Exponential stability in porous media problem saturated by multiple immiscible fluids

This paper concerns a continuum theory of porous media saturated by multiple immiscible fluids. The case of a porous media saturated by two immiscible fluid proposes some new mathematical difficulties. We study the exponential stability of the one-dimensional problem when the nonwetting fluid is trapped in the wetting fluid and the exponential stability of the anti-plane shear deformations when the two fluids saturate the elastic media.     

A continuum theory of superconductivity

Abstract. A continuum theory of superconductivity is formulated for a mixture consisting of three species: a superelectron fluid, a conducting fluid, and a conducting elastic solid. Each one of the three species is subject to their own electro-magnetic (E-M) fields and motions. Irreversible thermodynamics are used to obtain constitutive equations. Field equations, boundary and initial conditions are given. A special case is obtained, suitable for mathematical analysis and applications. The Pippard theory of superconductivity is shown to be a special case of the present theory.     

A parametric investigation of the harmonic vibrations of a poroelastic rod

The problems of calculating the forced harmonic vibrations of porous structures saturated with a fluid are considered. The equations of motion are obtained by starting from general relations of the Biot theory of poroelasticity and continuous mechanics after taking into account the anisotropy of the elastic and hydraulic properties of the medium. The influence of the mechanical constants of the material on its dynamical characteristics is investigated using the example of the flexural vibrations of a simple rod-shaped structure.     

On the ray method for Biot media

This paper is devoted to an investigation of wave propagation in a Biot porous medium, which consists of elastic and fluid phases. The space-time ray expansion of solutions of dynamic equations for a Biot medium is constructed (in the anisotropic inhomogeneous case). In the inhomogeneous isotropic case, a Rytov law analog is derived similarly to elasticity theory. Bibliography: 3 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 354, 2008, pp. 112–131.     

The propagation of elastic waves in a viscous-fluid-saturated porous medium

Elastic shock waves in a viscous-fluid-saturated porous medium are investigated. The porosity is only taken into account with respect to pores communicating with one another, and isolated pores are considered as elements of the elastic part of the porous skeleton. It is shown, using the theory of discontinuity, that in such a medium there are two types of vortex-free waves and one equivoluminal wave. Differential equations and their solution for determining the change in the wave-front intensity are obtained. The effect of the fluid viscosity and porosity on the propagation of spherical waves is demonstrated using an example.     

On existence and stability in the theory of swelling porous elastic soils

The aim of this paper is to introduce a new functional in thestudy of swelling porous elastic soils saturated by a fluid.This new functional is a useful tool; it allows us to provethe existence of solutions in the case of a compressible fluid.We also prove the stability of solutions and the exponentialdecay in the case of an incompressible fluid. We study as wellthe continuous dependence with respect to the initial time.     

Linear and non-linear magnetoconvection in a porous medium

Linear and non-linear magnetoconvection in a sparsely packed porous medium with an imposed vertical magnetic field is studied. In the case of linear theory the conditions for direct and oscillatory modes are obtained using the normal modes. Conditions for simple and Hopf-bifurcations are also given. Using the theory of self-adjoint operator the variation of critical eigenvalue with physical parameters and boundary conditions is studied. In the case of non-linear theory the subcritical instabilities for disturbances of finite amplitude is discussed in detail using a truncated representation of the Fourier expansion. The formal eigenfunction expansion procedure in the Fourier expansion based on the eigenfunctions of the corresponding linear stability problem is justified by proving a completeness theorem for a general class of non-self-adjoint eigenvalue problems. It is found that heat transport increases with an increase in Rayleigh number, ratio of thermal diffusivity to magnetic diffusivity and porous parameter but decreases with an increase in Chandrasekhar number.     

Investigation of normal waves in a porous layer surrounded by elastic half-spaces

The wave field and dispersion equations are found for a porous layer surrounded by two elastic half-spaces. The porous layer is described by the effective model of a medium in which elastic and fluid layers alternate. To investigate the normal waves, all real roots of dispersion equations are determined and their movements as the wave number increases are investigated. As a result, the dispersion curves of all normal waves are constructed and the dependence of normal waves on the parameters of the porous layer and elastic half-spaces is analyzed. Bibliography: 6 titles.     

