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.     

Effect of Chemical reaction on MHD flow of a visco-elastic fluid through porous medium.

The effect of heat and mass transfer on free convective flow of a visco-elastic incompressible electrically conducting fluid past a vertical porous plate through a porous medium with time dependant oscillatory permeability and suction in the presence of a uniform transverse magnetic field, heat source and chemical reaction has been studied in this paper. The novelty of the present study is to analyze the effect of chemical reaction, time dependant fluctuative suction and permeability of the medium on a visco-elastic fluid flow. It is interesting to note that presence of sink contributes to oscillatory motion leading to flow instability. Further it is remarked that presence of heat source and low rate of thermal diffusion counteract each other in the presence of reacting species.     

Multiple blow-up for a porous medium equation with reaction

The present paper is concerned with the Cauchy problem

A fast precipitation and dissolution reaction for a reaction–diffusion system arising in a porous medium

This paper is devoted to the study of a fast reaction–diffusion system arising in reactive transport. It extends the articles [R. Eymard, T. Gallouët, R. Herbin, D. Hilhorst, M. Mainguy, Instantaneous and noninstantaneous dissolution: Approximation by the finite volume method, ESAIM Proc. (1998); J. Pousin, Infinitely fast kinetics for dissolution and diffusion in open reactive systems, Nonlinear Anal. 39 (2000) 261–279] since a precipitation and dissolution reaction is considered so that the reaction term is not sign-definite and is moreover discontinuous. Energy type methods allow us to prove uniform estimates and then to study the limiting behavior of the solution as the kinetic rate tends to infinity in the special situation of one aqueous species and one solid species.     

Buoyancy and chemical reaction effects on MHD mixed convection heat and mass transfer in a porous medium with thermal radiation and Ohmic heating

The combined effect of mixed convection with thermal radiation and chemical reaction on MHD flow of viscous and electrically conducting fluid past a vertical permeable surface embedded in a porous medium is analyzed. The heat equation includes the terms involving the radiative heat flux, Ohmic dissipation, viscous dissipation and the internal absorption whereas the mass transfer equation includes the effects of chemically reactive species of first-order. The non-linear coupled differential equations are solved analytically by perturbation technique. The results obtained show that the velocity, temperature and concentration fields are appreciably influenced by the presence of chemical reaction, thermal stratification and magnetic field. It is observed that the effect of thermal radiation and magnetic field is to decrease the velocity, temperature and concentration profiles in the boundary layer. There is also considerable effect of magnetic field and chemical reaction on skin-friction coefficient and Nusselt number.     

Finite element analysis of hydromagnetic flow and heat transfer of a heat generation fluid over a surface embedded in a non-Darcian porous medium in the presence of chemical reaction

An analysis is carried out to study the flow, chemical reaction and mass transfer of a steady laminar boundary layer of an electrically conducting and heat generating fluid driven by a continuously moving porous surface embedded in a non-Darcian porous medium in the presence of a transfer magnetic field. The governing partial differential equations are converted into ordinary differential equations by similarity transformation and are solved numerically by using the finite element method. The results obtained are presented graphically for velocity, temperature and concentration profiles, as well as the Sherwood number for various parameters entering into the problem.     

Blow-up and global existence for the porous medium equation with reaction on a class of Cartan–Hadamard manifolds

We consider the porous medium equation with power-type reaction terms $u^p$ on negatively curved Riemannian manifolds, and solutions corresponding to bounded, nonnegative and compactly supported data. If $p>m$, small data give rise to global-in-time solutions while solutions associated to large data blow up in finite time. If $p<m$, large data blow up at worst in infinite time, and under the stronger restriction $p\in (1,(1+m)/2]$ all data give rise to solutions existing globally in time, whereas solutions corresponding to large data blow up in infinite time. The results are in several aspects significantly different from the Euclidean ones, as has to be expected since negative curvature is known to give rise to faster diffusion properties of the porous medium equation.     

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.     

