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.     

Two compressible immiscible fluids in porous media

We consider a model of flow of two compressible and immiscible phases in a three-dimensional porous media. The equations are obtained by the conservation of the mass of each phase. This model is treated in its general form with the whole nonlinear terms. The only assumption concerns the dependence of densities on a global pressure. We obtain the existence of weak solutions under different kinds of degeneracies of the capillary terms.     

On the mushy region arising between two fluids in a porous medium

We study the mushy region arising between two fluids in a porous medium. We prove that the interior of the mushy region is an epigraph in the horizontal direction. Moreover, when the interior of the mushy region is empty, we give a necessary and sufficient condition to claim that the Lebesgue measure of this mushy region is zero.     

Nonlocal problems for filtration equations for nonNewtonian fluids in a porous medium

The following nonlocal problems are studied for the filtration equations of 1) an Oldroyd fluid and 2) of a Kelvin-Voigt fluid: the existence of solutions for the initial boundary problems for the semiaxis t>0 with free terms and the existence of solution periodic in t with period with free terms which are also periodic in t with period .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 189, pp. 82–100, 1991.     

Interfaces with a corner point in one-dimensional porous medium flow

We prove that for a large class of initial distributions the solutions of the initial value problem for the one-dimensional porous medium equation have interfaces which start to move abruptly after a positive waiting time. This happens, for example, if the initial pressure is o(|x|²) as x → 0. We also give sufficient conditions for the smoothness of the interface which improve previous results.     

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.     

On existence and uniqueness for a coupled system modeling immiscible flow through a porous medium

We consider a system of nonlinear coupled partial differential equations that models immiscible two-phase flow through a porous medium. A primary difficulty with this problem is its degenerate nature. Under reasonable assumptions on the data, and for appropriate boundary and initial conditions, we prove the existence of a weak solution to the problem, in a certain sense, using a compactness argument. This is accomplished by regularizing the problem and proving that the regularized problem has a unique solution which is bounded independently of the regularization parameter. We also establish a priori estimates for uniqueness of a solution.     

The stability of an interface between miscible fluids in a porous medium

Résumé On étudie la nature de stabilité de l'interface horizontale de deux fluides miscibles qui se filtrent verticalement à travers une couche poreuse. Initialement, le profile de densité vertical est une fonction de Heaviside; on admet l'existence d'un déplacement ondé d'une amplitude infinitésimale à l'interface. Subséquemment, la diffusion moléculaire et la dispersion mécanique font décroître les gradients de densité et des autres caractéristiques des fluides. La vitesse d'accroissement d'amplitude de la perturbation dépend de la répartition de densité variable et du nombre d'onde et de l'amplitude initiale de chaque composante de Fourier de cette perturbation. On obtient une expansion en séries d'une composante typique, et de résultants numériques pour la perturbation symétrique qui se trouve par suite des conditions initiales.     

Uniqueness results for monopolar fluids

1 FormulationLet fi be a bounded set of R", n 2 2, tv'ith a sufficiently smooth boundary off. ac'e considerthe motion of an incompressible, viscous fluid in fi, which is described by the following systemll]with the initial conditionand the Dirichlet boundary conditionwhere v = (yi, .". u.) is the velocity, p is the pressure, the tensor r = (n j) is defined asp is a. positive constant. Notice that when p = 2 the s}l'steln (1), (2), (5) turns to the N*avierStokes system.A scalar potential to t…     

Uniqueness of weak solutions for a pseudo-parabolic equation modeling two phase flow in porous media

In this paper, we prove the uniqueness of weak solutions for a pseudo-parabolic equation modeling two-phase flow in a porous medium, where dynamic effects are included in the capillary pressure. We transform the equation into an equivalent system, and then prove the uniqueness of weak solutions to the system which leads to the uniqueness of weak solutions for the original model.     

