An effective model of a porous-fluid medium

For a medium containing alternating porous Biot layers and fluid layers, an effective model is established by the method of matrix averaging. An investigation of equations of this effective model shows that the wave field consists of a leading front and two triangular fronts. The velocities of these fronts along the axes are determined. If the thicknesses of the fluid layers are very small, then the second triangular front is converted into a back concave front, and a slow wave arises. This slow wave is of interest for seismology. Bibliography: 11 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 354, 2008, pp. 190–211.     

The Magnus embedding as a special case of a homological construction

Let E be a group extension with Abelian kernel. Then it can be assigned an extension E′ of modules over the group ring of the quotient group. As a consequence, an embedding of the initial extension in some splitting extension arises. We prove that the celebrated Magnus embedding is a special case of this general construction.     

The effective conductivity of a randomly inhomogeneous medium

Using a representation of the solution to the diffusion equation in a randomly inhomogeneous medium in the form of a Feynman path integral an explicit expression is obtained for the effective conductivity in a space of arbitrary dimension. A calculation of the path integral only turns out to be possible in the case of a large-scale limit. In particular, it is shown that in the three-dimensional case the expression for the effective conductivity does not admit of an expansion in terms of the conductivity variance. This indicates that the use of standard perturbation theory in the form of an expansion in terms of the conductivity fluctuations is incorrect.     

Investigation of the wave field in an effective model of a layered elastic-fluid medium

For a medium that consists of alternating elastic and fluid layers, an effective model is constructed and investigated. This model is a special case of the Biot medium. The wave field is represented as Fourier and Mellin integrals. In the Mellin integral, the contour of integration is replaced by a stationary contour. In the expressions obtained, the order of integration is changed and the inner integral is calculated. The outer integral is equal to two residues. The corresponding poles are roots of two equations of fourth order. These roots lie on the right half-plane and may be complex or real. The representation obtained for the wave field is in agreement with the expressions derived by the Smirnov-Sobolev method. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 175–192.     

Body forces equivalent to jumps of stresses and displacements in the Biot model of an anisotropic porous medium

Formulas for the body forces equivalent to given jumps of stresses and displacements in the anisotropic Biot model are obtained. Similarly to the case of elastic media, Betti's identity, the reciprocity relations, and the representation integrals for a porous media are derived first. The equivalent body forces are obtained by applying the representation integrals to a volume with surfaces of discontinuity. Some physical consequences are indicated, and examples of equivalent body forces for the isotropic Biot model are provided.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 33–44.The author wishes to express his sincere gratitude to V. M. Babich and L. A. Molotkov for helpful discussions. This work was supported in part by the Soros Foundation under grant A96-378 and the Russian Foundation for Basic Research under grant 96-05-65904.     

Asymptotic analysis of a transport model of organic material through a stratified porous medium: Concentration effect

We consider the transport through capillarity of an organic material inside a porous medium, using Leverett’s model. We first prove an existence result for a weak solution of this nonlinear evolution problem, using a regularization process. We then describe the asymptotic behavior of the solution, when the permeability _kε$k_{\epsilon }$ of the porous medium is associated to a scalar function which only depends on the third variable, assuming that _kε$k_{\epsilon }$ (resp. the inverse of _kε$k_{\epsilon }$) converges to some measure _λ∗${\lambda }_{\ast }$ (resp. λ^∗${\lambda }^{\ast }$). We use Γ-convergence arguments in order to describe this asymptotic behavior. We finally characterize the asymptotic behavior of the problem, considering special choices of the permeability _kε$k_{\epsilon }$, which correspond to stratified porous media, and give a numerical test for a 1D model.     

Some remarks on the Biot system of equations describing wave propagation in a porous medium

The ill-posedness of the Cauchy problem for the Biot system of equations in the case of energy dissipation is shown (the Cauchy problem does not have a unique solution).     

