Some uniqueness and continuous dependence results in micropolar fluid dynamics

Thanks to a preliminary lemma, which is a strengthened version of an analogous one given in a previous article [1], some uniqueness theorems for micropolar incompressible flows are derived. These results seem to improve the previous ones even in the classical scheme of Newtonian fluids. Subsequently, some continuous dependence theorems are given by extending the methods previously introduced in [2, 3].     

Some uniqueness and continuous dependence results in the theory of swelling porous elastic soils

This paper is concerned with the isothermal linear theory of swelling porous elastic soils. Initial-boundary value problems are formulated for the linear dynamic theory of an isothermal mixture consisting of three components: an elastic solid, a viscous fluid and a gas. Then the uniqueness and continuous dependence problems are discussed in connection with the solutions of such initial-boundary value problems. The uniqueness results are established under mild positive semi-definiteness assumptions or with no definiteness assumptions upon the internal energy. Various estimates are established for describing the continuous dependence of solutions with respect to the external given data. In this aim the Lagrange identity and the logarithmic convexity methods are used.     

Wave propagation in micropolar mixture of porous media

This paper is concerned with the possible propagation of waves in an infinite porous continuum consisting of a micropolar elastic solid and a micropolar viscous fluid. Micropolar mixture theory of porous media developed by Eringen [A.C. Eringen, Micropolar mixture theory of porous media, J. Appl. Phys. 94 (2003) 4184–4190] is employed. It is found that there exist four coupled longitudinal waves (two coupled longitudinal displacement waves and two coupled longitudinal microrotational waves) and six coupled transverse waves in a continuum of this micropolar mixture. All the waves are found to attenuate and dispersive in nature. A problem of reflection of coupled longitudinal waves from a free boundary surface of a half-space consisting the mixture of a micropolar elastic solid and Newtonian liquid, is investigated. The expressions of various amplitude ratios and surface responses are derived. Numerical computations are performed to find out the phase velocity and attenuation of the waves. The variation of amplitude ratios, energy ratios and surface responses are also computed for a specific model. All the numerical results are depicted graphically. Some limiting cases have also been discussed.     

On uniqueness and continuous dependence for a linear micropolar fluid

In this paper logarithmic convexity arguments are employed to establish the uniqueness and Holder continuous dependence on initial data of a certain class of solutions of an initial boundary value problem for a linear micropolar fluid.     

On the uniqueness and continuous data dependence of solutions in the theory of swelling porous thermoelastic soils

This paper studies the uniqueness and continuous data dependence of solutions of the initial-boundary value problems associated with the linear theory of swelling porous thermoelastic soils. The formulation belongs to the theory of mixtures for porous elastic solids filled with fluid and gas with thermal conduction and by considering the time derivative of temperature as a variable in the set of constitutive equations. Some uniqueness and continuous data dependence results are established under mild assumptions on the constitutive constants. Thus, it is shown that the general approach of swelling porous thermoelastic soils is well posed. The method of proof is based on some integro-differential inequalities and some Lagrange–Brun identities.     

On the uniqueness and existence of Stokes flows in micropolar fluid theory

Summary The present paper addresses itself to the problem of existence and uniqueness of solutions of Stokes flows in micropolar fluid theory using the methods of potential theory. Integral representations for the velocity and microrotation vectors are derived and they lead naturally to the introduction of single-layer and double-layer potentials whose properties are stated. With the aid of these results, necessary and sufficient conditions are generated for the Stokes problem in this microcontinuum fluid mechanics theory.

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Zur Eindeutigkeit und Existenz von Stokesschen Strömungen mikropolarer Flüssigkeiten

Zusammenfassung Die gegenständliche Arbeit befaßt sich mit der Existenz und Eindeutigkeit der Lösungen Stokesscher Strömungen mikropolarer Flüssigkeiten unter Verwendung der Methoden der Potentialtheorie. Integraldarstellungen werden für die Geschwindigkeits-und Mikrorotationsvektoren abgeleitet. Diese führen zur Einführung von Einschicht-und Zweischichtpotentialen, deren Eigenschaften festgelegt werden. Mit Hilfe dieser Ergebnisse werden notwendige und hinreichende Bedingungen für das Stokes-problem der Mikrokontinuumsflüssigkeitsmechanik abgeleitet.

     

Some contributions to the theory of flow of fluids through porous deformable media

Summary The present paper is based onBiot's theory of flow of fluids through a porous deformable medium. The presence of a distribution of sources of fluids in the medium gives rise to an additional term in the generalizedDarcy's law. The present note is concerned with solutions of the displacement equations ofBiot's theory for the case of fluid sources. The regular solution of these equations is given in terms of special functions. The singular solution — connected with the action of fluid sources — is obtained with the aid of the principle of reciprocity of displacements.

