Electroconvective stability of a weakly conducting liquid

In inhomogeneous electric fields, at sufficiently high field strengths, a weakly conducting liquid becomes unstable and is set in motion [1–4]. The cause of the loss of stability and the motion is the Coulomb force acting on the space charge formed by virtue of the inhomogeneity of the electrical conductivity of the liquid [4–13]. This inhomogeneity may be due to external heating [4–6], a local raising of the temperature by Joule heating [2, 7, 8], and nonlinearity of Ohm's law [9–13]. In the present paper, in the absence of a temperature gradient produced by an external source, a condition is found whose fulfillment ensures that the influence of Joule heating on the stability can be ignored. Under the assumption that this condition is satisfied, a criterion for stability of a weakly conducting liquid between spherical electrodes is obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 137–142, July–August, 1979.     

Modeling of the transient processes in the channels of an EHD pump

The nonstationary process of development of weakly conducting electrohydrodynamic flow in the channel of an EHD pump with plane permeable electrodes, between which a potential difference is created from an external source, is considered in the hydraulic approximation within the framework of the model proposed in [1, 2]. It is shown that the efficiency of the EHD device can be improved if in the fluid the spatial process of ion formation is retarded, while the recombination process is intense. The effect of the flow velocity on the formation of a space charge region in the interelectrode gap is investigated. On a certain range of the problem parameters the flow induced in the channel substantially modifies the space charge distribution as a result of the blowoff of narrow diffusion electrode ion layers. This creates a nonmonotonic dependence of the fluid velocity on the applied potential difference.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.3, pp. 30–41, May–June, 1994.     

Electrical breakdown in air in a transverse magnetic field

The electrical breakdown of gases in a transverse magnetic field is discussed in references [1–16]. Attention has mainly been concentrated on the case of coaxial electrode geometry [1–10]. The existing experimental data on breakdown between plane-parallel electrodes [11–14] relate to a narrow range of variation of the parameters characterizing breakdown (P, d, H, U). The author has made an experimental study of the process of electrical breakdown in air in a transverse magnetic field between plane-parallel electrodes of finite size in the pressure interval from 650 to 5·10^–3 mm Hg at gap lengths of from 1 to 140 mm and magnetic inductions from 0 to 10 600 G.     

On nonisothermal electroconvection

The flow from the tip of a needle electrode is caused by the Coulomb force acting on the space charge [1–3]. This charge is formed because of the dependence of the conductivity on the temperature, nonuniformity of which is due to Joule heating [1] and the electric field intensity [2] or processes near the electrode [3–5]. The present paper considers the stability of a dielectric liquid between spherical electrodes in order to elucidate the possibility of a thermoelectrohydrodynainic flow due to Joule heating. In the presence of external heating, the possibility of such a flow has been demonstrated both experimentally and theoretically [6–8].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 133–137, March–April, 1980.     

Stability of convective flow in a vibration field with respect to three-dimensional perturbations

A plane-parallel convective flow in a vertical layer between boundaries maintained at different temperatures becomes unstable when the Grashof number reaches a critical value (see [1]). In [2, 3] the effect of high-frequency harmonic vibration in the vertical direction on the stability of this flow was investigated. The presence of vibration in a nonisothermal fluid leads to the appearance of a new instability mechanism which operates even under conditions of total weightlessness [4]. As shown in [2, 3], the interaction of the usual instability mechanisms in a static gravity field and the vibration mechanism has an important influence on the stability of convective flow. In this paper the flow stability is considered for an arbitrary direction of the vibration axis in the plane of the layer and the stability characteristics with respect to three-dimensional normal perturbations are determined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 116–122, March–April, 1988.     