Fluid flow over a thin deformable porous layer

The flow of a viscous fluid over a thin, deformable porous layer fixed to the solid wall of a channel is considered. The coupled equations for the fluid velocity and the infinitesimal deformation of the solid matrix within the porous layer are developed using binary mixture theory, Darcy's law and the assumption of linear elasticity. The case of pure shear is solved analytically for the displacement of the solid matrix, the fluid velocity both in the porous medium and the fluid above it. For a thin porous layer the boundary condition for the fluid velocity at the fluid-matrix interface is derived. This condition replaces the usual no slip condition and can be applied without solving for the flow in the porous layer.     

Effective models of stratified media containing porous biot layers

Periodic stratified media in which either two porous Biot layer, or an elastic and a porous layers, or a fluid and a porous layer alternate are considered. The effective models of these media are constructed and investigated. In the case of alternating porous layers, the effective model is a generalized transversely isotropic Biot medium. In this medium, the density of the fluid phase and the mean density acquire tensor character. It is shown that the effective model of a porous-fluid medium is, on the one hand, a generalized transversely isotropic Biot medium of special type and, on the other hand, a generalization of the effective model of a stratified elastic-fluid medium.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 140–163.This work was supported by the Russian Foundation for Basic Research under grant Nos. 96-01-00666 and 96-05-66207.     

Analogues of the Kolosov–Muskhelishvili General Representation Formulas and Cauchy–Riemann Conditions in the Theory of Elastic Mixtures

Analogues of the well-known Kolosov–Muskhelishvili formulas of general representations are obtained for nonhomogeneous equations of statics in the case of the theory of elastic mixtures. It is shown that in this theory the displacement and stress vector components, as well as the stress tensor components, are represented through four arbitrary analytic functions.The usual Cauchy–Riemann conditions are generalized for homogeneous equations of statics in the theory of elastic mixtures.     

Nguetseng’s two-scale convergence method for filtration and seismic acoustic problems in elastic porous media

A linear system is considered of the differential equations describing a joint motion of an elastic porous body and a fluid occupying a porous space. The problem is linear but very hard to tackle since its main differential equations involve some (big and small) nonsmooth oscillatory coefficients. Rigorous justification under various conditions on the physical parameters is fulfilled for the homogenization procedures as the dimensionless size of pores vanishes, while the porous body is geometrically periodic. In result, we derive Biot’s equations of poroelasticity, the system consisting of the anisotropic Lamé equations for the solid component and the acoustic equations for the fluid component, the equations of viscoelasticity, or the decoupled system consisting of Darcy’s system of filtration or the acoustic equations for the fluid component (first approximation) and the anisotropic Lamé equations for the solid component (second approximation) depending on the ratios between the physical parameters. The proofs are based on Nguetseng’s two-scale convergence method of homogenization in periodic structures.     

An analysis of the swimming problem of a singly flagellated micro-organism in an MHD fluid flowing through a porous medium

The focus of the present work is concerned with the study of the swimming of microscopic organisms that use a single flagellum for propulsion in a magnetohydrodynamic (MHD) fluid flowing through a porous medium. The flow is modelled by appropriate equations and the organism is modelled by an infinite flexible but inextensible transversely waving sheet, which represents approximately the flagellum. The governing equations subject to appropriate boundary conditions are solved analytically. Expressions for the velocity of propulsion of the microscopic organism are obtained. We show that as the MHD character of the fluid is removed the results match those of an earlier analysed problem of propulsion through a fluid in a porous medium. In addition, in the final case of a simple viscous fluid (absence of magnetic field), we show that as the permeability becomes large the results reduce to the swimming of such organisms in a viscous fluid (discounting the pores and the MHD character).     

Derivation of a contact law between a free fluid and thin porous layers via asymptotic analysis methods

We describe the asymptotic behaviour of an incompressible viscous free fluid in contact with a porous layer flow through the porous layer surface. This porous layer has a small thickness and consists of thin channels periodically distributed. Two scales are present in this porous medium, one associated to the periodicity of the distribution of the channels and the other to the size of these channels. Proving estimates on the solution of this Stokes problem, we establish a critical link between these two scales. We prove that the limit problem is a Stokes flow in the free domain with further boundary conditions on the basis of the domain which involve an extra velocity, an extra pressure and two second-order tensors. This limit problem is obtained using Γ-convergence methods. We finally consider the case of a Navier–Stokes flow within this context.     

