Convective instability of a fluid in two-layer saturated strata

The problem of convective instability of a fluid in a system consisting of two horizontal porous strata with different permeabilities and a permeable common boundary is considered. The problem is investigated in parametric form as a function of the stratum thickness ratio and stratum permeabilities. As distinct from a uniform stratum, in this case the neutral curve can have one or two minima depending on the relationship between the parameters. The case of two minima is characterized by the condition of loss of stability of the fluid in the system as a whole and in the thinner stratum with greater permeability. These minima correspond to significantly different wave numbers. Makhachkala. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 165–169, January–February, 1999.     

Law of flow of a viscoplastic fluid through a porous medium with allowance for inertial losses

A macroscopic law of flow of a viscoplastic Schwedoff-Bingham fluid through a porous medium is obtained on the basis of percolation theory with allowance for viscous and inertial losses. The asymptotics of the flow law are estimated and expressions for determining the limiting pressure gradient as a function of the microinhomogeneity parameters are given. Satisfactory qualitative agreement between the theoretical and known experimental data is observed. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 68–73, January–February, 1999.     

Stability of a nonuniformly heated fluid in a porous horizontal layer

We investigate the stability of a nonuniformly heated fluid in the gravitational field in a plane horizontal porous layer through which vertical forced motion is effected. A similar system was studied in [1, 2]. In the present paper, the nonuniformity of the permeability of the porous layer with respect to the depth and the dependence of the viscosity of the saturating fluid on the temperature are taken into account in addition. As a result of the application of the linear stability theory, an eigenvalue problem arises, which is solved numerically. A family of curves representing the dependence of the critical modified Rayleigh number Ra _k ^⋆ on the injection parameter (the Péclet number Pe) for different degrees of inhomogeneity of the permeability and the viscosity is obtained. It is found that although Pe=0 corresponds to Ra _k ^⋆ for uniform permeability and viscosity and the stability increases monotonically as Pe increases, the presence of nonuniformity of the permeability or the viscosity leads to the appearance of a stability minimum in the region Pe≈1, while under the simultaneous influence of these two factors, the minimum is shifted into the region Pe≈2. The results of the paper can be used, for example, in the investigation of heat transfer in the case of forced fluid motion in the fissures of a permeable rock mass, when, in the case of pumping through a horizontal fissure, the fluid penetrates vertically across its permeable walls into the stratum. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 3–7, November–December, 1986.     

Crack opening regimes associated with the injection of fluid into a fractured porous medium

The development of the crack opening process and the dimensions of the open-crack zone are determined by the dynamics of the pressure variation in the injected fluid. Peaking regimes, corresponding to the unbounded growth of one of the characteristics of the process in a finite time, are of special practical interest. These regimes are examined within the framework of the nonlinear one-dimensional problem on the basis of a continuum model of flow through fractured porous media. Sverdlovsk. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 115–120, September–October, 1988.     

A colorimetric reaction to quantify fluid mixing

We found the colorimetric reaction of Tiron (1,2-dihydroxybenzene-3,5-disulfonic acid) and molybdate suitable for optical quantification of chemical reaction during fluid–fluid mixing in laboratory chambers. This reaction consists of two colorless reagents that mix to rapidly form colored, stable, soluble products. These products can be digitally imaged and quantified using light absorbance to study fluid–fluid mixing. Here we provide a model and equilibrium constants for the relevant complexation reactions. We also provide methods for relating light absorbance to product concentrations. Practical implementation issues of this reaction are discussed and an example of imaged absorbances for fluid–fluid mixing in heterogeneous porous media is given.     

The effect of a transition layer between a fluid and a porous medium: shear flow in a channel

Flow in a three-layer channel is modeled analytically. The channel consists of a transition layer sandwiched between a porous medium and a fluid clear of solid material. Within the transition layer, the reciprocal of the permeability varies linearly across the channel. The Brinkman model is used for the momentum equations for the porous medium layer and the transition layer. The velocity profile is obtained in closed form in terms of Airy, exponential, and polynomial functions. The overall volume flux and boundary friction factors are calculated and compared with values obtained with a two-layer model employing the Beavers–Joseph condition at the interface between a Darcy porous medium and a clear fluid.     

Effect of a drainage slit on the hydrodynamic stabilization of flow through porous media with lateral barotropic fluid injection

The effect of the scheme of fluid injection into a stratum on the length and the hydraulic drag of the initial portion of the flow through the porous medium and on the flow rate intercepted by a drainage slit separating the stratum from and end wall is investigated. The asymptotics of large and small values of the hydraulic conductivity coefficient of this slit are constructed. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 60–67, January–February, 1999. The work was carried out at the Moscow State Chemical Industry Academy.     

