Self-similar solution to a problem of axisymmetric motion of gas through a porous medium in the case of a quadratic law of resistance

The results of solution of the self-similar problem of planar flow of gas through a porous medium in the case of a quadratic law of resistance [1] are generalized to the case of axisymmetric motion. The equation in similarity variables for the velocity of isothermal gas flow is reduced to an equation having cylindrical functions as solution. Analytic dependences of the pressure and the gas velocity on the coordinate and time are obtained for a given flow rate of the gas at the coordinate origin and for zero Initial gas pressure in the porous medium.Translated from Izvestlya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza., No. 4, pp. 168–171, July–August, 1982.     

Equations of nonlinear anisotropic flow through a porous medium

A general law of nonlinear anisotropic flow through a porous medium is proposed. A corresponding equation for the pressure of the fluid is obtained in velocity hodograph variables. The conditions of ellipticity of this equation are expressed in terms of the dissipative function.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 158–160, September–October, 1980.I thank V. M. Entob for discussing the work.     

Calculation of two-dimensional dispersion in a gas in unsteady flow in a porous medium

A two-dimensional problem of the flow of a gas containing an impurity through a porous medium is considered. At the initial time, the gas containing a uniformly distributed impurity is at a high pressure in a spherical cavity in a porous medium at a certain distance from a flat surface. It is assumed that for t > the motion of the carrier gas is described by the system of equations for flow in a porous medium and the dispersion of the impurity is described by the equations of convective diffusion and nonequilibrium adsorption. A numerical method for solving the problem is discussed. Some results of calculations are given. The influence of the flat surface on the flow of the gas and the dispersion of the impurity is analyzed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 61–67, September–October, 1982.We thank V. N. Nikolaevskii for comments which permitted a significant improvement in the paper.     

The Use of Fick's Law for Modeling Trace Gas Diffusion in Porous Media

Two models for combined gas-phase diffusion and advection in porous media, the advective-diffusive model (ADM) and the dusty-gas model (DGM), are commonly used. The ADM is based on a simple linear addition of advection calculated by Darcy's law and ordinary diffusion using Fick's law with a porosity–tortuosity–gas saturation multiplier to account for the porous medium. The DGM applies the kinetic theory of gases to the gaseous components and the porous media (or dust) to develop an approach for combined transport due to diffusion and advection that includes porous medium effect. The ADM and Fick's law are considered to be generally inferior for gas diffusion in porous media, and the more mechanistic DGM is preferred. Under trace gas diffusion conditions, Fick's law overpredicts the gas diffusion flux compared to the DGM. The difference between the two models increases as the permeability decreases. In addition, the difference decreases as the pressure increases. At atmospheric pressure, the differences are minor (<10%) for permeabilities down to about 10^–13 m². However, for lower permeabilities, the differences are significant and can approach two orders of magnitude at a permeability of 10^–18 m². In contrast, at a pressure of 100 atm, the maximum difference for a permeability of 10^–18 m² is only about a factor of 2. A molecule–wall tortuosity coefficient based on the DGM is proposed for trace gas diffusion using Fick's law. Comparison of the Knudsen diffusion fluxes has also been conducted. For trace gases heavier than the bulk gas, the ADM mass flux is higher than the DGM. Conversely, for trace gases lighter than the bulk gas, the ADM mass flux is lower than the DGM. Similar to the ordinary diffusion variation, the differences increase as the permeability decreases, and get smaller as the pressure increases. At atmospheric pressure, the differences are small for higher permeabilities (>10^–13 m²) but may increase to about 2.7 for He at lower permeabilities of about 10^–18 m². A modified Klinkenberg factor is suggested to account for differences in the models.     

Shear flow over a porous particle

Linear axisymmetric Stokes flow over a porous spherical particle is investigated. An exact analytic solution for the fluid velocity components and the pressure inside and outside the porous particle is obtained. The solution is generalized to include the cases of arbitrary three-dimensional linear shear flow as well as translational-shear Stokes flow. As the permeability of the particle tends to zero, the solutions obtained go over into the corresponding solutions for an impermeable particle. The problem of translational Stokes flow around a spherical drop (in the limit a gas bubble or an impermeable sphere) was considered, for example, in [1,2]. A solution of the problem of translational Stokes flow over a porous spherical particle was given in [3]. Linear shear-strain Stokes flow over a spherical drop was investigated in [2].Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 113–120, May–June, 1995.     

