Self-similar solutions of non-Newtonian fluid boundary layer in MHD

The self-similar solutions of the boundary layer for a non-Newtonian fluid in MHD were considered in [1, 2] for a power-law velocity distribution along the outer edge of the layer and constant electrical conductivity through the entire flow. However, the MHD flows of many conducting media, which are solutions or molten metals, cannot be described by the MHD equations for non-Newtonian fluids.The self-similar solutions of the boundary layer for a non-Newtonian fluid without account for interaction with the electromagnetic field were studied in [3].In the following we present the self-similar solutions for the boundary layer of pseudoplastic and dilatant fluids with account for the interaction with an electromagnetic field for the case of a power-law velocity distribution along the outer edge of the layer, when the conductivity of the fluid is constant throughout the flow and the magnetic Reynolds number is small.Izv. AN SSSR. Mekhanika Zhidkosti i Gaza, Vol. 2, No. 6, pp. 77–82, 1967The author wishes to thank S. V. Fal'kovich for his interest in this study.     

Nonstationary flow of a conducting gas in crossed electric and magnetic fields

The plasma is inviscid, cool, and not thermally conducting; it flows in a channel of constant cross section. The solution is derived by the small parameter method, for which purpose the magnetic interaction N is used. There have been previous studies of the transient-state flow of an inviscid and thermally nonconducting plasma in crossed electric and magnetic fields [1–3]. A plasma of infinite conductivity has been considered [1], as well as flow involving entropy change in an MHD system with strong electromagnetic fields [2, 3].     

Method of narrow strips for nonlinear problems of a stratified fluid

Plane steady flow is considered for an ideal incompressible stratified fluid in a gravitational field of force. It is a characteristic feature of these flows that the density is constant and Bernoulli's constant remains the same along a streamline. Internal waves arise because of ponderability in the stratified fluid; they are not due to the presence of a free surface. These wave motions are studied in detail in the linear formulation, but flows of the solitary wave type can be described only by nonlinear equations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 174–178, March–April, 1986.     

Steady-state wave flows of a weakly conducting gas in transverse electromagnetic fields

Gas flow in a MHD channel in transverse magnetic and electric fields is considered. The steady-state flows associated with the establishment of equilibrium between the hydraulic resistence and ponderomotive forces are investigated. The conditions of existence and the properties of such a steady-state flow regime are analyzed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 123–129, July–August, 1990.     

Motion of two pulsating spheres in an ideal incompressible fluid

The problem of the interaction of two pulsating spheres in an ideal incompressible fluid was first investigated in detail by Bjerknes [1]. However, his and subsequent studies on this subject [2–5] were restricted to the interaction forces between the spheres, whereas the law of their motion was not considered because of the much greater complexity of the corresponding problem. The aim of the present paper is to find an approximate analytic solution to the problem of the motion of two pulsating spheres in an ideal incompressible fluid filling the entire space exterior to the spheres under the assumption that the flow of the fluid is irrotational.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 159–162, May–June, 1983.     

Stability of two-dimensional curvilinear flows of an ideal barotropic fluid

It was established by Arnol'd [1] that the conservation laws for the energy and vorticity can be used to establish sufficient conditions of stability of two-dimensional curvilinear flows of an ideal incompressible fluid in the exact nonlinear formulation. It is shown below that one can obtain similarly conditions of stability of two-dimensional curvilinear steady flows of an ideal barotropic fluid in the linear approximation. One of the conditions has a significance similar to Rayleigh's criterion and its generalization by Arnol'd [1]; the other is the condition of subsonic flow. In addition, a variational principle is established and an expression found for the second variation of the corresponding functional; these can be used to prove the stability of these flows in the exact nonlinear formulation.Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 5, pp. 19–25, September–October, 1981.I am sincerely grateful to V. L. Berdichevskii and A. G. Kulikovskii for constructive advice.     

Solution of a variational problem for the flow in an MHD generator

The most complete study and construction of extremal plasma flow regimes in the channel of an MHD generator may be accomplished using the methods of variational calculus. The variational problem of conducting-gas motion in an MHD channel was first discussed in [1]. The general formulation of the problem for the MHD generator was considered in [2]. Solutions of variational problems for particular cases of extremal flows are given in [2–5].The present study obtains the solution of the variational problem of the flow of a variable conductivity plasma in an MHD generator which has maximal output power for given channel length or volume. An analysis of the solution is made, and a comparison of the extremal flows with optimized flow in a generator with constant values of the electrical efficiency and flow Mach number is carried out.     

