Current distribution of permeable electrodes in the presence of the hall effect in a stream of electrically conducting medium

The two-dimensional problem of the current distribution on the surface of permeable electrodes contiguous with a stream of incompressible medium with Hall effect is considered. An electrically conducting medium with the same physical properties as those of the main stream is pumped in (out) through the electrodes.This problem was solved in [1] for one particular case when the electrodes are impermeable. It was established that due to the Hall effect in magnetohydrodynamic channels the current is distributed non-uniformly on the electrodes; for values of the Hall parameter of the order of several units or greater, the current flows into an isotropically conducting medium mainly from a small portion on the edge of the electrode. It was also noted that this phenomenon creates unfavorable conditions for the operation of electrodes in magnetohydrodynamic devices.It is shown in what follows that the current distribution on the electrodes may be controlled, and in particular made more uniform, by injecting an electrically conducting medium.     

Adiabatic heating (cooling) of a supercritical fluid with variation in its physical properties

The piston effect (adiabatic heating or cooling) is investigated numerically for a fluid with parameters in the neighborhood of the thermodynamic critical point. The fluid represents a plane layer on one of whose boundaries a jump in temperature takes place. The temperature of the medium is higher than the critical temperature and the mean density is equal to the critical density. The mathematical model includes the Navier–Stokes and energy equations for a heat-conducting compressible gas and the Van der Waals equation of state. The characteristics of the linear piston effect (properties of fluid are almost invariable) are compared the analytic data. The features of the nonlinear piston effect accompanied by significant variation in the physical properties of the medium are investigated.     

On the possibility of shock-induced cavitation in submerged cylindrical shell systems

A circular cylindrical shell loaded by one or two fluids and responding to an external shock wave is analyzed in the context of the possible inception of shock-induced cavitation. Several scenarios of fluid contact are considered including a submerged evacuated shell and a submerged fluid-filled shell for three different combinations of the parameters of the internal and external fluids. A semi-analytical shell-shock interaction model is employed in order to predict the regions of the fluids where cavitation is likely to occur, and the respective cavitation development is hypothesized about. The most interesting and practically important finding is that when fluid is present both inside and outside the shell, there exist conditions when cavitation is expected to occur in both the internal and external fluid, resulting in a particularly complex and violent structural re-loading occurring upon the collapse of the respective cavitation regions. The inception of cavitation in the internal fluid alone and in the external fluid alone is also possible. The findings are summarized in a manner that is suitable for use at the pre-design stage as a guide for preliminary assessment of the possibility of shock-induced cavitation in fluid-interacting industrial systems.     

Two-dimensional simulation of a hydraulic fracture crack cleaning process with consideration of lateral inflow of a fluid from the surrounding porous saturated rock

The cleaning of a hydraulic fracture crack filled with a fluid injected through a well is studied as one of the stages of oil extraction. A crack is considered as a porous medium whose permeability is much higher than that of the surrounding rock and whose length is several times larger than its width and is many times larger than its thickness. A two-dimensional model of this process is used; in this model it is assumed that a less viscous fluid displaces a more viscous fluid in a porous medium with consideration of inflow through the lateral surface of the crack.     

Throughflow Effects on Penetrative Convection in Superposed Fluid and Porous Layers

The effect of vertical throughflow on the onset of penetrative convection simulated via internal heating in a two-layer system in which a layer of fluid overlies and saturates a layer of porous medium is studied. Flow in the porous medium is governed by Forchheimer-extended Darcy equation, and Beavers?CJoseph slip condition is applied at the interface between the fluid and the porous layers. The boundaries are considered to be rigid, however permeable, and insulated to temperature perturbations. The eigenvalue problem is solved using a regular perturbation technique with wave number as a perturbation parameter. The ratio of fluid layer thickness to porous layer thickness, ??, the direction of throughflow, and the presence of volumetric internal heat source in fluid and/or porous layer play a decisive role on the stability characteristics of the system. In addition, the influence of Prandtl number arising due to throughflow is also emphasized on the stability of the system. It is observed that both stabilizing and destabilizing factors can be enhanced because of the simultaneous presence of a volumetric heat source and vertical throughflow so that a more precise control (suppress or augment) of thermal convective instability in a layer of fluid or porous medium is possible.     

Two-dimensional problem of electrochemical metal machining

The two-dimensional problem of the electrochemical dimensional machining of a metal is investigated within the framework of the model of an ideal stationary process, which makes it possible to use the analogy with the problems of fluid flows with free surfaces. In the problem considered the cathode (machining tool) takes the form of two parallel semi-infinite rectangular electrodes. The blank (anode) is a half-plane whose boundary is perpendicular to the cathodes. Depending on the relationship between the physical and geometrical parameters of the problem, on the machined part (anode) a projection symmetrical about the center line between the cathodes may be formed. Additional mechanical machining of the part is then required. In order to exclude such solutions, a condition is obtained for the mathematical parameters which determine the solution of the problem in the auxiliary complex plane. General and particular limiting cases are considered. For the cases considered the calculation results are presented in the form of plots of the shape of the part machined.     