Propagation of sound pulses in a semiinfinite stratified medium

We consider the propagation of sound pulses due to a line source in the inhomogeneous, semi-infinite mediumy ≥ 0 with the boundary conditionφ=0 or?φ/(?y)=0 aty=0, whereφ is the acoustic velocity potential. We suppose that the velocity of wave propagation,c, is given byc ^?2=p ?qe ^?ay , wherep, q, α are real and positive andp>q. The method of dual integral transformation is used. The solution in terms of pulse propagation modes yields the diffracted pulse and the method of steepest descents gives the geometrical acoustic field.     

Investigation of low-frequency normal waves in a Biot layer surrounded by an elastic medium

The porous Biot layer surrounded by two elastic half-plates is considered. For this medium, two dispersion equations are established. These dispersion equations correspond to symmetric and antisymmetric parts of the medium. The investigation of these equations is conducted in the region of low frequencies. The imaginary roots of these equations in this region determine low-frequency normal waves. Bibliography: 6 titles.     

On an effective two-phase model of a medium with cracks of finite length

An effective two-phase model of media with cracks filled with liquid is generalized to the case of finite cracks. Wave propagation in a half-space and in a free layer, described by the new model, is investigated. The results are compared with experimental data and with corresponding results in the case of media with infinite cracks. Bibliography: 11 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 203, 1992, pp. 137–155. Translated by L. A. Molotkov.     

The effective conductivity equation for a highly heterogeneous periodic medium

We consider the homogenization of a conductivity equation for a medium made up of a set ${F_\varepsilon}$ ( ${\varepsilon}$ being the size of the period of the medium) of highly conductive vertical fibers surrounded by another material (the matrix) assumed to be a poor conductor. The conductivity coefficients in the fibers behave as ${\frac{1}{\varepsilon^2}}$ while whose of the matrix behave as ${\varepsilon^2}$ . We show that the homogenized problem consists of an equality of the kind u(x) = m(x) f (x) where u denotes the macroscopic temperature, f the source term and m(x) a coefficient given by solving some cell equation.     

Smouldering front shapes for steady propagation in a stratified medium

The steady propagation of a smouldering reaction front parallelto the faces of a solid reactive slab has been considered withthe density of the reactive material changing with distancefrom one surface. Such slow steady propagation is known to existin peat bogs where smoulder can take place over many monthsPage et al.(2002, Nature, 420, 61–65). As smoulder progressesit leaves behind a porous matrix through which oxidizer is ableto diffuse to the reaction front. At the front, oxidizer andfuel combine stoichiometrically and the concentration of oxidizeris reduced to zero. In the analysis to be presented the Pecletnumber, based on an assumed constant smoulder speed, is small.The equations and boundary conditions for the oxidizer in theporous region are solved to first order by a complex variablemethod and hodograph transformation. The solution allows theshape of the smouldering front and the oxidizer distributionbehind the front to be determined. Example curves for particularfuel distributions are given. An analysis shows how furtherterms in the expansion of oxidizer concentration in powers ofthe Peclet number may be obtained.     

High-frequency asymptotic behavior of the electromagnetic field on the surface of a stratified medium

Translated from Metody Matematicheskogo Modelirovaniya i Vychislitel'noi Diagnostiki, pp. 174–180, Izd. Moskovskogo Universiteta, Moscow, 1990.     

On an effective model describing a layered periodic elastic medium with slide contacts on the interfaces

Effective models are derived for layered periodic elastic media'with slide contacts on all interfaces. In the case where each period consists of n layers with different plate velocities, the effective model has n phases. These models are investigated for typical media. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1994, pp. 192–212. Translated by L. A. Molotkov.     

Posynomial geometric programming as a special case of semi-infinite linear programming

This paper develops a wholly linear formulation of the posynomial geometric programming problem. It is shown that the primal geometric programming problem is equivalent to a semi-infinite linear program, and the dual problem is equivalent to a generalized linear program. Furthermore, the duality results that are available for the traditionally defined primal-dual pair are readily obtained from the duality theory for semi-infinite linear programs. It is also shown that two efficient algorithms (one primal based and the other dual based) for geometric programming actually operate on the semi-infinite linear program and its dual.