The Maxwell and Lame systems are considered in the case when the electromagnetic (EM) field is generated by elastic oscillations. We neglect the reverse influence of the EM field on the elastic oscillations. The influence of the EM field on the deformation field is considered as a result of the Lorentz forces.We consider some numerical approaches to solving the direct and inverse problems for a weakly coupled linearized set of equations of electromagnetoelasticity.
The effective coefficients for the problem of propagation of acoustic waves in multifractal elastic media using the subgrid modeling approach are obtained. The maximum scale of heterogeneities of the medium in question is assumed to be small as compared with the wavelength. If a isotropic medium is assumed to satisfy the improved Kolmogorov similarity hypothesis, the term for the effective coefficient of the elastic stiffness coincides with the Landau–Lifshitz–Matheron formula. Both isotropic and anisotropic media are considered. The numerical testing for the wave propagating at a distance, which is of the same order as a typical wavelength of a source, illustrates the efficiency of the approach proposed.
Equations for the effective coefficients of random permeability fields for fluid flow through a porous medium with log-stable distributions are derived using the Wilson renormalization group approach. Results of the theoretical modeling are compared with data of numerical modeling.Introduction. Full-scale and laboratory observations have shown that the fluid permeability field is nonuniform and that the spread of the nonuniformity scale increases with an increase in the number of measurements in a bounded interval. This has led to the development of fractal permeability models with log-stable permeability distributions [1]. Shvidler [2] and Dagan [3] studied statistical models with lognormal permeability distributions. Kuz'min and Soboleva [4,5], using the Wilson renormalization group (RG) approach [6], derived subgrid formulas for the effective permeability coefficients and studied the diffusion of the interface between the fluids for their joint flow in a multiscale porous medium for lognormal permeability and porosity distributions. Teodorovich [7] derived the Landau-Lifshits formula for the effective permeability within the framework of a rigorous field renormalization group and analyzed previous studies of the field RG. In particular, mention is made of the arguments of [8], according to which renormalization group methods partially allow for the highest orders of perturbation theory and should improve the accuracy of the formulas derived. The same arguments are also applicable to subgrid modeling. If the medium is assumed to satisfy the improved Kolomogorov similarity hypothesis [9,4], the effective coefficients take especially simple form. In the present paper, the ideas of the Wilson RG method are employed to find subgrid modeling formulas for solving filtration problems for fractal porous media with a lognormal permeability distribution. Differential equations for obtaining effective constants are also derived for media that do not satisfy the improved similarity hypothesis. The derived formulas are verified by direct numerical modeling.Formulation of the Problem. At small Reynolds numbers, the filtration velocity v and the pressure p are related by the Darcy law v = −ε(x)∇p, where ε(x) is the permeability (a random function of the coordinates). For an incompressible fluid, we have the equation
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