Linear,hydrodynamic flow in a rotating saturated porous medium

Linear, steady, axisymmetric flow of a homogeneous fluid in a rigid, bounded, rotating, saturated porous medium is analyzed. The fluid motions are driven by differential rotation of horizontal boundaries. The dynamics of the interior region and vertical boundary layers are investigated as functions of the Ekman number E(=v/ΩL ²) and rotational Darcy 3 numberN(=kΩ/v) which measures the ratio between the Coriolis force and the Darcy frictional term. IfN≫E ^−1/2, the permeability is sufficiently high and the flow dynamics are the same as those of the conventional free flow problem with Stewartson'sE ^1/3 andE ^1/4 double layer structure. For values ofN≤E ^−1/2 the effect of porous medium is felt by the flow; the Taylor-Proudman constraint is no longer valid. ForN≪E ^−1/3 the porous medium strongly affects the flow; viscous side wall layer is absent to the lowest order and the fluid pumped by the Ekman layer, returns through a region of thicknessO(N ⁻¹). The intermediate rangeE ^−1/3≪N≪E ^−1/2 is characterized by double side wall layer structure: (1)E ^1/3 layer to return the mass flux and (ii) (NE)^1/2 layer to adjust the interior azimuthal velocity to that of the side wall. Spin-up problem is also discussed and it is shown that the steady state is reached quickly in a time scaleO(N).     

Steady flow in a deformable porous medium

A nonlinear model for a steady flow in a deformable porous medium is considered. The flow is governed by the poroelasticity system consisting of an elasticity equation for the displacement of the porous medium and Darcy's equation for the pressure in the fluid. This poroelasticity system is nonlinear when the permeability in Darcy's equation is assumed to depend on the dilatation of the porous medium. Existence and uniqueness of a weak solution of this poroelasticity system is established under rather weak assumptions on the regularity of the data. Convergence of a finite element approximation is proved and verified through numerical experiments. Copyright © 2013 John Wiley & Sons, Ltd.     

Flow of oil and water in a porous medium

The flow of two immiscible and incompressible fluids in a porous medium is described by a system of quasilinear degenerate partial differential equations. In this paper the existence of a weak solution by regularization is shown.     

A numerical study of non-linear convection in a compressible medium

In this paper we consider the problem of non-linear convection in a compressible layer with polytropic structure. After deriving the appropriate forms of the basic conservation equations a single mode expansion is used to obtain a simpler model, within the framework of the anelastic approximation. These equations are integrated, using a band-matrix algorithm, for two characteristic density stratifications. The results are given in the form of graphs and are discussed in detail. An outline of the numerical algorithm is given in an appendix together with a discussion of its effectiveness.     

The porous medium equation in a two-component domain

We consider a system of two porous medium equations defined on two different components of the real line, which are connected by the nonlinear contact condition

     

Wave propagation in a fluid-saturated inhomogeneous porous medium

A closed system of constitutive equations for the dynamical and geometric quantities in a fluid- saturated inhomogeneous elastic porous medium is constructed within the framework of the three-dimensional theory of elasticity. The geometrical characteristics of the wave front and of the ray in a fluid-saturated inhomogeneous medium are obtained from the Fermi's principle.     

A three-dimensional inverse problemfor a physically non-linear inhomogeneous medium

A three-dimensional linearly elastic (viscoelastic) domain (finite or infinite) containing a physically non-linear inclusion of arbitrary shape is considered. The possibility of creating a prescribed uniform stress-strain state in the inclusion by a suitable choice of loads on the outer boundary of the domain is considered. A solution is constructed in closed form. Some examples are considered, including, in particular, the case of an ellipsoidal inclusion with the property of non-linear creep.     