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Zusammenfassung Die vorliegende Arbeit benützt dieBiotsche Theorie der Flüssigkeitsströmung durch ein poröses deformierbares Medium. Das Vorhandensein einer Quellverteilung gibt Anlaß zu einem zusätzlichen Glied im verallgemeinertenDorcyschen Gesetz. Für diesen Fall werden Lösungen derBiotschen Verschiebungs-gleichungen angegeben. Die reguläre Lösung wird in speziellen Funktionen angegeben. Die singuläre Lösung — mit den Flüssigkeitsquellen verknüpft — wird mit Hilfe der Reziproxität der Verschiebungen erhalten.

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With 1 Figure     

Existence and uniqueness of the traveling front in premixed combustion of porous media

We study a mathematical model of combustion processes in an inert porous media filled with a combustible gaseous mixture. We focus on the phenomenon of a combustion wave driven by a local pressure elevation. In this article, we are concerned with subsonic pressure-driven flames and with the case of a quadratic dependence of the friction force on the velocity of the gaseous mixture. After a suitable non-dimensionalization, the resulting mathematical model includes three nonlinear ordinary differential equations (ODEs). The system contains an unknown parameter V that represents the traveling wave speed. The existence of the traveling wave is proven in this study. It means that the parameter V can be chosen so that the corresponding phase trajectory satisfies the boundary conditions. Moreover, under reasonable assumptions about the monotonicity of the flame front, we prove the uniqueness of the pressure-driven wave.     

Some new results in porous silicon

Efforts have been made to see the effect of some standard microelectronic processing steps on porous silicon. Our diffusion experiments for making p-n junctions confirm that this material can withstand high temperatures of the order of 800°C to 1000°C. A new technique for photolithography has been suggested to obtain porous silicon in selected areas. Etch stop method to control the thickness of the porous layer and an organic protective layer for porous silicon have also been suggested. Models proposed by other workers to explain luminescence in porous silicon are not sufficient to explain many experimental observations. A hybrid model is suggested.     

Some peculiarities of heat exchange in porous media

Results are presented of investigations of the heat-exchange intensity in a porous material as a function of the coolant properties.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 20, No. 4, pp. 588–591, April, 1971.     

Existence and uniqueness result for two-dimensional porous media flows with porous inclusions based on Brinkman equation

The purpose of this paper is to use the potential theory in order to obtain the existence and uniqueness result of the classical solution to a boundary value problem which describes the flow of an unbounded viscous incompressible fluid in the presence of a porous body embedded in another porous medium.     

Some results in the dynamics of porous elastic mixtures

The present work describes a method for studying the spatial and temporal behaviour of the dynamic processes in porous elastic mixtures taking into account memory effects for the intrinsic equilibrated body forces. The method is based on a set of properties for an appropriate time-weighted surface power function associated with the process at issue. We get the domain of influence and the spatial decay estimates with time-independent decay rate inside the domain of influence. We establish the spatial decay estimates with time-independent decay rate for the interior of the domain of influence. Using the Cesáro means associated with the kinetic and strain energies, we establish the asymptotic partition of the total energy.     

Dynamics of acoustic waves in porous media saturated with gas-vapor mixture

Wave processes in wet porous media saturated with gas-vapor mixture have been studied with allowance for interphase interaction forces and heat and mass exchange between the skeleton of the porous medium and the gas-vapor mixture. Dispersion relations have been obtained for a porous medium saturated with gas-vapor mixture, vapor, or gas.     

A reciprocal theorem for a mixture of interacting continuous media

Using a linearized theory of interacting continuous media composed of linear elastic solid and linear viscous fluid, a reciprocal theorem is derived which relates the solution of one initial-boundary value problem to another. As an illustration of the reciprocal theorem, we obtain the early time displacement and velocity field of a mixture occupying an infinite region and subjected to an impulsively applied moving point load acting on the solid constituent.     

Compressible porous media models by use of the theory of mixtures

This article utilizes the usual formalism of the theory of mixtures to formulate compressible porous media models. The formulation allows for the effects of immiscibility and variable volume fractions. In addition, it allows for the effects of pore relaxation by treating the volume fractions as internal state variables. An attempt is made to relate the general results to classical porous media models. In particular, circumstances are presented which allow one to properly define a pore pressure for each fluid constituent. This article also contains the linear poroelasticity model which results from a formal linearization of the field and constitutive equations. One brief section is devoted to the specialization of this linear model to the one first proposed by Biot.