An inverse problem arising in fluid flow in a porous medium

In testing samples from a large inhomogeneous porous bed, it is necessary to determine the permeabilities of the samples. This leads to an elliptic partial differential equation with constant coefficients, in a circle, with mixed boundary conditions given on the perimeter. An extra piece of information (the total flow of fluid though the sample under a fixed pressure difference) is given and the problem requires the determination of the coefficients. This is then an inverse problem and an attempt is made to show how the coefficients may be found.     

Incompressible fluid flow and heat transfer through a nonsaturated porous medium

This work studies a nonsaturated flow and the heat transfer associated phenomenon of a newtonian fluid through a rigid porous matrix, using a mixture theory approach in its modelling. The mixture consists of three overlapping continuous constituents: a solid (porous medium), a liquid and an inert gas, included to account for the compressibility of the system as a whole. A set of four nonlinear partial differential equations describe the problem whose hydrodynamical part is approximated by means of a Glimm's scheme combined with an operator splitting technique.     

Polar fluid through a porous medium

Summary An analysis of the steady flow of a polar fluid through a porous medium bounded by an infinite plate by using the generalized Forchheimer's model is considered. The velocity profiles are shown for different values of the permeability parameter.     

A numerical model for immiscible two-phase fluid flow in a porous medium and its time domain solution

The governing equations for the interaction of two immiscible fluids within a deforming porous medium are formulated on the basis of generalized Biot theory. The displacement of the solid skeleton, the pressure and saturation of wetting fluid are taken as primary unknowns of the model. The finite element method is applied to discretize the governing eqations in space. The time domain numerical solution to the coupled problem is achieved by using an unconditionally stable direct integration procedure. Examples are presented to illustrate the performance and capability of the approach.     

Start-up flow in a porous medium channel

Summary The transient fluid motion in a porous medium channel is considered. Frictional resistance induced by the solid matrix and the channel walls is accounted for by a Darcian body force and a viscous shear stress, respectively. The adopted mathematical model leads to a one-parameter problem, in which the channel half-widthh, the porosity and the permeabilityK combine into a shape parameterA=(h ²/K)^1/2. Exact analytical solutions in terms of infinite series expansions are provided both for the start-up flow following the sudden imposition of a constant pressure gradient and for the transient motion induced by an instantaneously imposed flow rate. Time histories of the centerline velocity and the wall friction are presented, together with time-varying velocity profiles. It is observed that the start-up time required to reach a steady state is significantly reduced in the less porous channels, and this reduction is more pronounced when the start-up flow is driven by a pressure gradient than if the transient motion is forced by an imposed flow rate.     

Multiphase flow in a capillary porous medium

Based on the theory of porous media, a calculation concept for the multiphase flow in a capillary porous medium will be presented. We will exclusively investigate the rise of liquids in porous bodies due to the capillarity phenomenon. The field equations used consist of the mechanical balance equations and the physical constraint conditions. The treatment of the capillary problem, based on thermomechanical investigations, yields the result that the capillarity force is a volume interaction force and depends on the free Helmholtz energy functions of the phases and the density gradient of the liquid. With respect to the aforementioned outcome, further constitutive relations for the ternary model are developed. The aim of this investigation is the numerical simulation of the behavior of liquid and gas phases in a rigid porous body at rest. Therefore, the needed weak formulations of the governing field equations, i.e. the balance equations of mass and the balance equations of momentum of the liquid and gas phases, will be given. The usefulness of the proposed theory will be demonstrated with an example.     

A two-scale model for fluid flow in an unsaturated porous medium with cohesive cracks

A two-scale model is developed for fluid flow in a deforming, unsaturated and progressively fracturing porous medium. At the microscale, the flow in the cohesive crack is modelled using Darcy’s relation for fluid flow in a porous medium, taking into account changes in the permeability due to the progressive damage evolution inside the cohesive zone. From the micromechanics of the flow in the cavity, identities are derived that couple the local momentum and the mass balances to the governing equations for an unsaturated porous medium, which are assumed to hold on the macroscopic scale. The finite element equations are derived for this two-scale approach and integrated over time. By exploiting the partition-of-unity property of the finite element shape functions, the position and direction of the fractures are independent from the underlying discretization. The resulting discrete equations are nonlinear due to the cohesive crack model and the nonlinearity of the coupling terms. A consistent linearization is given for use within a Newton–Raphson iterative procedure. Finally, examples are given to show the versatility and the efficiency of the approach. The calculations indicate that the evolving cohesive cracks can have a significant influence on the fluid flow and vice versa.     