Stability of unsteady viscous flows

A study is made of the linear stability of plane-parallel unsteady flows of a viscous incompressible fluid: in the mixing layer of two flows, in a jet with constant flow rate, and near a wall suddenly set in motion [1]. The slow variation of these flows in time compared with the rate of change of the perturbations makes it possible to use the method of two-scale expansions [2]. The stability of nonparallel flows with allowance for their slow variation with respect to the longitudinal coordinate was investigated, for example, in [3–6]. The unsteady flows considered in the present paper have a number of characteristic properties of non-parallel flows [1], but in contrast to them are described by exact solutions of the Navier-Stokes equations. In addition, for unsteady planeparallel flows a criterion of neutral stability can be uniquely established by means of the energy balance equation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, 138–142, July–August, 1981.I thank G. I. Petrov for suggesting the problem, and also S. Ya. Gertsenshtein and A. V. Latyshev for assisting in the work.     

Nonlinear stability of two-dimensional steady flows of an ideal fluid with constant vorticity and their axisymmetric analogs

Flows with constant vorticity are widely used as local models of more complicated flows [1]. In many cases, such flows are stable against finite two-dimensional perturbations. In particular, the inviscid plane-parallel Couette flow has the property of nonlinear stability. Similar treatment of a class of axisymmetric flows yields nonlinear stability of a spherical Hill vortex and inviscid Poiseuille flow in a circular tube with respect to axisymmetric perturbations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 16–21, October–December, 1981.     

Internal rotation in the hydrodynamics of weakly conducting dielectric suspensions

Hydrodynamic phenomena in weakly conducting single-phase media due to interphase electric stresses are reviewed in [1]. In the present paper, a model is constructed of a dielectric suspension with body couples due to the field acting on free charges distributed on the surface of the particles of the suspension. Averaging of the microscopic fields yields macroscopic equations for the field and the polarization of the dielectric suspension with allowance for the finite relaxation time of the distribution of the free charge on the phase interface. The developed model is used to consider the occurrence of spontaneous rotation of a dielectric cylinder in a weakly conducting suspension in the presence of an electric field; compared with the case of single-phase media [2], this is characterized by a significant reduction in the threshold intensity of the electric field with increasing concentration of the particles [3]. In the present model of a dielectric suspension, the destabilization of the cylinder is due to the occurrence of rotations of the particles of the suspension due to the interaction between the polarization and the motion of the medium. The relaxation equation for the polarization for the given model is analogous to the corresponding equation for media which can be magnetized [4–6].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 86–93, March–April, 1980.     

Stability of nonplane-parallel three-dimensional jet flows

The problem of the stability of nonplane-parallel flows is one of the most difficult and least studied problems in the theory of hydrodynamic stability [1]. In contrast to the Heisenberg approximation [1], the basic state whose stability is investigated depends on several variables, and the stability problem reduces to the solution of an eigenvalue problem for partial differential equations in which the coefficients depend on several variables [2–7]. In the case of a periodic dependence of these coefficients on the time [2] or the spatial coordinates [3, 4], the analog of Floquet theory for the partial differential equations is constructed. With rare exceptions, the case of a nonperiodic dependence has usually been considered under the assumption of weak nonplane-parallelism, i.e., a fairly small deviation from the plane-parallel case has been assumed and the corresponding asymptotic expansions in the linear [6] and nonlinear [7] stability analyses considered. The present paper considers the case of an arbitrary dependence of the velocity profile of the basic flow on two spatial variables. The deviation from the plane-parallel case is not assumed to be small, and the corresponding eigenvalue problem for the partial differential equations is solved by means of the direct methods of [5], which were introduced for the first time and justified in the theory of hydrodynamic stability by Petrov [8].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 21–28, May–June, 1987.     

Numerical Investigation of Unipolar Injection in the Presence of Electroconvective Motion in a Plane Layer of Transformer Oil

The effect of unipolar injection of charges on the electroconvective motion of a weakly conducting liquid in a plane-parallel electrode system is investigated. By means of a numerical experiment it is confirmed that the crisis in the stability of a plane layer of weakly conducting liquid with unipolar injection conductivity depends significantly on the electrochemical processes in the electrode layer. A~dependence of the unipolar charge injection on the initial conductivity for a solution of molecular iodine in transformer oil is obtained.     

