Buoyant plume above a line source of heat in an anisotropic porous medium

Buoyancy-induced convection arising from a horizontal line heat source embedded in an anisotropic porous medium is investigated analytically. The porous medium is anisotropic is permeability with its principal axes oriented in a direction that is oblique to the gravity vector. Assuming the boundary layer approximation, closed-form exact similarity solutions for both flow and temperature fields are presented and compared with those of isotropic case. Scale analysis is applied to predict the order of magnitudes involved in the boundary layer regime for which the conditions of validity are obtained. Effects of both anisotropic parameters (K* and %) and Rayleigh number Ra_L are observed to be strongly significant. It is demonstrated that a minimum (maximum) intensity of the thermal convective plume above the line source of heat can be obtained if the porous matrix is oriented with its principal axis with higher permeability parallel (perpendicular) to the vertical direction.     

An eigen theory of static electromagnetic field for anisotropic media

Static electromagnetic fields are studied based on standard spaces of the physical presentation, and the modal equations of static electromagnetic fields for anisotropic media are derived. By introducing a new set of first-order potential functions, several novel theoretical results are obtained. It is found that, for isotropic media, electric or magnetic potentials are scalar; while for anisotropic media, they are vectors. Magnitude and direction of the vector potentials are related to the anisotropic subspaces. Based on these results, we discuss the laws of static electromagnetic fields for anisotropic media.     

Effect of finite conductivity in a medium on shock-wave interaction with a magnetic field

We consider the problem of shock-wave incidence on a magnetic wall, which has been studied in [1]. It is shown that the dynamics of the processes which take place in this case depend significantly on the behavior of the conductivity-temperature dependence (T) of the medium and also on the magnitude of the magnetic-field intensity H₀.An exact solution of the problem is constructed for a special form of the law (T). For an arbitrary law (T) the problem is studied numerically by means of digital computer computations; the results are compared with the exact results.Analysis of these solutions shows that the dissipative properties of the medium (electrical conductivity, viscosity), which determine the structure of the refracted wave front, affect the nature of the entire flow as a whole.The formulated problem also makes it possible to clarify the characteristic features of the decay of a discontinuity in a conducting medium.The authors wish to thank A. A. Samarskii, L. A. Zaklyaz'minskii, L. M. Degtyarev, and A. P. Favorskii for discussions of the study, D. A. Gol'dina and A. A. Ivanov for carrying out the numerical calculations, and also G. A. Lyubimov for several helpful comments.     

Interaction of a shock wave with a magnetic field in a medium with finite conductivity

We consider the process of the interaction of aplanar shock wave with a magnetic field (impact on a magnetic wall) in a medium having finite conductivity.The problem cannot be solved analytically in the general form. Numerical methods are used to study the problem. A computer is used to calculate the complete system of one-dimensional nonsteady equations of MHD with finite conductivity which depends on temperature in a nonlinear fashion. Results are also presented of particular analytic solutions obtained under simplifying assumptions.We discuss the dependence of the process dynamics on the magnitude of the magnetic field intensity and the law of variation of the medium conductivity with temperature.In the numerical calculations we note the formation of a T-layer, a phenomenon which occurs under definite conditions in unsteady MHD problems [1].In conclusion the authors wish to thank N. G. Basov, A. A. Samarskli, and O. N. Krokhin for posing the problem and for fruitful discussions, and also D. A. Gol'din and A. A. Ivanov for carrying out the numerical calculations.     

Propagation of electromagnetic disturbances above a plane surface

Under certain assumptions, it is shown that the propagation problem for an electromagnetic disturbance becomes self-similar, and the self-similarity parameters are determined. A basis is given for the absence of reflection, and it is shown that it is equivalent to the boundary conditions of M. A. Leontovich. Solutions of the propagation problem are obtained for the various components of a pulsed signal field from a dipole of arbitrary orientation, and their properties studied.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 22–30, September–October, 1970.     

Nonlinear electromagnetic phenomena in a liquid with nonequilibrium electrical conductivity in an alternating magnetic field

On the basis of [1] this note examines nonlinear electromagnetic phenomena in a dense plasma brought about by the variation in its electrical conductivity as the electrical field changes. It is well known that the electrical conductivity depends on the electric field strength due to the following causes. The electrons in moving in the electric field receive energy from the field which may be considerable over the free path length. However it is difficult for this energy to be transferred to the heavy particles. In monatomic gases the energy exchange between electrons and heavy particles comes about basically as a result of elastic collisions. Thus a noticeable difference in electron and ion temperature, determined by the electron energy balance taking radiation losses into account, turns out to be possible even for relatively weak electric fields. In molecular gases, on the other hand, the fundamental energy exchange mechanism is the excitation of the rotational and oscillatory degrees of freedom of the molecules. Thus the electron energy in these gases is dissipated relatively easily, and the electron temperature is not observed to be noticeably higher than the atomic temperature. The concept of the characteristic “plasma field” E_p is introduced in [2], which is determined for an Isotropic plasma by the relation

$$E_R = \sqrt {3kTme^{ - 2\delta } (\omega ^2 + v_0 ^2 )} .$$     

The problem of the electric field of an electrode with a pre-electrode potential drop in a medium with tensor conductivity

We investigate the nonlinear current distribution in an electrode of finite dimensions with a pre-electrode layer in which the potential locally depends on the current density. The electrode is in contact with a medium of anisotropic conductivity caused by the Hall effect. The problem is reduced to the solution of a nonlinear integrodifferential equation. It is shown that the structure of the field is determined by the Hall parameter and the form of the volt-ampere characteristic in the pre-electrode layer.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 19–23, January–February, 1971.     