Filtration of a magnetic fluid in a deformable porous medium

The equations of motion of a magnetizing fluid are obtained in a deformable non-magnetic porous medium.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 48, No. 1, pp. 49–54, January, 1985.     

Magnetohydrodynamic poiseuille flow through a porous medium

Abstract

This paper is concerned with formulating equations for the flow of an electrically conducting fluid through a non‐conducting porous medium with non‐porous and non‐conducting boundaries. Equations are developed for the general case of the flow of a solid‐fluid suspensions; flow through a porous medium is treated as a special case by letting the velocity of the particle phase goes to zero. Two cases are considered. Exact solution is obtained for the case of flow between parallel plates, but for flow in pipes of square and circular cross sections, the equations have to be solved numerically. The numerical technique developed can treat elliptical cross sections as well. The flow in all cases is assumed to be steady, laminar, incompressible, viscous, and fully developed. The results are presented in terms of a parameter which measures the resistance of the porous medium.     

Heat transfer in unsteady flow through a porous medium between two infinite parallel plates in relative motion

An approximate solution to the heat transfer in a flow of a viscous incompressible fluid through a porous medium bounded by two infinite parallel plates, the lower one stationary and the upper one oscillating in its own plane, is presented. Expressions for the mean temperature, the amplitude, and phase of the first and second harmonic of the rate of heat transfer and the mean rate of heat transfer are derived. The mean temperature is shown on graphs and the numerical values of the amplitudes and the phase are entered in a table. It is observed that the mean rate of heat transfer decreases with more ease of percolation but increases with increasing the frequency ω.     

Free convection effects on the oscillatory flow of a couple stress fluid through a porous medium

Summary Effects of free convection currents on the oscillatory flow of a polar fluid through a porous medium, which is bounded by a vertical plane surface of constant temperature, have been studied. The surface absorbs the fluid with a constant suction and the free stream velocity oscillates about a constant mean value. Analytical expressions for the velocity and the angular velocity fields have been obtained, using the regular perturbation technique. The effects of Grashof numberG; material parameters and ; Prandtl numberP; permeability parameterK and frequency parametern on the velocity and the angular velocity are discussed. The effects of cooling and heating of a polar fluid compared to a Newtonian fluid have also been discussed. The velocity of a polar fluid is found to decrease as compared to the Newtonian fluid.List of symbols C _p specific heat at constant pressure - g acceleration due to gravity - G Grashof number - K ⁺ permeability of the porous medium - K dimensionless permeability - P Prandtl number - t ⁺ time - t dimensionless time - T _w ⁺ mean temperature of the surface - T ⁺ temperature of the fluid - T ⁺ temperature of the fluid away from the surface - density of the fluid - viscosity - _r rotational viscosity - C _a ,C _d coefficients of couple stress viscosities - I a scalar constant of dimension equal to that of the moment of inertia of unit mass - x ⁺,y ⁺ coordinate system - u ⁺,v ⁺ velocity components in thex ⁺ andy ⁺ directions - u dimensionless velocity in thex ⁺-direction - ⁺ angular velocity component - dimensionless angular velocity - n ⁺ frequency of oscillations - n dimensionless frequency - perturbation parameter - U a constant velocity - u ₀ mean velocity - u ₁ fluctuating part of the velocity - ⁰ mean angular velocity - ¹ fluctuating part of the angular velocity - T ₀ mean temperature - T ₁ fluctuating part of the temperature - ⁰ coefficient of the volume expansion - kinematic viscosity - ^r rotational kinematic viscosity - , material parameters characterizing the polarity of the fluid - v ₀ suction velocity - density of the fluid far from the surface - y dimensionless coordinate normal to the surface     

State space formulation to viscoelastic fluid flow of magnetohydrodynamic free convection through a porous medium

