Solution of a problem of flow in a porous medium with limiting gradient

For the law of flow in a porous medium with limiting gradient studied previously in [1], an exact solution is found for the problem formulated in [2] of the plane steady motion of an incompressible fluid in a channel with a rectangular step. Particular cases of the solution obtained are given; these represent the solutions of the problem of flow past a broken wall and of motion from a point source in a strip.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 76–78, January–February, 1985.     

Nonstationary conducting free flow in a transverse magnetic field

In magnetohydrodynamic flow the viscous friction at the walls can be substantial. The role of viscous friction can be considerably reduced by using a free or a semirestricted flow of the conducting fluid. Nonstationary phenomena in one-dimensional motion of a free plane incompressible fluid flow in a transverse magnetic field are examined. The narrow sides of the flow come into contact with the sectional electrodes connected through external circuits with an active-inductive load. The magnetic Reynolds number and the magnetody-dynamic interaction parameter are assumed to be large. When the electric field due to electromagnetic induction in the channel is much smaller than the field due to the external circuits, the problem can be reduced to the characteristic Cauchy problem for a quasilinear hyperbolic system of first-order equations which can be solved by the method of characteristics using a computer.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 34–39, July–August, 1970.     

On the theory of flow of an incompressible viscous fluid in a channel of arbitrary section

An exact solution is constructed to the problem of the stationary flow of an incompressible viscous fluid in a straight tube whose section is formed by two connected rectangles.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 124–128, March–April, 1980.     

Flow between a porous rotating disk and a plane

It is shown that when a viscous incompressible fluid is sucked through a stationary porous disk spontaneous rotation of the fluid sets in at a certain Reynolds number. This is consistent with the results of a specially designed experiment. Another unusual result is the existence of multicell regimes, corresponding to suction, when the force acting on the porous, rapidly rotating disk is a lift force and, moreover, anomalously large. Charts of the possible steady-state flow regimes, stable and unstable, have been constructed. In the case of fairly intense suction and rotation a stable self-oscillating regime is observed. In the limit of vanishingly small viscosity unusual boundary layer properties associated with suction are noted.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 53–65, November–December, 1989.     

Flow of a viscous fluid in a closed cavity in the presence of slip effects

Slip at the wall is observed in the flow of non-Newtonian fluids [1–4] and rarefied gases [5]. The most complete information on the phenomenon is obtained in capillary viscosimetry. For small radii of the capillaries and in porous media the slip effect is manifested even for Newtonian fluids (water, kerosene, for example) [6]. Experiments [2, 4] show that the influence of the entrance section can be ignored if the length of the capillary exceeds its radius by about 100 times. For the measurement of the rheological characteristics of high-viscosity fluids the use of long capillaries is difficult, and it is necessary to calculate the two-dimensional flow at the entrance section with allowance for slip. The need for such calculations also arises, for example, when one is choosing the optimal parameters of the screw devices employed in the processing of polymers [7]. Two-dimensional flows of a viscous incompressible fluid are frequently calculated with the flow function and vorticity =– used as variables [8–14]. The expressions for the vorticity on the boundary are usually obtained from the viscous no-slip condition [8, 9]. In the present paper, expressions are obtained for the vorticity on a wall in the presence of slip. The obtained expressions are used to solve a test problem on the flow of a viscous incompressible fluid in a cavity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 10–16, January–February, 1980.     

Laminar flow through porous channels with an applied magnetic field

Summary The two-dimensional steady flow of an incompressible and electrically conducting viscous fluid through a porous channel with a transverse magnetic field is discussed. It is assumed that there is constant suction at one wall and constant blowing at the other wall. A perturbation solution is obtained where the perturbation parameter is equal to the difference between the two wall velocities. The behaviour of the solution for various suction Reynolds numbers and magnetic Reynolds number is considered. Finally, the skin friction at one wall is given and is found to increase with the increase of the magnetic field.     

Transient free convective flow in a vertical channel with constant temperature and constant heat flux on walls

This note presents transient motion of a viscous and incompressible fluid in a vertical channel due to free convective currents occuring as a result of application of constant heat flux at one wall and constant temperature on other wall. The method of Laplace transform is used to solve the problem. The transient behaviour of flow on velocity and temperature fields are shown on the graphs.     

Viscous flow in a partially-filled horizontal cylinder rotating at constant speed

The problem under consideration is that of the stationary shape of the free surface of a viscous fluid in a steadily rotating horizontal cylinder. In the majority of investigations of this problem the thickness of the fluid layer coating the inner surface of the cylinder is assumed to be small [1–3]. The case of a near-horizontal free surface, with the bulk of the fluid at the cylinder bottom, was considered in [4], where, after considerable simplification, the governing equations were reduced to ordinary differential equations. In the present study the behavior of the free surface is investigated using a creeping flow approximation. The controlling parameters vary over a wide range. In the numerical computations a boundary element method was used. The numerical results have been confirmed experimentally.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 25–30, May–June, 1993.     