Thermohaline instability in a rotating anisotropic porous medium

The onset of thermohaline convection in an anisotropic rotating porous layer of infinite horizontal extent is investigated. Numerical computations are made assuming horizontal isotropy in permeability. It is observed that (i) for certain parameters a bottom-heavy arrangement destabilises a rotating anisotropic porous layer, (ii) the lower the anisotropy parameter, the higher the range of bottom-heavy solute gradient for which there is destabilisation, (iii) increase in the anisotropy parameter stabilises the system, and (iv) for some values of the parameters rotation destabilises the system, though in general, it has a stabilising effect.     

Equations of nonlinear anisotropic flow through a porous medium

A general law of nonlinear anisotropic flow through a porous medium is proposed. A corresponding equation for the pressure of the fluid is obtained in velocity hodograph variables. The conditions of ellipticity of this equation are expressed in terms of the dissipative function.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 158–160, September–October, 1980.I thank V. M. Entob for discussing the work.     

Inhomogeneous waves at the boundary of a generalized thermoelastic anisotropic medium

The Christoffel equation is derived for the propagation of plane harmonic waves in a generalized thermoelastic anisotropic (GTA) medium. Solving this equation for velocities implies the propagation of four attenuating waves in the medium. The same Christoffel equation is solved into a polynomial equation of degree eight. The roots of this equation define the vertical slownesses of the eight attenuating waves existing at a boundary of the medium. Incidence of inhomogeneous waves is considered at the boundary of the medium. A finite non-dimensional parameter defines the inhomogeneity of incident wave and is used to calculate its (complex) slowness vector. The reflected attenuating waves are identified with the values of vertical slowness. Procedure is explained to calculate the slowness vectors of the waves reflected from the boundary of the medium. The slowness vectors are used, further, to calculate the phase velocities, phase directions, directions and amounts of attenuations of the reflected waves. Numerical examples are considered to analyze the variations of these propagation characteristics with the inhomogeneity and propagation direction of incident wave. Incidence of each of the four types of waves is considered. Numerical example is also considered to study the propagation and attenuation of inhomogeneous waves in the unbounded medium.     

Stress state of a transversely isotropic medium with an anisotropic inclusion. Arbitrary linear force field at infinity

Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 27, No. 7, pp. 25–35, July, 1991.     

Asymptotic analysis of the non-linear behavior of long anisotropic tubes

An asymptotically correct beam model is obtained for a long, thin-walled, circular tube with circumferentially uniform stiffness (CUS) and made of generally anisotropic materials. By virtue of its special geometry certain small parameters cause unusual non-linear phenomena, such as the Brazier effect, to be exhibited. The model is constructed without ad hoc approximations from 3D elasticity by deriving its strain energy functional in terms of generalized 1D strains corresponding to extension, bending, and torsion. Large displacement and rotation are allowed but strain is assumed to be small. Closed-form expressions are provided for the 3D non-linear warping and stress fields, the 1D non-linear stiffness matrix and the bending moment–curvature relationship. In bending, failure could be caused by limit-moment instability, local buckling or material failure of a ply. A procedure to determine the failure load is provided based on the non-linear response, neglecting micro-mechanical failure modes, post-failure behavior, and hygrothermal effects. Asymptotic considerations lead to the neglect of local shell interlaminar and transverse shear stresses for the thin-walled configuration. Results of the theory are illustrated for a few symmetric, antisymmetric angle-ply and unsymmetric layups and show that some previously published theories are not asymptotically correct.     

Convection in a rotating medium above a thermally inhomogeneous horizontal surface

The linear stationary problem of convection in a medium rotating about a vertical axis above a thermally inhomogeneous horizontal surface is theoretically investigated. Attention is mainly focused on the case of a homogeneous medium, but certain stratification effects and especially the convection characteristics in binary mixtures (for example, in saline sea water) are also considered. When the rotation is rapid (large Taylor numbers) the convective cells are strongly elongated in the vertical direction, though they also contain a thin Ekman boundary layer. The importance of the boundary conditions on the horizontal surface (in parallel with the no-slip conditions, more general conditions that may follow from the quadratic turbulent friction model are considered) is shown. In the case of binary mixtures, the differential diffusion and rotation effects may together result in the appearance of “induced salt fingers”, the deep penetration of convection into an arbitrarily stably stratified medium. The convective motions may then have a considerable effect on the background vertical temperature and admixture distributions. Attention is drawn to an original manifestation of the analogy between the rotation and stratification effects: in a non-rotating, stably stratified medium, near a thermally inhomogeneous vertical surface, the convection also penetrates deep into the medium, but in the horizontal direction, so that, when the coordinate system is rotated through 90°, the solution coincides with the case of a rotating non-stratified fluid considered here.     

Eshelby tensor construction for a weakly anisotropic elastic medium

A method of constructing the interior Eshelby tensor for a weakly anisotropic elastic medium is proposed.     

Tensor-linear determinative equations for a nonlinear elastic anisotropic medium

International Applied Mechanics -     

Effective thermal conductivity of a medium with ellipsoidal particles

The principal components of the effective thermal conductivity tensor, characterizing stationary heat macrotransfer in a dense medium with dispersed ellipsoidal particles of a different material are calculated by a method suggested in [1]. The case of equally oriented ellipsoids and of isotropically distributed ones are considered as examples.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 107–111, January–February, 1974.The author is grateful to Yu. A. Buevich for his interest.