Nonlinear and linear hyperbolic systems with dynamic boundary conditions

We consider first order hyperbolic systems on an interval with dynamic boundary conditions. The well-posedness for linear systems is established by using a variationalmethod. The linear theory is used to analyze the local-in-timewell-posedness for nonlinear systems. The results are applied to a model describing the flow of an incompressible fluid inside an elastic tube whose ends are attached to tanks. Global existence and stability for data that are smooth enough and close to the steady state are obtained by using energy and entropy methods.     

Mathematical modeling of flows through inelastic porous solids

We propose a framework, based on classical mixture theory, to describe the isothermal flow of an incompressible fluid through a deformable inelastic porous solid. The modeling of the behavior of the inelastic solid takes into account changes in the elastic response due to evolution in the microstructure of the material. We apply the model to a compression layer problem. The mathematical problem generated by the model is a free boundary problem.     

Peristatcally induced MHD slip flow in a porous medium due to a surface acoustic wavy wall

The influences of Hall current and slip condition on the MHD flow induced by sinusoidal peristaltic wavy wall in two dimensional viscous fluid through a porous medium for moderately large Reynolds number is considered on the basis of boundary layer theory in the case where the thickness of the boundary layer is larger than the amplitude of the wavy wall. Solutions are obtained in terms of a series expansion with respect to small amplitude by a regular perturbation method. Graphs of velocity components, both for the outer and inner flows for various values of the Reynolds number, slip parameter, Hall and magnetic parameters are drawn. The inner and outer solutions are matched by the matching process. An interesting application of the present results to mechanical engineering may be the possibility of the fluid transportation without an external pressure.     

An Effective Model of a Layered Medium in Which Alternating Porous and Elastic Layers Are in Slide Contact

For a medium in which porous and elastic layers alternate and there is slide contact on the interfaces, an effective model is established. This model is of three phases and includes two elastic phases and one fluid phase. Specific features of this effective model are that two waves with triangular front sets propagate and the second (slow) longitudinal wave is absent in it. In the special case where the thickness of elastic layers is very small but they continue to work as barriers for fluid particles from porous layers, the effective model is of two phases, and one of the triangular front sets disappears. Bibliography: 14 titles.     

Fluid flow in an inhomogeneous granular medium

The seepage of a compressible fluid in an inhomogeneous undeformable granular medium is investigated. It is assumed that the fluid flow in a porous space is described by the Navier–Stokes equations. It is shown that, in the case of an inhomogeneous velocity field, a tensor of additional effective stresses occurs in connection with the transfer of fluid particles in a transverse direction when flow occurs around the granules of the medium in a longitudinal direction. Using the fundamental propositions of Reynolds’ averaging theory and Prandtl's mixing path, the structure of the effective viscosity coefficient is determined and hypotheses are formulated which enable it to be assumed to be independent of the flow velocity. It is established by comparison with experimental data that the effective viscosity coefficient can exceed the viscosity coefficient of the flowing fluid by an order of magnitude. The equations of average motion are obtained, which in the case of an incompressible fluid have the form of the Navier–Stokes equations with body forces proportional to the velocity. It is established that, in addition to the well-known dimensionless flow numbers, there is a new number which characterizes the ratio of the Darcy porous drag forces to the effective viscosity forces. The proposed equations are extended to the case of the flow of an aerated fluid. The components of the angular momentum vector are used as the required functions instead of the components of the velocity vector. This enables a solving system of equations to be obtained, which, apart from the notation, is identical with the similar equations for the case of an incompressible fluid. The solution of a new problem of the fluid flow in a plane channel with permeable walls is presented using three models: Darcy's law for an incompressible and aerated fluid, and also of an aerated fluid taking the effective viscosity into account. It is established that, for the same pressure drop, the maximum flow rate corresponds to Darcy's law. Compressibility leads to its reduction, but by simultaneously taking into account the compressibility and the effective viscosity one obtains minimum values of the flow rate. The effective viscosity and aeration of the fluid has a considerable effect on the flow parameters.     

Mathematical analysis of variable density flows in porous media

We consider a simple model describing the motion of a two-component mixture through a porous medium. We discuss well posedness of the associated initial-boundary value problem, in particular, with respect to the choice of boundary and far-field conditions. The existence of global-in-time solutions is proved in the ideal case when the fluid occupies the whole physical space. Finally, similar results are obtained also for the boundary value problems in the simplified 1-D geometry.