Laminar flow through channels with porous walls and with an applied transverse magnetic field

Summary The problem of two-dimensional steady laminar flow of a viscous incompressible and electrically conducting fluid through a channel with two equally porous walls in the presence of a transverse magnetic field has been extended to include all values of Hartmann number and small suction velocity at the walls. Expressions for the velocity components, the pressure and the wall friction in terms of the Hartmann number and the suction Reynolds number are given. It is found that the pressure drop in the major flow direction and the wall friction decrease with the increase in suction and increase with the increase in the strength of the magnetic field.     

Structure of wall flow through a channel with a granular bed

The problem of viscous incompressible fluid flow through a plane channel with one linear and one sinusoidal boundary is considered. Using the representation of the system of Stokes equations in terms of the stream function in a region including a single periodic element, we obtain a boundary-value problem for the biharmonic operator. Its solution is found by the mixed Galerkin method - the straight line method. The near-degenerate matrix of unknown coefficients was calculated on a computer. The velocity vector component, pressure and streamline fields are found as functions of the curvature of the boundary. The features of the flow structure resulting from the asymmetry of the walls are established. The distortion of the pore space required to refine the dependence of the permeability on the structure is introduced. The results are of interest for analyzing the wall effect of increased flow velocity in a channel with a fixed granular bed.Ufa. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 9–13, November–December, 1994.     

Influence of surface tension on damping of the oscillations of a viscous incompressible fluid in a cylindrical channel

A numerical calculation is made of small oscillations of a viscous incompressible fluid that fills half of a horizontal cylindrical channel. The calculation is made with and without allowance for surface tension. The results of the calculation show that allowance for surface tension increases the damping of the oscillations. The general properties of problems of the normal oscillations of a heavy and capillary viscous incompressible fluid were studied in [1–3], in which the possibility of applying the Bubnov-Galerkin method to these problems was pointed out. A method for calculating the oscillations of a viscous incompressible fluid that partly fills an arbitrary vessel at large Reynolds numbers was developed in [3–5].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 128–132, May–June, 1980.     

Form of a free surface during steady flow of a capillary fluid in a rectangular channel

The two-dimensional problem of the form of a free surface of an ideal incompressible fluid during steady flow from a rectangular channel through a thin slot with simultaneous uniform delivery of fluid through the side walls is examined. Forces of gravity and surface tension are taken into account. The nonlinear problem of the simultaneous determination of the free surface and velocity field of the fluid is solved by the iteration method. Convergence of the iterations to the solution of the problem for small values of the parameters is investigated. The solution of the linearized problem is obtained in a closed form for a small depth of the discharge and small width of the channel, which is compared with the solution of the problem in a complete formulation. Graphs of the free surface of the fluid for different values of the parameters, obtained as a result of numerical solution of the nonlinear problem, are presented.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 67–75, January–February, 1977.     

Flow past a thin profile in a channel with permeable working part

An exact solution is obtained to the problem of flow of an ideal incompressible fluid past a thin profile in a straight channel. The channel walls are continuous except for the working part, where they are permeable.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 180–185, July–August, 1982.I thank Yu. B. Lifshitz for constant interest in the work.     

Bifurcation of a Main Steady-State Viscous Fluid Flow in a Plane Divergent Channel

The evolution of steady-state viscous incompressible fluid flows in a plane divergent channel is investigated. For the classical formulation of the Jeffery-Hamel problem a complete solution is given as a function of the determining parameters. For a fixed angle of divergence the behavior of the main unimodal flow is determined as a function of the Reynolds number. Critical values at which the flow pattern bifurcates and the steady-state unimodal flow ceases to exist are found. The mechanism of bifurcation is established and its diagram is constructed. This mechanism and the diagram were not previously known in the scientific literature in connection with the investigation of the Jeffery-Hamel problem. The critical Reynolds number at which bifurcation occurs is given as a function of the channel divergence angle. The results may be of interest for hydromechanical, technological, and geophysical applications.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, 2005, pp. 25–36.Original Russian Text Copyright © 2005 by Akulenko and Kumakshev.     