Some problems in metallurgical fluid mechanics

Application of computational fluid mechanics in two areas of metallurgy is considered: solidification of liquid alloys and MHD turbulence. In the first class of problems, where the technical issue is to obtain a reasonably homogeneous composition of the cast, one has to consider not only the completely molten and solidified regions but also the “mushy zone” that is made up of small-scale dendrites, which appear between the first two regions. In the completely molten region, the composition is practically constant and the fluid is set into motion due to the inhomogeneous temperature field. In the mushy zone, on the other hand, solutal convection often dominates strongly over thermal convection. It is shown that laminar convection is of prime importance for the composition of the solidified alloy. In the second class of problems, two cases of turbulent MHD flows in cylindrical containers are considered: an electromagnetic furnace and an electromagnetic stirrer. In the electromagnetic furnace, the mean flow consists of two toroidal vortices. The mean motion in the electromagnetic stirrer is a swirling motion that is accompanied by a weak meridional circulation, which is reminiscent of that occuring in spin down phenomena. The MHD flows are computed by using large eddy simulation methodology with a new subgrid model of the Smagorinsky type that accounts for a variable mesh. Predictions from the all model computations are compared with experimental observations. In general, the agreement between theory and experiments is satisfactory.     

AN UPPER BOUND ON RANDOM BUFFETING FORCES CAUSED BY TWO-PHASE FLOWS ACROSS TUBES

This paper investigates the random buffeting excitation forces that apply to tubes in two-phase cross-flows. The spectral magnitude of these forces is studied with the help of a database that includes most of the available experimental data on direct or indirect force measurements. Different fluid mixtures, namely steam–water, air–water and various types of Freon, as well as different thermohydraulic or geometrical situations are included in the database. Using a formalism similar in principle to that used successfully in the modelling of buffeting in single-phase flows, the scaling of the data is undertaken. While dynamic pressure, viscosity or surface tension are found not to be relevant parameters, gravity forces allow us to define appropriate dimensionless spectra for all cases. The meaning of these parameters as well as the effects of flow regimes or fluid mixtures are discussed. Finally, an upper bound on the magnitude of these forces, which is of practical applicability, is proposed.     

Development and instability of steady convective flows in a square cavity heated from below and a field of vertically directed vibrational forces

The present paper is devoted to numerical investigation of the spatial structure and stability of secondary vibrational convective flows resulting from instability of the equilibrium of a fluid heated from below. Vibrations parallel to the vector of the gravitational force (vertical vibrations) are considered. As in earlier work [7–9], a region of finite size is used — a square cavity heated from below. It is shown that enhancement of the vibrational disturbance of the natural convective flow may either stabilize or destabilize flows with different spatial structures; it may also stabilize certain solutions of the system of convection equations that are unstable in the absence of vibrational forces. In addition, increase of the vibrational Rayleigh number can lead to a change of the mechanisms responsible for equilibrium instability and oscillatory instability of the secondary steady flows.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 9–18, March–April, 1991.I thank G. Z. Gershuni for assistance and extremely fruitful discussions of the results of the paper.     

Unsteady flows of an inhomogeneous incompressible viscous fluid

This paper deals with a theoretical analysis of the transfer of reactive impurities by open and filtration flows of an incompressible viscous fluid. The first section of the paper studies the model of an inhomogeneous incompressible viscous fluid, which is widely used in meteorology and oceanology, with additional allowance for the drag of the magnetic field or porous medium. Another object of research in this paper is the model of filtration of an inhomogeneous incompressible fluid in porous media proposed by V. N. Monakhov (1977) (Section 2). In both models, hydrodynamic flows determine the motion of the mixture as a whole and the temperature and concentration distributions of the components of an inhomogeneous fluid are described by a common nonlinear system of equations of diffusive heat and mass transfer.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 2, pp. 44–51, March–April, 2005.     

Thermomechanical Behavior of Electrically Conducting Solids Exposed to an External Electromagnetic Field

This paper describes a procedure for the mathematical and numerical modeling of the thermomechanical behavior of electrically conducting solid bodies exposed to an external electromagnetic field. The constitutive equations for the electromagnetic field are the Maxwell equations written for the region of the solid body and the ambient medium. The stress-strain state of the solid is described using the relations for nonisothermal elastoplastic flow. The effects of the electromagnetic field on the heat-transfer and deformation processes are taken into account via heat release and ponderomotive forces, respectively. The relations between the electric and magnetic inductions and the corresponding field strengths are considered nonlinear. All physicomechanical parameters of the body material are temperature dependent.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 5, pp. 14–26, September–October, 2005.     

Convection in an oscillating force field and microgravity

Convective flows in a plane layer of viscous fluid in the presence of an oscillating external force are investigated numerically [1 – 8]. The layer is assumed to be placed in a gravitational field. The cases in which the external field oscillations are generated by rotation about the horizontal axis or by vibration in the longitudinal direction are considered. The Navier-Stokes equations and the Boussinesq approximation are used for describing the fluid motion. The flows developing in the layer in the presence of a transverse temperature gradient are determined, the stability boundaries of these flows are found, and the supercritical motion regimes are studied. These investigations are carried out using the averaging method (in order to find the stability limits for high rotation velocities and vibration frequencies) and the Galerkin method.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 99–106, September–October, 1994.     