Extremal geometry of a Brownian porous medium

The path \(W[0,t]\) of a Brownian motion on a \(d\) -dimensional torus \(\mathbb T ^d\) run for time \(t\) is a random compact subset of \(\mathbb T ^d\) . We study the geometric properties of the complement \(\mathbb T ^d{{\setminus }} W[0,t]\) as \(t\rightarrow \infty \) for \(d\ge 3\) . In particular, we show that the largest regions in \(\mathbb T ^d{{\setminus }} W[0,t]\) have a linear scale \(\varphi _d(t)=[(d\log t)/(d-2)\kappa _d t]^{1/(d-2)}\) , where \(\kappa _d\) is the capacity of the unit ball. More specifically, we identify the sets \(E\) for which \(\mathbb T ^d{{\setminus }} W[0,t]\) contains a translate of \(\varphi _d(t)E\) , and we count the number of disjoint such translates. Furthermore, we derive large deviation principles for the largest inradius of \(\mathbb T ^d{{\setminus }} W[0,t]\) as \(t\rightarrow \infty \) and the \(\varepsilon \) -cover time of \(\mathbb T ^d\) as \(\varepsilon \downarrow 0\) . Our results, which generalise laws of large numbers proved by Dembo et al. (Electron J Probab 8(15):1–14, 2003), are based on a large deviation estimate for the shape of the component with largest capacity in \(\mathbb T ^d{{\setminus }} W_{\rho (t)}[0,t]\) , where \(W_{\rho (t)}[0,t]\) is the Wiener sausage of radius \(\rho (t)\) , with \(\rho (t)\) chosen much smaller than \(\varphi _d(t)\) but not too small. The idea behind this choice is that \(\mathbb T ^d {{\setminus }} W[0,t]\) consists of “lakes”, whose linear size is of order \(\varphi _d(t)\) , connected by narrow “channels”. We also derive large deviation principles for the principal Dirichlet eigenvalue and for the maximal volume of the components of \(\mathbb T ^d {{\setminus }} W_{\rho (t)}[0,t]\) as \(t\rightarrow \infty \) . Our results give a complete picture of the extremal geometry of \(\mathbb T ^d{{\setminus }} W[0,t]\) and of the optimal strategy for \(W[0,t]\) to realise extreme events.     

On a filtration problem through a porous medium

Summary We prove the existence of a unique solution for a free boundary problem relative to the stationary flow between two water reservoirs of different levels separated by a dam of a non-homogeneous porous medium. Entrata in Redazione il 5 maggio 1973.     

The critical pressure for flow in a porous medium

Bingham flow in a porous medium is considered. This can be modelled by a random structure whose dimensions are large compared with the local scale. The principal term of the asymptotic form of the critical pressure at which the liquid starts to move in this limit is computed explicitly.     

Mathematical modeling of infiltration metasomatism in a porous medium

We construct a mathematical model describing the processes of dissolution and redeposition of minerals in a medium with a nonhomogeneous distribution of acidity. The dynamics of extraction of a mineral from a leaching solutions is investigated. We show that filtration of solutions through reduced acidity regions induces deposition, increasing the concentration of the target mineral in the solid phase; in high pH regions, on the other hand, the mineral dissolves. The stratum may retain certain reserves of the target mineral after leaching depending on the size of the reduced pH region and its proximity to the extraction borehole. __________ Translated from Prikladnaya Matematika i Informatika, No. 26, pp. 5–17, 2007.     

Pair correlations in a multicomponent fluid placed in a porous medium

We consider a multicomponent fluid placed in a porous medium. The Ornstein—Zernike approximation is used to calculate the pair correlation functions for density fluctuations in the mixture components. We show that light scattering in the neighborhood of the critical state of the system is determined (in the single-scattering approximation) by the commonly known Ornstein—Zernike formula. We investigate the shift in the critical parameters due to the porous medium. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 3, pp. 525–528, March, 2006.     

Nonlinear rotating convection in a sparsely packed porous medium

We investigate linear and weakly nonlinear properties of rotating convection in a sparsely packed Porous medium. We obtain the values of Takens–Bogdanov bifurcation points and co-dimension two bifurcation points by plotting graphs of neutral curves corresponding to stationary and oscillatory convection for different values of physical parameters relevant to rotating convection in a sparsely packed porous medium near a supercritical pitchfork bifurcation. We derive a nonlinear two-dimensional Landau–Ginzburg equation with real coefficients by using Newell–Whitehead method [16]. We investigate the effect of parameter values on the stability mode and show the occurrence of secondary instabilities viz., Eckhaus and Zigzag Instabilities. We study Nusselt number contribution at the onset of stationary convection. We derive two nonlinear one-dimensional coupled Landau–Ginzburg type equations with complex coefficients near the onset of oscillatory convection at a supercritical Hopf bifurcation and discuss the stability regions of standing and travelling waves.     

Subgrid modeling of filtration and dispersion in a fractal porous medium

The equations are obtained for effective coefficients of correlated random fields of permeability and porosity in a fractal porous medium. The fields have log-normal distributions. The refined perturbation theory is formulated that uses some ideas of the Wilson renormalization group. The theoretical results are compared with the results of a direct numerical modeling and the results of the conventional perturbation theory. The advantages of the refined perturbation theory over the conventional perturbation theory are demonstrated.     

Linear and non-linear stability for general linear methods

We explore the interrelation between a number of linear and non-linear stability properties. The weakest of these,A-stability, is shown by counterexample not to imply any of the various versions ofAN-stability introduced in the paper and two of these properties, weak and strongAN-stability, are also shown not to be equivalent. Finally, another linear stability property defined here, EuclideanAN-stability, is shown to be equivalent to algebraic stability.