Gravitional penetration of a nonwetting fluid into a porous medium. Classification of regimes

The existence of capillary-gravitational equilibrium is detected in the problem of the penetration of a nonwetting fluid into a porous medium from the top. Using a numerical simulation method, three qualitatively feasible regimes of operation of the system are distinguished: total penetration of the soil; penetration to a finite depth, i.e., starting from a certain moment of time the gravitational head is weaker than the capillary resistance of the medium; no penetration of the soil. The existence of these three regimes makes it possible to distinguish critical parameters of the process expressed in terms of the Bond number. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 95–103, March–April, 1998.     

Statistical characteristics of the motion of a fluid particle in a medium with random porosity

In the study of flow of a neutral admixture in a porous medium, it is most often assumed in the stochastic formulation that the porosity is constant and a determinate quantity, and the velocity is a random function [1–4]. The velocity distribution is usually regarded as known. Flow in a porous medium with random porosity has been studied to a far lesser extent. We note [5], which studies the averaged equations obtained within the framework of the correlation approximation. We consider the model problem of one-dimensional motion of a fluid particle (position of the front for flow of a neutral admixture in a porous medium) in a medium with random porosity. For a particular form of random porosity field, expressions are obtained for the one- and two-point densities of the distribution of the position of the particle. A study is made of the dependences of the first four moments and the correlation function of the position of the particle as functions of the time. It is shown that for large values of the time the motion of the particle is asymptotically similar to Brownian motion. It is shown by means of numerical modeling that the results obtained transfer to the case of an arbitrary random porosity field. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 59–65, November–December, 1986.     

Convective mass transfer across fluid interfaces in straight angular pores

Steady convective mass transfer to or from fluid interfaces in pores of angular cross-section is theoretically investigated. This situation is relevant to a variety of mass transport process in porous media, including the fate of residual non-aqueous phase liquid ganglia and gas bubbles. The model incorporates the essential physics of capillarity and solute mass transfer by convection and diffusion in corner fluid filaments. The geometry of the corner filaments, characterized by the fluid–fluid contact angle, the corner half-angle and the interface meniscus curvature, is accounted for. Boundary conditions of zero surface shear (‘perfect-slip’) and infinite surface shear (‘no-slip’) at the fluid–fluid interface are considered. The governing equations for laminar flow within the corner filament and convective diffusion to or from the fluid–fluid interface are solved using finite-element methods. Flow computations are verified by comparing the dimensionless resistance factor and hydraulic conductance of corner filaments against recent numerical solutions by Patzek and Kristensen (J. Colloid Interface Sci 236, 305–317 2001). Novel results are obtained for the average effluent concentration as a function of flow geometry and pore-scale Peclet number. These results are correlated to a characteristic corner length and local pore-scale Peclet number using empirical equations appropriate for implementation in pore network models. Finally, a previously published “2D-slit” approximation to the problem at hand is checked and found to be in considerable error.     

Natural convection in an inclined enclosure with a fluid layer and a heat-generating porous bed

Multiple steady-state solutions of natural convection in an inclined enclosure with a fluid layer and a heat-generating porous bed is investigated numerically by the finite volume method. The conservation equations for the porous layer are based on a general flow model which includes both the effects of flow inertia and friction. The flow in fluid layer is modeled by Navier–Stokes equations. The method of pseudo arc-length continuation is adapted in studying the effects of tilt angle on flow pattern and heat transfer. It is found that, in the whole domain of tilt angle, there exist two groups of solutions with quite different flow pattern and heat transfer behavior. The effects of aspect ratio on flow pattern and heat transfer have also been studied. Received on 04 March 1997     

Linear and nonlinear stability analysis of binary Maxwell fluid convection in a porous medium

The stability analysis of the quiescent state in a Maxwell fluid-saturated densely packed porous medium subject to vertical concentration and temperature gradients is presented. A single phase model with local thermal equilibrium between the porous matrix and the Maxwell fluid is assumed. The critical Darcy–Rayleigh numbers and the corresponding wave numbers for the onset of stationary and oscillatory convection are determined. A Lorenz like system is obtained for weakly nonlinear stability analysis.     

Effects of Hall current on f lows of a Burgers’ fluid through a porous medium

This paper generalizes the analysis of four magnetohydrodynamic (MHD) flow problems of an Oldroyd-B fluid discussed by Asghar et al. [Int. J. Non-linear Mech. 40, 589–601 (2005)] into three directions: (i) to discuss the problems in a porous medium using modified Darcy’s law (ii) to see the influence of Hall current (iii) to determine the effect of rheological parameter of Burgers’ fluid. Analytical solutions of velocity distribution valid at large and small times are given in each problem. Comparison has been provided for Oldroyd-B and Burgers’ fluids through graphs. The physical interpretation is also included.     