Predicting dispersion in porous media

We present a fundamental theory of solute dispersion in porous using (i) critical path analysis and cluster statistics of percolation theory far from the percolation threshold and (ii) the tortuosity and structure of large clusters near the percolation threshold. We use the simplest possible model of porous media, with a single length scale of heterogeneity in which the statistics of local conductances are uncorrelated. This combination of percolation‐based techniques allows comprehensive investigation and predictions concerning the process of dispersion. Our predictions, which ignore molecular diffusion and make minimal use of unknown parameters, account for results obtained in a comprehensive set of nearly 1100 experiments performed on systems ranging in size from centimeters to 100 km. The success of our simple treatment overturns many existing notions about transport in porous media, such as (1) multiscale heterogeneity must be accounted for in predictions (single scale is sufficient), (2) geologic correlations are of great importance (the randomness of percolation theory is more appropriate for prediction than the most complicated models in other frameworks), (3) geologic complexity is more important than statistical physics (exactly the reverse), (4) knowledge of the subsurface is more important than knowledge of the initial conditions of the plume (the latter is critical, the former may be virtually irrelevant), (5) diffusion is dominant over advection (diffusion appears seldom to be relevant at all), (6) fracture networks are fundamentally different, and more complex, than porous media (the two are mostly equivalent), (7) the fractal structure of the medium is relevant to power‐law behavior of the dispersion (in fact, at short times it is the heterogeneity of the medium, while at long times it is the fractal structure of the critical paths), and (8) there is a relation between an increase in dispersion with scale and a similar increase in the hydraulic conductivity (in fact the present model is consistent with both a diminishing hydraulic conductivity and a diminishing solute velocity with increasing spatial scale). © 2009 Wiley Periodicals, Inc. Complexity, 16,43–55, 2010     

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.     

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

     

壁面马赫数分布规律可控的新型内收缩基准流场设计方法

根据有旋特征线理论,设计出了沿程马赫数下降规律可控的轴对称基准流场,分析了基准流场的几何参数(前缘压缩角及中心体半径)的影响规律,发现选取较小的前缘压缩角和中心体半径有利于得到性能优良的基准流场;然后在设计状态Ma=6时研究了三种典型的马赫数下降规律对这种轴对称流场性能的影响。最后考虑了粘性的影响,并进行了粘性修正探索,结果表明,采用附面层位移厚度修正方法后,基准流场的壁面压力分布和无粘情况吻合良好。      

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.     

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.     

Effect of rotation on the onset of double diffusive convection in a Darcy porous medium saturated with a couple stress fluid

The effect of rotation on the onset of double diffusive convection in a horizontal couple stress fluid-saturated porous layer, which is heated and salted from below, is studied analytically using both linear and weak nonlinear stability analyses. The extended Darcy model, which includes the time derivative and Coriolis terms, has been employed in the momentum equation. The onset criterion for stationary, oscillatory and finite amplitude convection is derived analytically. The effect of Taylor number, couple stress parameter, solute Rayleigh number, Lewis number, Darcy–Prandtl number, and normalized porosity on the stationary, oscillatory, and finite amplitude convection is shown graphically. It is found that the rotation, couple stress parameter and solute Rayleigh number have stabilizing effect on the stationary, oscillatory, and finite amplitude convection. The Lewis number has a stabilizing effect in the case of stationary and finite amplitude modes, with a destabilizing effect in the case of oscillatory convection. The Darcy–Prandtl number and normalized porosity advances the onset of oscillatory convection. A weak nonlinear theory based on the truncated representation of Fourier series method is used to find the finite amplitude Rayleigh number and heat and mass transfer. The transient behavior of the Nusselt number and Sherwood number is investigated by solving the finite amplitude equations using Runge–Kutta method.     

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.     

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.     

Homogenization of the Euler system in a 2D porous medium

We study the homogenization of the Euler system in a periodic porous medium (of period ɛ) by using the notion of two-scale convergence. At the limit, we recover a system which couples a cell problem with the macroscopic one.