Summary In this work we formulate the state space approach for one-dimensional problems of viscoelastic magnetohydrodynamic unsteady free convection flow through a porous medium past an infinite vertical plate. Laplace transform techniques are used. The resulting formulation is applied to a thermal shock problem and to a problem for the flow between two parallel fixed plates both without heat sources. Also a problem with a distribution of heat sources is considered. A numerical method is employed for the inversion of the Laplace transforms. Numerical results are given and illustrated graphically for the problem considered.Notation C specific heat at constant pressure - g acceleration due to gravity - density - time - u velocity component parallel to the plate - H _x induced magnetic field - x, y coordinates system - T temperature distribution - T _o temperature of the plate - T temperature of the fluid away from the plate - ₀ limiting viscosity at small rates to shear - v _o ^* / - v _m magnetic diffusivity - Alfven velocity - ^* coefficient of volume expansion - thermal conductivity - ^* thermal diffusivity - G Grashof number - Pr Prandtl number - L some characteristic length - k _o the elastic constant - K permeability of the porous medium     

On the electro-osmotic flow in a saturated porous medium

Electro-osmosis in a saturated porous rigid medium is studied by both theoretical and experimental approach. Relationships giving liquid and electrical coupled macroscopic flows are obtained from the homogeneization method. They point out possible non symmetrical cross effects that explain deviations from Onsager reciprocal relations. Experiments on a kaolinite are reported which exhibit an important non symmetrical coupling.     

Identification of the coefficients in the equations of motion for a fluid-saturated porous medium

Summary Main aim of the paper is a detailed interpretation of the mass coupling coefficient occurring in the equation of motion for a fluid-saturated porous medium. Biot's (1956) and Darski's (1978) equations are discussed. A new kinematical model is presented which allows transformation of Derski's equations into those of Biot (and vice versa). The interpretation of their coefficients is given in detail and boundary conditions for these equations are discussed.With 5 Figures     

Unsteady free convective flow through a porous medium

Unsteady 2-dimensional free-convective flow through a porous medium bounded by an infinite vertical plate is considered, when the temperature of the plate is oscillating with the time about a constant non-zero mean value. An analytical solution for the velocity field is derived and the effects of K (permeability parameter) and ω (parameter of frequency) on the velocity field are discussed.     

Free convective oscillatory flow of a polar fluid through a porous medium in the presence of oscillating suction and temperature

The aim of this work is to analyze a two-dimensional oscillatory free convective flow of an incompressible polar fluid through a porous medium bounded by an infinite vertical porous plate with oscillating suction and temperature at the wall. The governing equations are based on the volume averaging technique. Analytical expressions for the velocity, angular velocity, and temperature fields are obtained by using the regular perturbation technique. The analysis reveals a multiple boundary layer structure near the wall for the fields mentioned.     

Determination of the macrodispersive parameters of a motion in a two-layer porous medium

Summary This paper develops a method for deriving good estimates of macrodispersive transport parameters describing the asymptotic evolution of a non-reactive chemical pollutant injected into a two-layer porous medium. These parameters are extracted from the coefficients of asymptotic time-polynomial expansions of some adequately chosen integral transforms performed upon the solution of the original transport problem.     

Nonsteady flow of non-Newtonian fluids through a porous medium

In this paper, the nonsteady flow of power-law fluids with a yield stress through a porous medium is investigated. In order to illustrate the rheological behaviour effects on pressure and flow rate distributions, a flow system of practical interest was analysed. The approximate solutions in a closed form were obtained by using the integral method. For a short time, these have been formulated in terms of a moving boundary problem. The flow deviation from Newtonian behaviour was emphasized by means of several dimensionless groups. These have been found to be relevant in evaluating the rheological effects on steady and nonsteady flow behaviour through a porous medium.     

Dynamic behaviour of a porous medium saturated by a newtonian fluid

The macroscopic dynamical behaviour of a porous elastic medium the cavities of which are filled with a Navier-Stokes liquid can be described with equations obtained from certain assumptions and the homogenization method, based on the use of periodic solutions of space variables. The so-determined linear behaviour is a generalization of Biot and Levy works: particularly, strain rates of the solid are taken into account for the flow law of the liquid. Then, some special relationships are determined for coefficients which are involved in the macroscopic equations and determined by particular solutions of the boundary problem at the pore scale.