Effects of Viscous Dissipation on Unsteady Free Convection in a Fluid Past a Vertical Plate Immersed in a Porous Medium

The effects of viscous dissipation on unsteady free convection from an isothermal vertical flat plate in a fluid saturated porous medium are examined numerically. The Darcy–Brinkman–Forchheimer model is employed to describe the flow field. A new model of viscous dissipation is used for the Darcy–Brinkman–Forchheimer model of porous media. The simultaneous development of the momentum and thermal boundary layers are obtained by using a finite difference method. Boundary layer and Boussinesq approximation have been incorporated. Numerical calculations are carried out for various parameters entering into the problem. Velocity and temperature profiles as well as local friction factor and local Nusselt number are shown graphically. It is found that as time approaches infinity, the values of friction factor and heat transfer coefficient approach steady state.     

The onset of buoyancy-driven convection in a ferromagnetic fluid saturated porous medium

The onset of buoyancy-driven convection in an initially quiescent ferrofluid saturated horizontal porous layer in the presence of a uniform vertical magnetic field is investigated. The Brinkman-Lapwood extended Darcy equation with fluid viscosity different from effective viscosity is used to describe the flow in the porous medium. The lower boundary of the porous layer is assumed to be rigid-paramagnetic, while the upper paramagnetic boundary is considered to be either rigid or stress-free. The thermal conditions include fixed heat flux at the lower boundary, and a general convective–radiative exchange at the upper boundary, which encompasses fixed temperature and fixed heat flux as particular cases. The resulting eigenvalue problem is solved numerically using the Galerkin technique. It is found that increase in the Biot number Bi, porous parameter σ, viscosity ratio Λ, magnetic susceptibility χ, and decrease in the magnetic number M ₁ and non-linearity of magnetization M ₃ is to delay the onset of ferroconvection in a porous medium. Further, increase in M ₁, M ₃, and decrease in χ, Λ, σ and Bi is to decrease the size of convection cells.     

On the MHD flow of fractional generalized Burgers＇ fluid with modified Darcy＇s law

This work is concerned with applying the fractional calculus approach to the magnetohydrodynamic (MHD) pipe flow of a fractional generalized Burgers’ fluid in a porous space by using modified Darcy’s relationship. The fluid is electrically conducting in the presence of a constant applied magnetic field in the transverse direction. Exact solution for the velocity distribution is developed with the help of Fourier transform for fractional calculus. The solutions for a Navier–Stokes, second grade, Maxwell, Oldroyd-B and Burgers’ fluids appear as the limiting cases of the present analysis.     

Forces and torques exerted on bodies sliding in a fluid

The motion of a system of bodies along a plane in a viscous fluid in the presence of flow shear is considered. It is demonstrated that a main torque, linearly proportional to the velocities of the bodies, is exerted by the fluid on the system of the bodies and the plane. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 89–96, January–February, 1997.     

Analysis of matching conditions at the boundary surface of a fluid-saturated porous solid and a bulk fluid: the use of Lagrange multipliers

The compatibility conditions matching macroscopic mechanical fields at the contact surface between a fluid-saturated porous solid and an adjacent bulk fluid are considered. The general form of balance equations at that discontinuity surface are analyzed to obtain the compatibility conditions for the tangent and normal components of the velocity and the stress vector fields. Considerations are based on the procedure similar to that used in the phenomenological thermodynamics for derivation of constitutive relations, where the entropy inequality and the concept of Lagrange multipliers are applied. This procedure made possible to derive the compatibility conditions for the viscous fluid flowing tangentially and perpendicularly to the boundary surface of the porous solid and to formulate the generalized form of the so called slip condition for the fluid velocity field, postulated earlier by Beavers and Joseph, J. Fluid. Mech. 30, 197–207 (1967). PACS 47.55.Mh Communicated by Y.D. Shikhmurzaev     

Analytic solution for flow of Sisko fluid through a porous medium

This paper presents the analytic solution for flow of a magnetohydrodynamic (MHD) Sisko fluid through a porous medium. The non-linear flow problem in a porous medium is formulated by introducing the modified Darcy’s law for Sisko fluid to discuss the flow in a porous medium. The analytic solutions are obtained using homotopy analysis method (HAM). The obtained analytic solutions are explicitly expressed by the recurrence relations and can give results for all the appropriate values of material parameters of the examined fluid. Moreover, the well-known solutions for a Newtonian fluid in non-porous and porous medium are the limiting cases of our solutions.