On a model of structured medium fracture under compression conditions

Extension is necessary in order to initiate brittle fracture in a structured medium. One possible version of the fracture scenario under compression conditions is the development of extension near local concentrators. The dimensions of such regions depend on the level of hydrostatic compression (pressure). As the hydrostatic compression increases, these dimensions decrease approaching the dimension of individual structure elements of the medium (e.g., grains or pores). Under these conditions, the mechanisms of brittle fracture of the medium differ from those in ordinary structural materials. We consider mechanisms of brittle fracture in compression. It is assumed that the sources of the local extension required for the development of discontinuities are the response of the heterogeneous structure elements of the medium (pores) at which the external compressive stresses can transform into local extensions sufficient for crack propagation. In this case, a characteristic cell representing the scale of the leading process of local fracture is a volume containing two pores. The coalescence of these pores is an elementary fracture process. An increase in the pressure level increases the role of such fracture processes and leads to the development of ordered fracture structures. Several examples of fracture scenarios under loads that are combinations of homogeneous compression and a concentrated action are used to illustrate the conditions for the development of discontinuities, in whose end regions an elementary act of pore coalescence takes place, and to demonstrate the appearance of linear structures such as curtains or echelons of microcracks.     

Small and Moderate Prandtl Number Chaotic Convection in Porous Media in the Presence of Feedback Control

Feedback control on thermal convection in a fluid-saturated porous medium is investigated based on the dynamical systems approach. A low dimensional Lorenz-like model was obtained using the Galerkin-truncated approximation. The possible suppression or enhancement of chaotic convection is demonstrated when the fluid layer is subjected to feedback control in a low-dimensional framework.     

Vibrational Convection in an Inclined Fluid Layer Heated from Below

The stability of mechanical equilibrium of an inclined fluid layer with respect to three-dimensional perturbations under the action of high-frequency vibration is studied. It is shown that under heating from below the spiral perturbations are always the most dangerous for vibration transverse to the layer. For vertical vibration the stability limit is determined by three-dimensional perturbations whose shape depends in a complicated way on the angle of inclination of the layer and the vibrational Rayleigh number. In the limiting case of a thin vertical layer supercritical vibrational-convective motions are calculated numerically and analytically and scenarios of transition from quasi-equilibrium to irregular motions are studied.     

Nonlinear effects in gas filtration

The existence of considerable deviations from the linear Darcy filtration law has been established for numerous systems consisting of a fluid and a porous medium. One of the manifestations of this nonlinearity is the existence of a limiting (initial) pressure gradient—the minimum value of the pressure gradient for which fluid motion occurs. (As a rule, fluid motion still takes place for subcritical values of the pressure gradient, but very slowly; on reaching the limiting value of the pressure gradient there is a marked acceleration of the filtration. The limiting-gradient concept thus provides a good approximation for velocities which are not too low.)We would also expect nonlinear effects in the motion of a Newtonian liquid or gas in a porous medium containing some amount of fluid which does not participate in the main motion. These effects can take the form of layers enclosing the particles of the porous medium and partly or completely blocking the pore channels. For sufficiently high pressure gradients rearrangement of these layers must begin, accompanied by a change of the hydrodynamic resistance of the porous medium.As a result of this restructuring it is natural to expect a disproportionately fast increase of the filtering-fluid flow rate with increase of the pressure differential; i.e., the filtration law of a Newtonian fluid in a medium containing a. layer of bound fluid having elasticity will have the form which is characteristic for pseudoplastic non-Newtonian fluids. In particular, if the initial attached-fluid content is so large that all the pore channels are blocked in the initial state, then the motion of the liquid (gas) being externally pumped begins only after the attached fluid layers are partly ruptured. Therefore, under these conditions the appearance of a limiting (initial) pressure gradient for filtration of a Newtonian fluid is possible. This can occur in filtration of a gas in argillous rocks containing connate water, since the water and the clay particles form a colloidal suspension which has some shear strength.This phenomenon may be of importance in the development of gas deposits which are associated with argillous rocks, particularly in determining the possible current and final gas yield. In fact, the most characteristic property of flows with an initial gradient is the formation of impermeable blocks; if the deposits have the usual nonuniformity the incompleteness of the gas extraction from the reservoir will manifest itself in both microscopic and macroscopi scales.In the following study we present relations describing gas filtration under such conditions and results of laboratory experiments which confirm the concepts described. Hydrodynamic estimates are also made of the possible effects of the initial gradient.The authors wish to thank I. I. Eremina for assistance in making the calculations, A. Sh. Asadov and Sh. S. Aslanov for assistance in conducting the experiments.     

