Dispersion of a filtration flow

In this paper we shall consider the transport of a dynamically neutral impurity in a porous medium containing random inhomogeneities. The original versions of the equations for the mean impurity concentration [1, 2] were based on the hyphothesis that the random motions obeyed the Markov principle, use being made of the diffusion equations of A. N. Kolmogorov. Later [3, 4] the method of perturbations was used to study the complete system of equations for the impurity concentration and random filtration velocity in the case of a constant, nonrandom porosity; after an averaging process this yields a generalized equation for the average concentration. In the limiting cases of small- and large-scale inhomogeneities in the permeability of the medium, the basic integrodifferential equation may be, respectively, reduced to parabolic and hyperbolic equations of the second order. In the present analysis we shall use the perturbation method to study the transport of an impurity by a flow when the filtration velocity of the latter fluctuates around inhomogeneities in the permeability field, the porosity of the medium in which the flow is taking place also constituting a random field, correlating with the field of permeability. We shall derive equations for the average concentration and should formulate the corresponding boundary-value problems for these equations; we shall also calculate the components of the dispersion tensor and shall consider the equilibrium sorption of an impurity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 65–69, July–August, 1976.The author is grateful to A. I. Shnirel'man for useful discussions.     

Effect of the shape of a blunt nose on the drag coefficient of a body in a hypersonic rarefied gas flow

The effect of the shape of a blunt nose of a body located in a hypersonic rarefied gas flow on the field of flow and on the aerodynamic characteristics is studied in the example of flow round ellipsoids of revolution at a zero angle of attack. The problem of the flow in the transition regime is solved on the basis of numerical analysis of the model kinetic Bhatnagar—Gross—Krook (BGK) equation for a monatomic gas. The good agreement of the results of the numerical calculations with the experimental data in a broad range of Mach numbers has shown [1, 2] that the numerical solution of the model kinetic equations is a reliable and effective means for studying flow problems. In the case when the problem is posed of determining the laws of the purely force interaction of a flow with the body, sufficiently good accuracy is given by the use of the model BGK equation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 190–192, March–April, 1985.     

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.     

Theory of thermal instability of the flow of a viscoelastic fluid

The problem of the stability of the flow of viscoelastic fluids has fundamental importance for the technology of the production of polymer products and viscosimetry. This problem is not reduced only to classical inertial turbulence. A number of other mechanisms leading to flow instability are known [1, 2]. A thermal mechanism based on the allowance for dissipative heating and elastic properties within the framework of a linear model of a viscoelastic fluid was drawn upon to explain this phenomenon in [1]. The possibility of a self-oscillatory mode of flow was demonstrated on the basis of a qualitative analysis of the theological equation and the equation of heat balance in application to simple shear flow and uniform stretching. A theoretical analysis of the self-heating of flowing systems possessing viscoelastic properties is carried out in the present report. The main laws of the thermal instability of viscoelastic fluids discovered in [1] are described.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 115–122, May–June, 1979.     

Exact solutions of boundary-value problems for nonlinear flow in porous media

Exact solutions for flow problems in porous media with a limiting gradient in the case when the flow region in the hodograph plane is a half-strip with a longitudinal cut [1] are known only for two models of the resistance law [2–6]. The present study gives a one-parameter family of flow laws, and argues the possibility of effective determination of exact and approximate analytical solutions on the basis of successive reduction to boundary-value problems for the Laplace equation or for the equation studied in detail in [1]. It should be noted that the characteristics of the flow are determined without additional quadratures.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 107–112, May–June, 1985.     

Steady-state particle distributions in shear flows of colloids and finely dispersed suspensions

On the basis of an analysis of the pseudoturbulent motion of both the suspended particles and the carrier fluid, the normal stress components in the dispersed phase are obtained for the problem of inclined confined flows of finely dispersed suspensions and colloids. These hydrodynamic pulsations are due to the shear and the work done by the average relative flow of the fluid phase on random concentration fluctuations of the disperse system because of the substantial slip of the phases of the suspension under gravity. The momentum conservation equations for the particles are obtained with allowance for the angle of inclination of the flow to the vertical and on the basis of these equations the suspension capacity of the flow as a function of the angle of inclination, particle size, Galileo number and other parameters is illustrated.Ekaterinburg. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 78–84, January–February, 1996.     

Resistance matrix for a particle in a nonuniform gas flow

The slow (relative to the medium) motion of a heat-conducting particle in a gas flow described by the Navier-Stokes equations is examined. The particle is small as compared with the macroscopic scale of the flow and the Local perturbation it creates is therefore described by a Linear Boltzmann equation. A system of thermodynamic fluxes corresponding to the thermodynamic forces is established, and the transport coefficients forming the resistance matrix are shown to be symmetric.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 101–108, January–February, 1988.     

