Derivation of an averaged model of isothermal acoustics in a heterogeneous medium in the case of two different poroelastic domains

We consider some mathematical model of isothermal acoustics in a composite medium consisting of two different porous soils (poroelastic domains) separated by a common boundary. Each of the domains has its own characteristics of the solid skeleton; the liquid filling the pores is the same for both domains. The differential equations of the exactmodel contain some rapidly oscillating coefficients. The averaged equations (i.e., without rapidly oscillating coefficients) are derived.     

Squeeze flow of multiply-connected fluid domains in a Hele-Shaw cell

Summary. The theory of algebraic curves and quadrature domains is used to construct exact solutions to the problem of the squeeze flow of multiply-connected fluid domains in a Hele-Shaw cell. The solutions are exact in that they can be written down in terms of a finite set of time-evolving parameters. The method is very general and applies to fluid domains of any finite connectivity. By way of example, the evolution of fluid domains with two and four air holes are calculated explicitly. For simply connected domains, the squeeze flow problem is well posed. In contrast, the squeeze flow problem for a multiply connected domain is not necessarily well-posed and solutions can break down in finite time by the formation of cusps on the boundaries of the enclosed air holes. Received September 20, 2000; accepted September 10, 2001 Online publication November 5, 2001     

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.     

Propagation of weak perturbations in cracked porous media

A three-velocity, three-pressure mathematical model is proposed which enables one to study wave processes in the case of a double porosity, deformable, fluid-saturated medium. This model takes account of the differences in the velocities and pressures in pore systems of different characteristic scales of the pores, fluid exchange between these pore systems and the unsteady forces due to interphase interactions. It is established that a single transverse and three longitudinal waves: one deformation wave and two filtration waves, propagate in such a medium. The existence of two filtration waves is associated with the two different characteristic scales of the pores and the difference in the velocities and pressures of the fluid in these pore systems. The filtration waves decay considerably more rapidly than the deformation and transverse waves. The velocities of the deformation and transverse waves are mainly determined by the elastic moduli of the skeleton. The velocity and decay of the first filtration wave depend strongly on the intensity of the interphase interaction force while the velocity of the second filtration wave depends strongly on the rate of mass exchange between the pores and the cracks. The rate of decay of the second filtration wave is significantly higher than that of the first filtration wave.     

The effect of osmotic pressure on the flow of solutions through semi-permeable hollow fibers

We study the laminar flow of binary liquid mixture, whose components are a Newtonian fluid (solvent) and a solute, in a hollow fiber. The fiber walls are porous, but the pores size is small enough preventing the solute molecules to be transported across the membrane. This produces an osmotic pressure that offers resistance (in many cases non negligible) to the fluid cross flow.     

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).     

A multiphase single relaxation time lattice Boltzmann model for heterogeneous porous media

The lattice Boltzmann (LB) method has been shown to be a highly efficient numerical method for solving fluid flow in confined domains such as pipes, irregularly shaped channels or porous media. Traditionally the LB method has been applied to flow in void regions (pores) and no flow in solid regions. However, in a number of scenarios, this may not suffice. That is partial flow may occur in semi-porous regions. Recently gray-scale LB methods have been applied to model single phase flow in such semi-porous materials. Voxels are no longer completely void or completely solid but somewhere in between. We extend the single relaxation time LB method to model multiphase, immiscible flow (e.g., gas and liquid or water and oil) in a semi-porous medium. We compare the solution to test cases and find good agreement of the model as compared to analytical solutions. We then apply the model to real porous media and recover both capillary and viscous flow regimes. However, some deficiencies in the single relaxation time LB method applied to multiphase flow are uncovered and we describe methods to overcome these limitations.     

Boundary layer problem: Navier–Stokes equations and Euler equations

This work is concerned with the boundary layer turbulence, which is an outstanding problem in fluid mechanics. We consider an incompressible viscous fluid in 2D domains with permeable walls. The permeability is described by the Yudovich condition. The goal of the article is to study the fluid behavior at vanishing viscosity (large Reynold’s numbers). We show that the vanishing viscous limit is a solution of the Euler equations with the Yudovich condition on the inflow region of the boundary.     