Magnetohydrodynamic laminar boundary-layer flow over a porous wedge with thermophoresis particle deposition in the presence of variable stream conditions

An analysis is presented to investigate the effects of thermophoresis variable viscosity on MHD mixed convective heat and mass transfer of a viscous, incompressible and electrically conducting fluid past a porous wedge in the presence of chemical reaction. The wall of the wedge is embedded in a uniform porous medium in order to allow for possible fluid wall suction or injection. The governing boundary layer equations are written into a dimensionless form by local non-similarity transformations. The transformed coupled nonlinear ordinary differential equations are solved numerically by using the R.K. Gill and shooting methods. Favorable comparison with previously published work is performed. Numerical results for the dimensionless velocity, temperature and concentration profiles are obtained and displayed graphically for pertinent parameters to show interesting aspects of the solution.     

Low-frequency model of a granular medium saturated with a viscous fluid

A mathematical model of a granular medium saturated with a viscous homogeneous fluid is constructed. The steady-state one-dimensional oscillations of cylindrical granules and an incompressible fluid under the action of a plane sonic wave whose length is significantly greater than the cell dimensions are investigated. The steady-state flow of the medium across the cell cross-section and the mean fluid velocity (Darcy’s law) are determined by means of passages to the limits with respect to the frequency and granule mass. The expressions obtained for the soil permeability coefficient under the action of a gravitational hydraulic head are compared with the representations of other authors.     

Derivation of a Darcy’s Law for a Porous Medium Composed of Two Solid Phases Saturated by a Single- Phase Fluid: A Homogenization Approach

The objective of this article is to derive a macroscopic Darcy’s law for a fluid-saturated moving porous medium whose matrix is composed of two solid phases which are not in direct contact with each other (weakly coupled solid phases). An example of this composite medium is the case of a solid matrix, unfrozen water, and an ice matrix within the pore space. The macroscopic equations for this type of saturated porous material are obtained using two-space homogenization techniques from microscopic periodic structures. The pore size is assumed to be small compared to the macroscopic scale under consideration. At the microscopic scale the two weakly coupled solids are described by the linear elastic equations, and the fluid by the linearized Navier–Stokes equations with appropriate boundary conditions at the solid–fluid interfaces. The derived Darcy’s law contains three permeability tensors whose properties are analyzed. Also, a formal relation with a previous macroscopic fluid flow equation obtained using a phenomenological approach is given. Moreover, a constructive proof of the existence of the three permeability tensors allows for their explicit computation employing finite elements or analogous numerical procedures.     

Numerical modelling of the stability of loaded shells of revolution containing fluid flows

A mixed finite-element algorithm is proposed to study the dynamic behavior of loaded shells of revolution containing a stationary or moving compressible fluid. The behavior of the fluid is described by potential theory, whose equations are reduced to integral form using the Galerkin method. The dynamics of the shell is analyzed with the use of the variational principle of possible displacements, which includes the linearized Bernoulli equation for calculating the hydrodynamic pressure exerted on the shell by the fluid. The solution of the problem reduces to the calculation and analysis of the eigenvalues of the coupled system of equations. As an example, the effect of hydrostatic pressure on the dynamic behavior of shells of revolution containing a moving fluid is studied under various boundary conditions. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 2, pp. 185–195, March–April, 2008     

Numerical investigation of variable viscosities and thermal stratification effects on MHD mixed convective heat and mass transfer past a porous wedge in the presence of a chemical reaction

An analysis is presented to investigate the effects of variable viscosities and thermal stratification on the MHD mixed convective heat and mass transfer of a viscous, incompressible, and electrically conducting fluid past a porous wedge in the presence of a chemical reaction. The wall of the wedge is embedded in a uniform nonDarcian porous medium in order to allow for possible fluid wall suction or injection. The governing boundary layer equations are written into a dimensionless form by similarity transformations. The transformed coupled nonlinear ordinary differential equations are solved numerically with finite difference methods. Numerical calculations up to the thirdorder level of truncation are carried out for different values of dimensionless parameters. The results are presented graphically, and show that the flow field and other quantities of physical interest are significantly influenced by these parameters. The results are compared with those available in literature, and show excellent agreement.     