Impact of a strip of compressible liquid on a barrier

The exact solution of the plane problem of the impact of a finite liquid strip on a rigid barrier is obtained in the linearized formulation. The velocity components, the pressure and other elements of the flow are determined by means of a velocity potential that satisfies a two-dimensional wave equation. The final expressions for them are given in terms of elementary functions that clearly reflect the wave nature of the motion. The exact solution has been thoroughly analyzed in numerous particular cases. It is shown directly that in the limit the solution of the wave problem tends to the solution of the analogous problem of the impact of an incompressible strip obtained in [1]. A logarithmic singularity of the velocity parallel to the barrier in the corner of the strip is identified. A one-dimensional model of the motion, which describes the behavior of the compressible liquid in a thin layer on impact and makes it possible to obtain a simple solution averaging the exact wave solution, is proposed. Inefficient series solutions are refined and certain numerical data on the impact characteristics for a semi-infinite compressible liquid strip, previously considered in [2–4] in connection with the study of the earthquake resistance of a dam retaining water in a semi-infinite basin, are improved. The solution obtained can be used to estimate the forces involved in the collision of solids and liquids. It would appear to be useful for developing correct and reliable numerical methods of solving the nonlinear problems of fluid impact on solids often examined in the literature [5].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 138–145, November–December, 1990.The results were obtained by the author under the scientific supervision of B. M. Malyshev (deceased).     

The steady-state flow of a rarefied gas from a spherical source or sink

The method of characteristics is used to solve problems in the steady-state flows of a rarefied gas on the basis of approximating the kinetic equations. Numerical results are given for the solution of the problem of the flow from a spherical source or sink using the generalized Kruk equation [1] and approximating the Boltzmann equation by the method proposed by the author [2, 3], Various flow conditions are discussed: flow into a vacuum, flow into a flooded volume, flow from a sink.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 58–66, March–April, 1971.     

Asymptotic solution of the equations of a laminar multicomponent boundary layer with intensive suction

An asymptotic solution is obtained for the equations of the laminar multicomponent boundary layer encountered in the plane-parallel and axially symmetrical flow of a gas with large values of the suction parameter. It is shown that the roots of the characteristic equation to which the solution of the diffusion equations reduce in the first approximation may be found in the form of radicals when the external gas flow contains chemical components capable of being combined into r5 groups as regards their diffusion properties. The number of components in the groups and the number of components in the boundary layer may be arbitrary. Asymptotic equations are obtained for the coefficient of friction, the temperature and concentration gradients, and the diffusion flows of the components on the surface of the body. By way of example, formulas are given for the thermal flux passing to a body during the flow of dissociated air or a dissociated mixture of N₂ and CO₂. A numerical solution is given for the equations of the boundary layer in the case of the flow of dissociated air. The asymptotic solution is compared with the numerical result, and the range of applicability of the asymptotic equations is established.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 66–74, November–December, 1970.The author wishes to thank G. A. Tirskii for discussion of this analysis.     

The boundary layers of a fully ionized two-temperature plasma with given component temperatures at an electrode

The flow of a plasma with different component temperatures in the boundary layers at the electrodes of an MHD channel is investigated without any assumptions as to self-similarity. For the calculation of the electron temperature, the full energy equation for an electron gas [1] is solved with allowance for the estimates given in [2]. In contrast to [3, 4], the calculation includes the change in temperature of electrons and ions along the channel caused by the collective transport of energy, the work done by the partial pressure forces, and the Joule heating and the energy exchange between the components. The problem of the boundary layers in the flow of a two-temperature, partially ionized plasma past an electrode is solved in simplified form by the local similarity method in [5–7]. In these papers, either the Kerrebrock equation is used [5, 6] or the collective terms are omitted from the electron energy equation [7].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 3–10, September–October, 1972.The author thanks V. V. Gogosov and A. E. Yakubenko for interest in this work.     

Particle suspension in a simple shear flow

A hydrodynamic model describing the particle distribution over the cross-section of a finely dispersed flow is proposed. The model is constructed on the basis of notions concerning the diffusion of particles induced by their random displacements in the process of relative motion of neighboring layers at constant shear velocity. It is shown that the suspension capacity of the flow is large for small particles due to thermal fluctuations and for relatively large particles due to shear-induced particle pulsations. There are critical particle sizes for which the particles are suspended and transported by the flow less effectively than larger or smaller particles.Ekaterinburg. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 112–121, January–February, 1995.     

Strong recondensation in a one-component and a two-component rarefied gas for an arbitrary value of Knudsen number

As is known, surface phenomena such as evaporation, absorption, and reflection of molecules from the surface of a body depend strongly on its temperature [1–5]. This leads to the establishment of a flow of a substance between two surfaces maintained at different temperatures (recondensation). The phenomenon of recondensation was studied in kinetic theory comparatively long ago. However, up to the present, only the case of small mass flows in a onecomponent gas has been investigated completely [3,4]. Meanwhile it is clear that by the creation of appropriate conditions we can obtain considerable flows of the recondensing substance, so that the mass-transfer rate will be of the order of the molecular thermal velocity. Such a numerical solution of the problem with strong mass flows along the normal to the surface for small Knudsen numbers for a model Boltzmann kinetic equation was obtained in [7]. In this study we numerically solve the problem of strong recondensation between two infinite parallel plates over a wide range of Knudsen numbers for a one-component and a two-component gas, on the basis of the model Boltzmann kinetic equation [6] for a one-component gas and the model Boltzmann kinetic equation for a binary mixture in the form assumed by Hamel [8], for a ratio of the plate temperatures equal to ten. We also investigate the effect of the relative plate motion on the recondensation flow.Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 130–138, September–October, 1972.     