Non-steady interaction of fluids with free surfaces and rigid bodies

Interaction between fluids and rigid bodies is a phenomenon, which occurs for example in floating bearings or turbines. Mostly the focus is on domains with rigid boundaries on every side or defined influx or effusion of fluid over the boundaries. The interaction between fluids with free surfaces and a rigid body were mostly studied under the aspect of stability for steady-state conditions, e. g. for fluid-filled centrifuges. A characteristic property is the instability over non-empty intervalls of angular velocities, so the analysis of non-steady behaviour is essential to investigate the stability of the drive through these instable domains. A first approach to this topic is the qualitative investigation of a fluid domain with the shallow water theory. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

考虑毛管形状的幂律流体等效渗透率计算方法(英文)

While studying the flow of oil and gas in the reservoir, it is not realistic that capillary with circular section is only used to express the pores. It is more representative to simulate porous media pore with kinds of capillary with triangle or rectangle section etc. In the condition of the same diameter, when polymer for oil displacement flows in the porous medium, there only exists shear flow which can be expressed with power law model. Based on fluid flow-pressure drop equation in single capillary, this paper gives a calculation method of equivalent permeability of power law fluid of single capillary and capillary bundles with different sections.     

Identification of a coefficient in the leading term of a pseudoparabolic equation of filtration

The identification problem of a leading coefficient in a linear pseudoparabolic equation is examined under an integral overdetermination condition on the boundary of a domain. We prove a local existence and uniqueness theorem of a strong solution. In a particular case the results obtained allow us to solve a problem of determining the rate of fluid exchange between fissures and pores in a model filtration equation for a fissured medium.     

Modelling of displacement washing of pulp fibers using orthogonal collocation on finite elements

Flow of fluid through packed bed of porous particles is modelled with the help of Peclet number (Pe) and Biot number (Bi). Packed bed is divided into three zones, flowing liquor, intrapore solute present in pores of particles and solute adsorbed on particle surface. Langmuir isotherm is used to describe the relationship between intrapore solute concentration and concentration of solute adsorbed on particle surface, whereas the bulk fluid concentration and the intrapore solute concentration are related by linear adsorption isotherm. Model predicted values are also compared with the experimental values. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Measure-valued solution for non-Newtonian compressible isothermal monopolar fluid

The global existence of measure-valued solutions of initial boundary-value problems in bounded domains to systems of partial differential equations for viscous non-Newtonian isothermal compressible monopolar fluid and the global existence of the weak solution for multipolar fluid is proved.     

Modeling and analysis of reactive solute transport in deformable channels with wall adsorption–desorption

We show well posedness for a model of nonlinear reactive transport of chemical in a deformable channel. The channel walls deform due to fluid–structure interaction between an unsteady flow of an incompressible, viscous fluid inside the channel and elastic channel walls. Chemical solutes, which are dissolved in the viscous, incompressible fluid, satisfy a convection–diffusion equation in the bulk fluid, while on the deforming walls, the solutes undergo nonlinear adsorption–desorption physico‐chemical reactions. The problem addresses scenarios that arise, for example, in studies of drug transport in blood vessels. We show the existence of a unique weak solution with solute concentrations that are non‐negative for all times. The analysis of the problem is carried out in the context of semi‐linear parabolic PDEs on moving domains. The arbitrary Lagrangian–Eulerian approach is used to address the domain movement, and the Galerkin method with the Picard–Lindelöf theorem is used to prove existence and uniqueness of approximate solutions. Energy estimates combined with the compactness arguments based on the Aubin–Lions lemma are used to prove convergence of the approximating sequences to the unique weak solution of the problem. It is shown that the solution satisfies the positivity property, that is, that the density of the solute remains non‐negative at all times, as long as the prescribed fluid domain motion is ‘reasonable’. This is the first well‐posedness result for reactive transport problems defined on moving domains of this type. Copyright © 2015 John Wiley & Sons, Ltd.     