Derivation of the Forchheimer Law via Homogenization

In this paper we derive the Forchheimer law via the theory of homogenization. In particular, we study the nonlinear correction to Darcy's law due to inertial effects on the flow of a Newtonian fluid in rigid porous media. A general formula for this correction term is derived directly from the Navier–Stokes equation via homogenization. Unlike other studies based on the same approach that concluded for the nonlinear correction to be cubic in velocity for isotropic media, the present work shows that the nonlinear correction is quadratic. An example is constructed to illustrate our theory. In this example, the analytic solution to the Navier–Stokes equation is obtained and is utilized to show the validity of the quadratic correction. Both incompressible and compressible fluids are considered.     

Relaxation effects associated with the flow of a dense gas through a porous medium

Unsteady processes of gas and Newtonian liquid flow through porous media are usually described within the framework of the standard elastic regime model [1, 2]. At the same time, from general theoretical considerations it is clear that for fairly small characteristic times of variation of the pressure and the seepage velocity the elastic regime model loses its validity and more general relaxation flow models [3, 4] must be employed. It is therefore important to determine the limits of applicability of the elastic regime model. For this purpose the unsteady process of gas flow from one vessel to another through a porous medium has been investigated theoretically and experimentally for small pressure differences and absolute pressures up to 50 MPa. It is shown that the experimental results diverge sharply from the theoretical predictions based on the elastic regime model. It is therefore proposed that for unsteady processes a generalization of Darcy's law with a relaxation kernel be employed. From the results of the experiments it is possible to determine the parameters of the kernel characterizing the internal relaxation processes in the porous medium-dense gas system.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 105–113, January–February, 1991.     

Equations of the mechanics of porous media saturated with liquid and gas

The approach proposed by Podil'chuk [1] is used to derive a system of equations of motion for saturated porous media, allowance being made for the mutual influence of the solid, liquid, and gas phases. The permeabilities of the anisotropic porous medium are assumed to depend on the direction. It is shown that when there are no gas phases and the liquid is incompressible the system of equations reduces to the general equations of the theory of elasticity of an anisotropic body with fictitious stress components. For a porous medium saturated with liquid, the relationships between the permeabilities and the anisotropy constants are obtained. The motion of liquid in an elastic porous medium in the form of an orthotropic cylindrical region with a cavity in the form of a circular cylinder is considered as an example.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 82–87, July–August, 1981.     

Influence of the structure of a porous medium on the residual gas saturation in the case of capillary displacement by a liquid

A model of a porous medium consisting of randomly branching conical pores is used to investigate the quasistatic displacement of gas by a wetting liquid without application of an external pressure. Allowance is made for the circumstance that in the capillary process all the pores have at least one-sided permeability for the liquid phase. An expression is obtained that relates the residual gas saturation to the parameters which characterize the structure of the pores and the wetting properties of the system. Two new characteristics of the pore space are introduced — the branching parameter and the opening angle of the pores — and the influence of these parameters on the residual saturation is investigated. It is shown that for individual classes of natural media the residual gas saturation depends only on the porosity and the contact angle of wetting.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 128–133, September–October, 1981.     

Self-similar solutions of the problem of displacement of one gas by another in an axisymmetric case with a quadratic drag law

A problem of piston-induced displacement of one gas by another in cracks (porous media) in an axisymmetric case with a quadratic drag law is studied. Self-similar solutions for determining the dynamic characteristics (velocity and pressure) of the displacing and displaced gases are constructed in quadratures. The velocity and pressure are studied as functions of a self-similar variable for several initial conditions and parameters. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 5, pp. 87–92, September–October, 2008.     