Mixing of mutually soluble liquids in turbulent flow in a pipe

The problem concerned with mixing of mutually soluble liquids in turbulent flow in a pipe [1–11] is considered. To describe the distribution of concentration in the region of mixture, taken average across the section of the pipe, we use a model based on a one-dimensional model of the type of heat-conduction equation with an effective coefficient which, as tests show, is different from the coefficients of molecular and turbulent transfer. The dimensionless value of this coefficient depends on a number of parameters, such as the Reynolds number calculated for one of the liquids, roughness, ratio of the densities and viscosities of the liquids, as well as on the concentration, gradients of concentration, etc. These relationships can be established either by means of tests or on the basis of theoretical consideration of the mixing phenomenon. In this paper we theoretically derive a dispersion model with an effective diffusion coefficient which depends on Reynolds and Schmidt numbers, as well as on roughness.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 96–102, September–October, 1971.     

Mixing up to a molecular level and the development of a chemical reaction in a turbulent flow

Using the example of the process of ignition in flows of unmixed combustible components, an analysis is made of the effect of turbulence on the course of a chemical reaction. It is established that this process is conveniently described using the equation for the distribution of the probabilities of the temperature. This equation is derived and its solution is analyzed. In the case of mixing without chemical reactions, the intensity of the pulsations of the temperature is calculated without bringing in empirical constants. The ignition criterion is obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 32–41, May–June, 1977.     

Low-Dimensional Modelling of Turbulence Using the Proper Orthogonal Decomposition: A Tutorial

The proper orthogonal decomposition identifies basis functions or modes which optimally capture the average energy content from numerical or experimental data. By projecting the Navier–Stokes equations onto these modes and truncating, one can obtain low-dimensional ordinary differential equation models for fluid flows. In this paper we present a tutorial on the construction of such models. In addition to providing a general overview of the procedure, we describe two different ways to numerically calculate the modes, show how symmetry considerations can be exploited to simplify and understand them, comment on how parameter variations are captured naturally in such models, and describe a generalization of the procedure involving projection onto uncoupled modes that allow streamwise and cross-stream components to evolve independently. We illustrate for the example of plane Couette flow in a minimal flow unit – a domain whose spanwise and streamwise extent is just sufficient to maintain turbulence.     

The convective instability of Maxwell fluid-saturated porous layer using a thermal non-equilibrium model

A linear stability analysis is carried out to study viscoelastic fluid convection in a horizontal porous layer heated from below and cooled from above when the solid and fluid phases are not in a local thermal equilibrium. The modified Darcy–Brinkman–Maxwell model is used for the momentum equation and two-field model is used for the energy equation each representing the solid and fluid phases separately. The conditions for the onset of stationary and oscillatory convection are obtained analytically. Linear stability analysis suggests that, there is a competition between the processes of viscoelasticity and thermal diffusion that causes the first convective instability to be oscillatory rather than stationary. Elasticity is found to destabilize the system. Besides, the effects of Darcy number, thermal non-equilibrium and the Darcy–Prandtl number on the stability of the system are analyzed in detail.     

Flow of a relaxed gas in the vicinity of a solid surface

Behavior of a polyatomic relaxed gas in the vicinity of a solid surface was studied. The case in which the size of the relaxation zone exceeds considerably the mean free path between the elastic collisions (suppressed exchange of translational and internal energies) was considered. A smooth validity of the asymptotic expansion for the distribution function with defined assumptions was indicated. A solution for the zero-approximation equation and boundary conditions for the surface flow, based on this solution and generalized model for a diffusive gas reflection from the surface were derived. The latter problem was partially studied in [1] by the Grad method and in [2] in terms of an analysis of the temperature jump.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 33–39, July–August, 1973.     

Plane Couette flow of binary gaseous mixture in the whole range of the Knudsen number

The Couette flow of binary gaseous mixtures is studied on the basis of the McCormack model of the Boltzmann equation, which was solved numerically by the discrete velocity method. The calculations were carried out for three mixtures of noble gases: neon–argon, helium–argon, and helium–xenon. The stress tensor and bulk velocity of both species were calculated for several values of the gas rarefaction in the range from 0.01 to 40 for three values of the molar concentrations: 0.1,0.5 and 0.9. The numerical solution together with an analytical solution based on the slip boundary condition cover the whole range of the gas rarefaction. It was showed that the Couette flow is weakly affected by the intermolecular interaction law.     

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.