Application of p-Adic Wavelets to Model Reaction–Diffusion Dynamics in Random Porous Media

Fourier and more generally wavelet analysis over the fields of p-adic numbers are widely used in physics, biology and cognitive science, and recently in geophysics. In this note we present a model of the reaction–diffusion dynamics in random porous media, e.g., flow of fluid (oil, water or emulsion) in a a complex network of pores with known topology. Anomalous diffusion in the model is represented by the system of two equations of reaction–diffusion type, for the part of fluid not bound to solid’s interface (e.g., free oil) and for the part bound to solid’s interface (e.g., solids–bound oil). Our model is based on the p-adic (treelike) representation of pore-networks. We present the system of two p-adic reaction–diffusion equations describing propagation of fluid in networks of pores in random media and find its stationary solutions by using theory of p-adic wavelets. The use of p-adic wavelets (generalizing classical wavelet theory) gives a possibility to find the stationary solution in the analytic form which is typically impossible for anomalous diffusion in the standard representation based on the real numbers.     

On viscous incompressible flows of nonsymmetric fluids with mixed boundary conditions

In this paper we are concerned with the initial boundary value problem for the micropolar fluid system in nonsmooth domains with mixed boundary conditions. The considered boundary conditions are of two types: Navier’s slip conditions on solid surfaces and Neumann-type boundary conditions on free surfaces. The Dirichlet boundary condition for the microrotation of the fluid is commonly used in practice. However, the well-posedness of problems with different types of boundary conditions for microrotation are completely unexplored. The present paper is devoted to the proof of the existence, regularity and uniqueness of the solution in distribution spaces.     

On the use of vorticity-vector-potential with a spectral tau method in rotating annular domains

A new approach for the resolution of three-dimensional Navier-Stokes and energy equations is presented, using the vorticity-vector-potential formulation for multiply connected domains. The dependent variables are solved in a complex form to uncouple the equations in radial and azimuthal directions. A spectral tau method, with Chebyshev polynomials in radial and axial directions and Fourier series in the azimuthal one, is proposed for the prediction of flow structures and heat transfer of a Boussinesq fluid inside rotating annular domains. The integration in time utilizes a combination of second-order backward Euler and Adams-Bashforth schemes. The first results concern the stokes problem.     

On the mathematical transport theory in microporous media: The billiard approach

This paper is an expository of the main dynamical properties of billiards, which depend on the shape of the walls of the container, and the recent developments like the introduction of an external field, which mimic the coupling with a thermostat.The class of dynamical system dealt with in this paper exhibits characteristics of hybrid systems as it links discrete and continuous, deterministic and stochastic dynamics.The contents are focused on applications. Specifically, transport dynamics in highly-confined regions has been of interest in the last few decades because of industrial and medical applications. Aspects of confined transport remain elusive, considering that in microporous membranes, whose size pores is about that of the molecules, the transport is sometimes ballistic, and sometimes diffusive. The classical kinetic and macroscopic approach can not be directly applied because collisions of particle fluid with walls prevail. The microscopic mathematical billiard theory can be applied as a mathematical tool since the interstices between obstacles can be considered as the pores of the membranes.     

Toroidal magnetodynamic stability of cylindrical fluid jet

The hydromagnetic stability of a fluid jet under the combined influence of the electromagnetic (with toroidal varying field) and capillary forces has been developed. A general dispersion relation valid for all modes of perturbation is derived. The magnetic fields interior and exterior the fluid jet are stabilizing. The capillary force is destabilizing for small domain in the axisymmetric mode while it is stabilizing for all the rest. The magnetic fields increase the capillary stable domains and at the same time cause the shrinking those of instability. Above a certain value of the relative magnetic field strength, the capillary instability is depressed and then stability sets in. In spite of the field interior the fluid is non-uniform, however, we found that it has a strong stabilizing influence.