A coupled solid/fluids mixture theory that suffices for diffusion problems

In this paper, I begin with the general formulation of mixture theory by Bowen and present the derivation of a minimal set of field equations, constitutive relations, and material parameters suitable for the solutions of meaningful diffusion problems. The specific results are for a single solid and two fluids, and they may be extended to any number of fluids. I allude to the results of three problems, viz. (1) the injection of a fluid into a geological formation saturated with another fluid, (2) the drainage of two dissimilar fluids from a geological formation due to in-situ fluid pore pressures, and (3) the process of squeezing a sponge dry, in order to illustrate the general applicability of the derived theory.     

Analysis of the squeezing flow of dusty fluids

The current investigation deals with the study of the effect of introducing a small fraction of dust, by volume, to the fluid in a squeeze film on the viscous resistance to a steady moving disc. Expressions are obtained for the fluid-phase and the dust-phase velocity distributions and the dust particle number density. Analysis based on an iterative procedure indicates that the resistance to motion experienced by the moving disc increases due to the presence of dust.Nomenclature A arbitrary function of integration - B bulk concentration - F resistance to motion experienced by the disc (dusty fluid case) - F _c resistance to motion experienced by the disc (clean fluid case) - F* difference in resistance between the clean fluid and dusty fluid films - f mass concentration - h thickness of the squeeze film - K Stokes coefficient of resistance - m mass of a single dust particle - fluid viscosity coefficient - N dust particles number density - N ₀ dust particles number density at r=R - n iteration level - p fluid pressure in the squeeze film - P pressure in the surrounding - R radius of the disc - fluid density - (r, , y) cylindrical coordinates - t time - U fluid-phase velocity vector - V dust-phase velocity vector - ₁ fluid-phase radial velocity component - U ₂ dust-phase radial velocity component     

Orientation effects in extensional and squeezing flows of certain anisotropic fluids

The Ericksen-Leslie continuum theory of anisotropic fluids is here used to examine the behaviour of the orientation pattern within the bulk of a fluid that is undergoing an extensional-type of flow. The flow (which is irrotational and generally unsteady) is considered to be generated by application of prescribed normal stresses, and the orientation pattern (represented by a director field n) is taken to be spatially homogeneous but time-dependent. By means of a phase-plane analysis it is shown that, in stretching flows, the director eventually aligns parallel to the direction of imposed stretch, whereas in squeezing-type flows it eventually lies in the plane normal to the direction of squeezing. In both cases the lateral components of n may vary non-monotonically in time, before approaching their asymptotic values; also the lateral components of velocity may change sign during flow.A two-parameter classification is given of all possible modes of behaviour of these model fluids in these flows. Also analytical solutions are obtained for certain special cases, such as axisymmetric flow.     

A theory of hydroshock in a two-phase gas-liquid mixture

A hydroshock in a two-phase gas-liquid mixture is usually calculated by an analog of the Zhukovskii formula for a mixture (see, for example, [1, 2]) which establishes the relation of the hydro-shock intensity to the velocity of sound in the mixture. However, as the experimental data [1] show, the hydroshock intensity in the mixture can significantly exceed the calculated values, a fact which is explained by the increase in the propagation velocity of the perturbation wave in comparison to the velocity of sound in the mixture [3, 4]. In the present paper, an equilibrium model of shock transition, similar to [5, 6], is used to calculate the attenuation of the hydroshock as the gas content of the mixture increases in a bubble flow regime. It is shown that owing to the high compressibility of the mixture the effect of the elasticity of the pipe-line walls is small, and the dependence of the propagation velocity of the perturbation wave on the intensity and gas content becomes the main effect. A simple dependence of the hydroshock intensity on the gas content and two similarity parameters is obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 202–204, September–October, 1984.     

Variational formulation of pre-stressed solid–fluid mixture theory,with an application to wave phenomena

Fluid saturated porous media are modelled by the theory of mixtures and the placement maps of the solid and of the fluid are considered. The momentum balance equations are derived in the framework of a variational approach: We take an action functional and two families of variations and assume that the sum of the virtual work of the external forces and the variation of such an action along each variation are zero. Constitutive equations for the two Cauchy stress tensors and for the interaction force are derived taking into account a general state of pre-stress for the solid and for the fluid species. Governing equations are therefore formulated, however, for the sake of simplicity, only the case of pure initial pressure is further investigated. The propagation of bulk (transversal and longitudinal) waves and the influence of pre-stress are studied: In particular, stability analyses are carried out starting from dispersion relations and the role of pre-stress is investigated. Finally, a numerical example is established for a given state of pre-stress, deriving the phase velocities and the attenuation coefficients of transversal and longitudinal waves.     

Forced acoustical response of a cavity coupled with a semi-infinite space using coupled mode theory

This paper presents a study of the forced acoustical response of an open cavity coupled with a semi-infinite space by using acoustic coupled mode theory, an approach analogous to effective non-Hermitian Hamiltonian method. The reduced differential operator of the open cavities and associated frequency-dependent basis functions are obtained in order to describe the sound field in terms of modal expansion. The properties and physical behavior of this expansion are numerically investigated and explained using computational examples.     

A stability theorem for a mixture of two viscous fluids at different temperatures

Archive for Rational Mechanics and Analysis -     

Asymptotic theory of a wing moving near a solid wall

In this study we use the method of matched asymptotic expansions to obtain an approximate solution of the problem of the nonstationary motion of a lifting surface near a solid wall. The region of flow is provisionally subdivided into characteristic zones, in which, using the appropriate coordinates, we construct asymptotic expansions for the velocity potential, which thereafter coalesce in the regions of common validity. In the first approximation (extremely small heights of flight) the problem reduces to the solution of a Poisson equation in a plane region bounded by the contour of the wing in the horizontal plane with boundary conditions established from the coalescence.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 115–124, November–December, 1977.     

A universal relation in torsion for a mixture of solid and fluid

In his study of combined finite extension and torsion of a nonlinear, incompressible, isotropic elastic circular cylinder, Rivlin [1] established a relation for the torsional stiffness which depends only on the axial force, the axial extension ratio and the radius of the undeformed cylinder, in the case of small twist. The relationship did not depend on the structure of the stored energy function and is hence a universal relation. In this paper, we extend Rivlin's result to the case of combined extension and torsion of a cylindrical mixture of a nonlinear elastic solid and fluid.     

Hydroshock theory in a two-phase gas-liquid mixture in a slug flow regime

A study is made of the intensity of a hydroshock in a two-phase gas-liquid mixture in a slug flow regime in the case when a pipeline is shut off by a liquid slug. The intensity is studied as a function of the length of the shut-off section of the liquid slug, the content of gas bubbles in the liquid slugs, and the pipeline shut-off law, and with allowance for the shock-wave character of the process [1, 2]. The calculated data using the shock-wave theory agree well with the experimental data of [3] and, unlike the results of the linear theory of [3], make it possible to determine the intensity of the hydroshock not only in the case of weak waves, but also in the case of waves of moderate intensity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 188–190, September–October, 1985.