Rayleigh stability of shear flow in relaxing media

The stability of steady-state flow is considered in a medium with a nonlocal coupling between pressure and density. The equations for perturbations in such a medium are derived in the linear approximation. The results of numerical integration are given for shear motion. The stability of parallel layered flow in an inviscid homogeneous fluid has been studied for a hundred years. The mathematics for investigating an inviscid instability has been developed, and it has been given a physical interpretation. The first important results in flow stability of an incompressible fluid were obtained in the papers of Helmholtz, Rayleigh, and Kelvin [1] in the last century. Heisenberg [2] worked on this problem in the 1920's, and a series of interesting papers by Tollmien [3] appeared subsequently. Apparently one of the first problems in the stability of a compressible fluid was solved by Landau [4]. The first investigations on the boundary-layer stability of an ideal gas were carried out by Lees and Lin [5], and Dunn and Lin [6]. Mention should be made of a series of papers which have appeared quite recently [7–9]. In all the papers mentioned flow stability is investigated in the framework of classical single-phase hydrodynamics. Meanwhile, in recent years, the processes by which perturbations propagate in media with relaxation have been intensively studied [10–12].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 87–93, May–June, 1976.     

Nonlinear waves in a liquid film entrained by a turbulent gas stream

In the long-wavelength approximation and on the basis of a simplified system of equations analogous to the one considered by Shkadov and Nabil' [1, 2], an investigation is made into waves of finite amplitude in thin films of a viscous liquid on the walls of a channel in the presence of a turbulent gas stream. A bibliography on the linear stability of such plane-parallel flows can be found in [3–5]. The nonlinear stability is considered in [6]. A stationary periodic solution is sought in the form of a Fourier expansion whose coefficients are found near the upper curve of neutral stability by Newton's method and near the lower branch of the stability curve by the method of Petviashvili and Tsvelodub [7, 8].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No, 2, pp. 37–42, March–April, 1981.I thank V. Ya. Shkadov for supervising the work and all the participants of G. I. Petrov's seminar for a helpful discussion.     

Linear stability in the flow of a liquid film concurrent with a gas flow

The two-phase flow of liquid films are often encountered in practice, but the number of theoretical papers devoted to this problem is limited. The problem of the linear stability of a viscous liquid film subjected to a gas flow has been formulated in [1] and, in somewhat different form, in [2]. The linear stability of plane-parallel motion in films has been studied analytically in [1–8] for some limiting cases. The range of validity of the analytic approaches remains an open question. Therefore, an exact numerical analysis of flow stability over a fairly broad range is required. In the present paper a separate solution of the problem for the gas and the liquid is shown to be possible. The Orr-Sommerfeld equation has been integrated numerically, and the results are compared to the results of analytic calculations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 143–146, January–February, 1976.The author is grateful to É. É. Markovich for directing the work and to V. Ya. Shkadov for his interest in the work and many useful comments.     

Similarity autooscillations in a plane jet

Experimental investigations into the stability of a plane jet [1, 2] show that after the stationary flow has lost its stability a stable autooscillatory regime arises. In the present paper, an autooscillatory flow in a jet is studied theoretically on the basis of a plane-parallel flow in a fairly wide channel in the presence of a field of external forces. The external forces are such that at zero amplitude of the autooscillations they produce a Bickley—Schlichting velocity profile. The excitation of the secondary regimes is studied by the methods of bifurcation theory [3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 26–32, May–June, 1979.We thank M. A. Gol'dshtik and V. N. Shtern for discussing the formulation of the problem and the results.     

Secondary convective motions in a plane inclined layer

A linear theory of stability of a plane-parallel convective flow between infinite isothermal planes heated to different temperature was developed in [1–6]. At moderate Pr values the instability is monotonic and leads to the development of steady secondary motions. These motions for the case of a vertical layer have been investigated by the net [7, 8] and small-parameter [9] methods. In this paper steady secondary motions in an inclined layer are investigated. The small-parameter and net methods are used. The hard nature of excitation of secondary motions in a defined range of tilt angles is established. There are two types of secondary motions, whose regions of existence overlap — vortices at the boundary of countercurrent streams and convection rolls; the hard instability is due to the development of convection rolls. The analog of the Squire transformation obtained in [4] for infinitely small disturbances of a plane-parallel convective flow is extended to secondary motions of finite amplitude.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 3–9, May–June, 1977.I thank G. Z. Gershumi, E. M. Zhukhovitskii, and E. L. Tarunin for interest in the work and valuable discussion.     

