is shown that the establishment of equilibrium is accompanied by dissipation due to capillary forces, which does not lead to seepage dissipation or thermal dissipation. For the case when the deformation of the skeleton and the deviation of the mean porous pressure and the temperature from the initial values are small, while the saturation and the non-equilibrium parameter undergo finite changes, an approximation of the potential of the skeleton is proposed in the form of a quadratic expansion in small parameters. A feature of the expansion is the presence of an initial value of the potential, which depends on the saturation and non-equilibrium. The relationship between the thermodynamic potential and the non-equilibrium kinetics, related to the requirement that the dissipation by the capillary forces should be non-negative, is determined. A generalized Darcy's law is formulated, which takes cross terms into account. It is shown that the proposed approximations enable key effects, which accompany the motion of immiscible fluids in a porous medium, to be described.     

Archie''s law for a mathematical model of movement of a conducting liquid through a conducting porous medium

It is shown that Archie's law (which relates the conductivity of a porous body to the porosity and the consolidation index) for small values of partial density of a conducting liquid in the model of magnetoelasticity for a porous medium is obtained as a necessary condition of the existence of seismomagnetic waves.     

Propagation of axially symmetric elastic vibrations along a waveguide in an elastic medium

The scatter equation is derived for axially symmetric (longitudinal and torsional) normal vibrations of an infinitely long elastic cylindrical waveguide embedded in an elastic medium. The low-frequency vibrations are analyzed thoroughly. It is shown that low-frequency longitudinal vibrations follow either of two branches of the scatter curve. Recommendations are made concerning the use of waveguide methods for studying the physicomechanical properties of polymers.     

The kinetics equations in a multiplying medium with free boundaries

We study a mathematical model of neutron multiplication in a slab ??, by taking into account temperature feedback effects and considering one group of delayed neutrons. The thickness 2a of ?? is time dependent because of temperature variations due to the energy released by fissions. Starting from a quite detailed picture of the physical phenomena occurring in ??, we derive a system of three coupled ordinary differential equations for the total number of neutrons F? = F?(t), for the total number of precursors ? = ?(t), and for the half-thickness of ??, a = a(t). We finally examine some stability properties of such a system of ordinary differential equations.     

On wave propagation in a random conducting magneto-non-simple thermo-viscoelastic medium

The paper examines the problem of wave propagation in a random conducting magneto-non-simple thermo-viscoelastic medium. The medium has been assumed to be weakly conducting and weakly thermal. The thermomechanical coupling parameter and the conductivity are random functions, proportional to ε, with non-zero mean values, ε measuring the smallness of the scale of random fluctuation of inhomogeneities of the medium. The smooth perturbation technique enunciated by Keller (1964) has been employed to analyze the appropriate dispersion equation in non-simple thermoelastic medium. The longitudinal and transverse waves were discussed by using a particular form of thermomechanical coupling parameter representing the corresponding auto-correlation function. The effect of magnetic conductivity has been investigated. The phenomena of attenuation of waves and change of phase speed were discussed numerically in details.     

Self-similar convergence of a shock wave in a heat conducting gas

The problem of the convergence of a spherical shock wave (SW) to the centre, taking into account the thermal conductivity of the gas in front of the SW, is considered within the limits of a proposed approximate model of a heat conducting gas with an infinitely high thermal conductivity and a small temperature gradient, such that the heat flux is finite in a small region in front of the converging SW. In this model, there is a phase transition in the surface of the SW from a perfect gas to another gas with different constant specific heat and the heat outflow. The gas is polytropic and perfect behind the SW. Constraints are derived which are imposed on the self-similarity indices as a function of the adiabatic exponents on the two sides of the SW. In front of the SW, the temperature and density increase without limit. In the general case, a set of self-similar solutions with two self-similarity indices exists but, in the case of strong SW close to the limiting compression, there are two solutions, each of which is completely determined by the motion of the spherical piston causing the self-similar convergence of the SW.     

Scattering of elastic waves through a heterogeneous medium

In this paper we study the elastic wave diffraction in R ³ through a heterogeneous medium, with periodic structure, which occupies a bounded domain. We show that, as the period tends to zero, the solution tends, in some sense, to the solution corresponding to the diffraction by an obstacle made of the classical “homogenized medium”. An analogous result is also proved for the scattering frequencies and the associated scattering functions.