Intrapore convection when the average temperature gradient is vertical

The heat conduction of a porous medium saturated with a fluid is usually regarded as being purely molecular [1]. The assumption here is that in the case of heating from below the local temperature gradient within each of the pores, like the averaged gradient in the complete layer, is strictly vertical, and, since the pores are as a rule small, this local gradient is less than the critical. It is therefore assumed that in the absence of large-scale convection the fluid in the pores is in equilibrium. However, for different thermal conductivities of the fluid and the porous skeleton surrounding it a vertical temperature gradient in the fluid and, accordingly, equilibrium of the fluid are possible only if a cavity is a sphere or an ellipsoid with a definite orientation [1]. Since the pores do not have such shapes, the convective motion that arises in each of the pores or in several communicating pores can lead to an increase in the effective thermal conductivity of the fluid and, accordingly, the effective thermal conductivity of the complete medium. The present paper is devoted to study of this effect.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 93–98, January–February, 1984.     

Motion of a vortex near the boundary between two heavy fluids

The motion of a vortex beneath the surface of a heavy fluid has been discussed in both linear [1, 2] and nonlinear [3–5] formulation. The density of the upper medium is neglected, which makes it possible to replace the continuity of pressure during transition through the boundary between the media by constancy of the pressure at the boundary of the heavy fluid. In this paper, the problem is solved in a general nonlinear formulation, including the mutual effects of media motion, and the vortex can be in either the upper or lower medium. Steady-state motion of a vortex of given intensity near the boundary between two heavy fluids is discussed in terms of a model of an ideal and incompressible medium. Approximate expressions are obtained for the boundary.     

A model of the steady-state motion of suspensions

The authors examine the steady-state one-dimensional motions of suspensions whose particles have a density equal to that of the corresponding dispersion medium. As a whole, the mechanical behavior of such suspensions is described by equations of motion that coincide in form with the Navier-Stokes equations for a certain incompressible fluid whose viscosity is a known function of the particle concentration in the suspensions. To close these equations, the authors postulate a principle of minimum energy dissipation for steady-state motion, which plays the paxt of an equation of state for the suspension. This new equation permits the determination of the spatial distribution in the concentration of solids. Exact solutions are presented for certain variational problems associated with the Poiseuille flow of a fluid of this kind in circular tubes and Couette flows between concentric cylinders and parallel planes. It is shown that in most cases separation of the suspension takes place.     

On the problem of the stability of convective two-phase medium flows under high-frequency vibration

The principles and methods of constructing a model of vibrational convection in a medium consisting of a liquid (gas) and a solid admixture are discussed. A closed system of averaged equations is first obtained. The system admits passage to the limits of the equations of both vibrational convection in a homogeneous fluid and convection in a dusty medium in the static case. As a measure of the difference with respect to the homogeneous fluid, in addition to the sedimentation parameter, which also manifests itself in the absence of vibrational accelerations, it is possible to take the inhomogeneity parameter introduced in this study and responsible for the pulsatory transport of the average fields. The problem of the stability of plane parallel flow in a vertical layer of a two-phase medium under horizontal longitudinal vibration with respect to infinitesimal perturbations is considered. It is shown that the introduction of particles into the flow leads to qualitatively novel effects which cannot be predicted within the framework of the homogeneous fluid model.     

Influence of the hall effect on the characteristics of a MHD generator with two pairs of electrodes

The characteristics of segmented-electrode MHD generators with Hall currents are at present the object of considerable interest. Various types of electrode connections ate being examined: ordinary segmented-electrode generators, Hall generators, mixed-type generators, and Montardy generators. Research is being pushed in several directions. In some cases infinitely fine segmentation is assumed [1--4]. In these studies it is considered that the current density j in the duct is everywhere uniform; the net characteristics or the most favorable electrode connection angle are determined. In another group of studies periodic problems are solved, it being assumed that the processes taking place in a single elementary celi are repeated in the other ceils; fringe effects are not taken into account. In this case it is usually assumed that the lengths of the electrodes and insulators are finite, but small as compared with the duct height [5–7 ]. Finally, in a last group of studies nonperiodic problems are considered. In [3] Vatazhin solves the problem of the current distribution in a duct with a single pair of electrodes between infinitely long insulators. In [8] a general expression is found for the current density function in the case of an arbitrary number of electrode pairs and for any scheme of electrode connection at finite electrode and insulator dimensions. However, numerical calculations are made only for the periodic problem, whose solution is also obtained in [8]; the effect of segment pitch on the characteristics of a Montardy generator is studied.The present author has investigated the influence of the Hall effect on the characteristics of a MHD generator having two pairs of electrodes with symmetrical and crossed electrode connections. Although it is obvious that in practice only multisegment ducts will be employed, the examination of a generator with two pairs of electrodes makes possible the qualitative anatysis of the various effects observed in segmented-electrode ducts in which the electrodes are connected in different ways. Numerical calculations, based on formulas obtained by solving the corresponding problems, have been made on a M-20 computer. Integrated characteristics of the various generator systems have been obtained as a function of electrode and insulator length, external loads, and Hall parameter w.In conclusion, the author thanks A. B. Vatazhin and A. N. Kraiko for their helpful advice.