Transient-state flow of a conducting liquid in an MHD generator at constant flow rate in the presence of side walls

It is usual in studies of transient [nonsteady] flow for a viscous incompressible conducting fluid in an MHD channel to take the distance between the side walls as infinite, which allows the initial equations to be simplified, these reducing to a single equation for the velocity if the magnetic Reynolds number is small [1–3]. A real system has a finiteratio of the sides, so it is desirable to establish the effects of the side walls.     

Drag of porous cylinders in a viscous fluid at low reynolds numbers

The problem of flow past a permeable cylinder at low Reynolds numbers is of interest for the solution of a number of problems in chemical technology in, for example, the design of porous electrodes and porous catalysts and in the calculation of nonstationary filtration of aerosols by fibrous filters. In the present paper, we solve the problem of transverse flow of a viscous fluid past a continuous cylinder in a porous shell and, in particular, in the case of a porous cylinder under conditions of constrained flow (system of cylinders) and an isolated cylinder at arbitrary permeability. The analogous problem of Stokes flow past permeable spheres has been solved in a number of papers [1–3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 122–124, November–December, 1979.     

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.     

Stability of plane-parallel MHD flows in transverse magnetic field

We study within the framework of linear theory the stability of plane-parallel flows of a viscous, electrically conducting fluid in a transverse magnetic field. The magnetic Reynolds numbers are assumed small. The critical Reynolds number as a function of the Hartmann number is obtained over the entire range of variation of the latter. The small perturbation spectrum is studied in detail on the example of Hartmann flow. Neutral curves are constructed for symmetric and antisymmetric disturbances. The destablizing effect of a magnetic field is studied in the case of modified Couette flow. The results obtained agree with the calculations of Lock and Kakutani (where they meet) and are at variance with the results of Pavlov.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 11, No. 3, pp. 127–131, May–June, 1970.The authors wish to thank M. A. Gol'dshtik for his interest in this study.     

Stability of frontal displacement of fluids in a porous medium in the presence of interphase mass transfer and phase transitions

To preserve the stability of the front relative to small perturbations when one fluid is displaced by another the pressure gradient must decrease on crossing the front in the direction of displacement. Initially, this criterion was established for the piston displacement of fluids [1, 2], and later in the case of two-phase flow of immiscible fluids in porous media for the displacement front corresponding to the saturation jump in the Buckley—Leverett problem [3, 4]. Below it is shown that the same stability criterion remains valid for flows in porous media accompanied by interphase mass transfer and phase transitions [5, 6]. Processes of these kinds are encountered in displacing oil from beds using active physicochemical or thermal methods [7] and usually reduce to pumping into the bed a slug (finite quantity) of reagent after which a displacing agent (water or gas) is forced in. The slug volume may be fairly small, especially when expensive reagents are employed, and, accordingly, in these cases the question of the stability of displacement is one of primary importance. These active processes are characterized by the formation in the displacement zone of multiwave structures which, in the large-scale approximation (i.e., with capillary, diffusion and nonequilibrium effects neglected), correspond to discontinuous distributions of the phase saturations and component concentrations [5–10]. It is shown that the stability condition for a plane front, corresponding to a certain jump, does not depend on the type of jump [11, 12] and for a constant total flow is determined, as in simpler cases, by the relation between the total phase mobilities at the jump. An increase in total flow in the direction of displacement is destabilizing, while a decrease has a stabilizing influence on the stability of the front. Other trends in the investigation of the stability of flows in porous media are reviewed in [13].Translated fron Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 98–103, March–April, 1986.