Laminar flow and mass transfer in channels with a porous bottom wall and with fins attached to the top wall

 The steady incompressible, viscous, two- dimensional flow of a solution in a channel was considered. The bottom wall was porous and the fins were attached to the top wall. Employing control volume approach, a computer program based on SIMPLE algorithm was developed. Computations were carried out to investigate the effects of the inlet Reynolds number, the fin length, the suction Reynolds number and the slip coefficient on the flow structure and the concentration distribution. It was observed that the thickness of concentration boundary layer increases in the flow direction. The concentration on the porous wall and the concentration boundary layer thickness decrease with increasing fin length, the slip coefficient and the inlet Reynolds number. These results show that fins attached to the upper wall of the channel can be utilized to reduce the concentration polarization and hence improve the effectiveness of the separation process. Received on 24 February 1999     

Flow of a suspension in a flat channel with porous walls

In the flow of a suspension in a channel with porous walls, when the size of particles of a suspended phase is much less than the width of the channel but greatly exceeds the size of the pores (in particular, in the flow of blood in the plasma separator used in an artificial kidney system [1, 2]), phenomena are observed which apparently cannot be satisfactorily explained by means of the well-known solutions of problems on the motion of a Newtonian fluid [3]. For example, the flow rate of the liquid phase of the suspension through the walls depends on the velocity of the main flow and does not depend on the pressure drop on the wall at fairly high values of it [1, 2]. The present study gives below the formulation and an approximate solution, which explains this effect, of the problem of an incompressible two-phase suspension in a long slit with porous walls which are impermeable in relation to the suspended phase and through which the liquid phase is pumped. Certain effects are taken into account which are caused by the high volume concentration of the suspended phase.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 37–43, November–December, 1987.     

Asymptotic analysis of a pulsating flow of viscous fluid in a symmetric deformed channel

The time-periodic flow of a viscous incompressible fluid in a two-dimensional symmetric channel with slightly deformed walls is considered. The solution of the Navier-Stokes equations is constructed by means of the method of matched asymptotic expansions [1] at large characteristic Reynolds numbers. It is shown that in an unsteady flow a region of nonlinear perturbations surrounds the line of zero velocity inside the fluid. The formation and development of such nonlinear zones with respect to time is considered. An alternation of the topological features of the streamline pattern in the nonlinear perturbation zone is discovered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 17–23, July–August, 1987.The author is deeply grateful to V. V. Sychev for his formulation of the problem and his attentive attitude to my work.     

Effect of a periodic action on average flow in an inhomogeneous liquid-saturated porous medium

The problem of the average flow of a viscous incompressible fluid saturating a stationary porous incompressible matrix under a periodic action is considered. It is shown that a spatial inhomogeneity of the medium porosity leads to an average fluid flow, quadratically dependent on the action amplitude, in the direction of increase in porosity. In particular, this means that wave action on an oil reservoir could lead to fluid flow on the interfaces from low-porosity,weakly permeable collector regions into high-porosity regions, for example, to flow from blocks to fractures in fractured porous reservoirs, which makes it possible to enhance oil production. It is shown that in the presence of a constant pressure gradient the flow component generated by a periodic action can provide a substantial proportion of the total flow, especially on the boundaries with low-porosity strata or blocks. With increase in amplitude this may significantly exceed the component associated with the constant pressure gradient.     

Stability of stationary convective flow of a mixture in a vertical porous layer

In contrast to the corresponding viscous flow, the convective flow of a homogeneous liquid in a planar vertical layer whose boundaries are maintained at different temperatures is stable [1]. When a porous layer is saturated with a binary mixture, in the presence of potentially stable stratification one must expect an instability of thermal-concentration nature to be manifested. This instability mechanism is associated with the difference between the temperature and concentration relaxation times, which leads to a buoyancy force when an element of the fluid is displaced horizontally. In viscous binary mixtures, the thermal-concentration instability is the origin of the formation of layered flows, which have been studied in detail in recent years [2–4]. The convective instability of the equilibrium of a binary mixture in a porous medium was considered earlier by the present authors in [5]. In the present paper, the stability of stationary convective flow of a binary mixture in a planar vertical porous layer is studied. It is shown that in the presence of sufficient longitudinal stratification the flow becomes unstable against thermal-concentration perturbations; the stability boundary is determined as a function of the parameters of the problem.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 150–157, January–February, 1980.     

Laminar flow between porous disks for intense homogeneous asymmetric inflow and outflow

In [1–3], a class of self-similar solutions was considered for the flow of incompressible fluid in a plane channel with porous walls, through which there is homogeneous symmetric inflow or outflow. An analogous class of self-similar solutions for flow between porous disks with natural homogeneous conditions at the periphery was considered in [4], where the asymptotic behavior of these solutions at a small Reynolds number of the outflow R was investigated, and the limiting form of the solution for symmetric outflow with R= was noted. In the present paper, the boundary-function method is applied to the singular problem corresponding to the flow between porous disks for asymmetric inflow and outflow characterized by large R.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 13–19, November–December, 1976.