Viscous incompressible flow in a narrow channel with one free wall and fluid supply through a porous insert

The problem investigated relates the plane unsteady flow of a viscous incompressible fluid in a narrow channel one of whose walls is free and acted upon by a given load, while the other is rigidly fixed. The fluid enters the channel through a porous insert in the stationary wall. A model of the flow of a thin film of viscous incompressible fluid and Darcy's law for flow in a porous medium are used to find the distribution of fluid pressure and velocity in the channel and the porous insert in the two-dimensional formulation for fairly general boundary conditions in the case where the length of the porous insert exceeds the length of the free wall. In the particular case where the length of the porous insert is equal to the length of the free wall an exact stationary solution of the problem is obtained for a given value of the channel height. The stability of the equilibrium position of the free wall supported on a hydrodynamic fluid film is examined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 16–24, January–February, 1986.     

Linear problem of the dissociation of the hydrates of a gas in a porous medium

Self-similar solutions are obtained for the linearized equations of the flow of a gas in porous regions with a moving interface that describe the dissociation of gas hydrates when the pressure is reduced.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 174–177, January–February, 1980.     

Convergent shock wave in an ideal elastic nonhomogeneous medium

The boundary-value problem for symmetric focusing of a shock wave in a medium with variable density under a constant load (model of a porous body with variable initial velocity) is solved. The solution asymptotic is studied. Focusing in a homogeneous medium has been previously studied [1], One inverse problem related to the choice of the optimal pressure conditions is examined. Constraints on the applicability of the model are touched on.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 107–111, January–February, 1976.     

Flow regimes with temperature extrema is the motion of a gas through a porous medium

Self-similar solutions are obtained for the problem of the motion of a heated gas through a porous medium with allowance for heat transfer between the gas and the solid phase in accordance with Newton's law. It is shown that there exist flow regimes in which the gas temperature increases with distance in the direction of motion.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 71–77, July–August, 1987.     

So-called constants of turbulence in rough pipes

A direct analysis of Nikuradse's experiments on flow in rough pipes has shown that on the quadratic resistance interval the value of the second constant of turbulence depends on the relative roughness of the pipe. It is established that the universal logarithmic relation for the velocity profile in a pipe in the quadratic resistance regime is an approximation of the more accurate relation obtained by the authors, which takes into account the effect of the relative roughness of the pipe. Where the roughness is significant, the proposed expression for the skin-friction coefficient is more accurate than the well-known Prandtl-Nikuradse formula.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 182–185, March–April, 1990.     

Detonation Wave Propagation in Rotational Gas Flows

This paper studies the propagation of detonation and shock waves in vortex gas flows, in which the initial pressure, density, and velocity are generally functions of the coordinate — the distance from the symmetry axis. Rotational axisymmetric flow having a transverse velocity component in addition to a nonuniform longitudinal velocity is considered. The possibility of propagation of Chapman–Jouguet detonation waves in rotating flows is analyzed. A necessary conditions for the existence of a Chapman–Jouguet wave is obtained.     

Rheological properties of solutions of gases in a liquid in the saturation pressure zone

The flow of single-phase gas-liquid systems through a porous medium near the saturation pressure is accompanied by nonequilibrium effects [1, 2]. A series of experimental investigations has been carried out with the object of making a detailed study of the rheological properties of solutions of gases in a liquid at pressures close to the gas release pressure.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 172–175, January–February, 1988.     

Shock–Free Breakup of Droplets. Temporal Characteristics

In the present paper, we consider the shock–free breakup of droplets in their encounter with a layer (sheet) of a moving gas in the absence of pressure perturbations when the droplets are affected by a short U–shaped pulse of aerodynamic forces. Under a high pressure of the ambient gas medium p₀ = 20—80 bar, the droplets (ethanol or liquid oxygen) have a chance to break up after stay in a thing (2—5 mm thick) gas layer (jet) moving with a velocity of 1—10 m/sec. A distinctive feature of the process is that the characteristic time of droplet deformation and the period of natural oscillations coincide with the residence time for the droplets in the region of their interaction with the gas stream. Empirical formulas are proposed for determination of the total breakup time and the duration of the droplet disintegration stage in shock–free breakup.     

Filtration of a piecewise-homogeneous liquid

The nonsteady filtration of two immiscible liquids, in the absence of pressure, in a homogeneous and isotropic porous medium, is considered. A nonlinear hydrodynamic problem is formulated, and a number of its features are noted. The solution of the corresponding linearized problem is obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 62–69, November–December, 1976.