On the free boundary problem to the two viscous immiscible fluids

In this paper, we prove the local solvability of the free boundary problem describing the motion of two layers of immiscible, heavy, viscous, incompressible fluid lying above an infinite rigid bottom and with surface tension on the interfaces, and global solvability near the equilibrium state.     

Finite element methods for an acoustic well-logging problem associated with a porous medium saturated by a two-phase immiscible fluid

A mathematical model is presented concerning wave propagation in a domain that arises in geophysical well-logging problems. The domain consists of a borehole Ω_f surrounded by a porous medium Ω_p. Ω_f is filled with a compressible inviscid fluid, and Ω_p is saturated by a two-phase immiscible fluid. Absorbing boundary conditions for artificial boundaries and boundary conditions on the interface between Ω_f and Ω_p are used. The existence and uniqueness theorems are stated for the resulting initial-boundary value problem. Stability and convergence estimates for a finite element method are also studied. © 1993 John Wiley & Sons, Inc.     

Dusty effects on pulsating viscous flow of two immiscible fluids between two parallel plates

The flow of two immiscible incompressible dusty viscous fluids between two parallel plates generated by a pulsating pressure gradient is investigated. Velocity fields for the fluid-particle system along with the expressions for the skin friction drag at the plates are obtained and studied graphically. It is found that there is an immediate response to pressure fluctuations in the first stream at low frequency range 0<σ≤4 being maximum at σ=4. On the contrary, the second stream is more responsive to fluctuations at relatively higher frequencies. The maximum response in this case is shifted to σ=16.     

Global solutions for a one-dimensional problem in conducting fluids

A mathematical model of the motion of conducting fluids is studied in this paper. The dynamics of such fluids is described by the equations of compressible fluids coupled to the Maxwell’s equations. We prove global existence of strong solution for a one-dimensional initial-boundary value problem of this model (plane conducting flows) with general large data.     

The lift on a small sphere in a linear shear flow near the interface of two immiscible fluids

Analytical expressions have been derived which predict, to lowest order, the inertial lift and the lateral migration velocity of a rigid sphere translating and rotating in a linear shear flow field near the flat interface of two immiscible fluids. This asymptotic analysis is primarily based on the assumption that the two Reynolds numbers defined by the gap width between the interface and the sphere, the shear rate and the translational slip velocity with which the spherical particle moves parallel to the interface are small. Furthermore, the radius of the sphere is assumed to be small compared to the gap width. To leading order in this creeping flow regime, the linear Stokes equations are obtained and a symmetry argument can be used to show that the Stokes solution does not predict any lift force. The transverse force experienced by the sphere and its migration velocity are due to the small but finite inertial terms in the Navier-Stokes equations, which can be studied by perturbation techniques. By applying a Green's function approach and matched asymptotic methods, which also incorporate the effects of the outer Oseen-like flow regime, the three components comprising the lift velocity have been calculated in closed form: the one induced by the shear rate only, the purely slip induced one and the one due to the interaction of the slip velocity with the shear flow field. The thus obtained expressions for the case of two immiscible fluids with arbitrary density and viscosity ratios extend the results that already exist in the literature for other flow configurations, such as an unbounded shear flow field [1] or a wall-bounded one, where the wall lies either within the leading order Stokes region [2] or in the outer Oseen region [3]. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Uniqueness results for the one-dimensional p-Laplacian

This paper is concerned with positive solutions of the boundary value problem ^′(|y^′|^p−2y^′)+f(y)=0, y(−b)=0=y(b) where p>1, b is a positive parameter. Assume that f is continuous on (0,+∞), changes sign from nonpositive to positive, and f(y)/yp−1 is nondecreasing in the interval of f>0. The uniqueness results are proved using a time-mapping analysis.     

Traveling waves in a one-dimensional heterogeneous medium

We consider solutions of a scalar reaction–diffusion equation of the ignition type with a random, stationary and ergodic reaction rate. We show that solutions of the Cauchy problem spread with a deterministic rate in the long time limit. We also establish existence of generalized random traveling waves and of transition fronts in general heterogeneous media.