A case of jet flow of a gravity fluid

The problem of plane steady ideal heavy fluid flow bounded by an impermeable polygonal section, a curvilinear arc section, and a finite section of free surface is investigated in an exact nonlinear formulation. Hydrodynamic singularities may exist in the stream. A large class of captation problems of jet theory reduces to studying this kind of flow. The unique solvability of the problem under investigation is proved for sufficiently large Froude numbers and small arc curvature. A method of solution is given and an example is computed. Such problems have been solved earlier by numerical methods [1–3]. Some problems about jet flows of a gravity fluid with polygonal solid boundaries have been investigated by an analogous method in [4, 5].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 140–143, May–June, 1975.     

Viscous fluid flow near the line of intersection of curved surfaces

Viscous fluid flow near the line of intersection of curved surfaces at large Re numbers is a topic of considerable interest. The intersection of two fixed planes has been the subject of many experimental and theoretical studies. This case is characterized by very small transverse velocities and by the fact that the corner does not affect the remoter parts of the flow [1–4]. The flows near intersecting curved surfaces have received very little attention, except for the particular case of the intersection of a concave cylindrical surface and a plane in an incompressible fluid flow. With reference to this example it has been shown that the curvature qualitatively affects the flow pattern not only near the line of intersection but also at a distance from it [5]. The present article is concerned with viscous fluid flow at Re1 near the line of intersection of arbitrary, relatively smooth surfaces in the presence of external body forces and, moreover, in the noninertial coordinate system moving with the exposed surfaces (for example, rotating surfaces). On the basis of an analysis of the Navier-Stokes equations and the energy equation as Re sufficient conditions are obtained for the development of intense transverse flows near the line of intersection, which also lead to a qualitative change in the flow pattern; it is shown that depending on the external forces and the geometric parameters of the surfaces various types of flow are possible; the relations determining the occurrence of a particular type of flow and the equations and necessary boundary conditions describing some of these flows are obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 16–21, May–June, 1985.     

Integral characteristics of MHD channel with nonconducting baffles

The electromagnetic characteristics of an MHD channel with insulated walls, two quite long electrodes, and nonconducting baffles are obtained for arbitrary magnetic field distribution law along the channel. Some specific cases of magnetic field and baffle location specification are examined in detail.Translated from Zhurnal Prikladnoi Mekhaniki Tekhnicheskoi Fiziki, Vol. 10, No. 6, pp. 30–39, November–December, 1969.     

Nonlinear magnetoacoustic waves in a conducting liquid containing gas bubbles

The propagation of long waves in an incompressible conducting liquid saturated with nonconducting gas bubbles is considered on the basis of the equations of magnetohydrodynamics of a homogeneous gas-liquid medium. It is shown that the propagation of weakly nonlinear MHD waves in such a medium is described by the Burgers-Korteweg-de Vries (BKdV) equation. The influence of MHD interaction effects on the parameters of fast and slow weak magnetoacoustic shock waves is investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 142–147, March–April, 1991.     

Heat transfer enhancement with deformation of the velocity profile

The problem of enhancing the heat transfer in channels and boundary layers by the appropriate deformation of the fluid velocity profile is considered. The resulting additional hydraulic losses, the price of heat transfer enhancement, are determined. The possibilities of controlling heat transfer by redistributing the fluid velocity in channels are demonstrated with reference to flows at low Prandtl numbers. Laminar and turbulent liquid and gas flows with heat transfer in channels and boundary layers are numerically modeled on the basis of modern models of turbulence (flow development in channels with different initial velocity profiles, flows with wall roughness and boundary layer flows with forces acting on the flow to cause deformation of the velocity profile). In all cases it is found that the heat transfer can be enhanced only at the expense of a considerable increase in the hydaulic losses. A class of self-similar thermal problems for flows in plane diffusers is formulated. The eigenfunctions — temperature modes — for various velocity profiles are determined with allowance for the nonuniqueness of the solution of the classical dynamical problem for a plane diffuser and the corresponding heat transfer coefficients are found.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.4, pp. 94–105, May–June, 1993.The authors are grateful to A. Yu. Klimenko for useful discussions.     

On the asymptotic behavior of localized perturbations in the presence of Kelvin-Helmholtz instability

The asymptotic behavior of localized two-dimensional perturbations of the surface of a shear discontinuity separating two homogeneous steady flows of ideal incompressible fluid is studied in the linear approximation. The effect of surface tension and gravity forces is taken into account. Mathematically the problem reduces to the investigation by the method of steepest descent of the asymptotic behavior of a double integral for various values of parameters which are the components of the group velocity vector. In this problem the principal difficulty is to find the two-dimensional steepest descent contour in the space of two complex variables that determines which of the various saddle points gives the asymptotic form. First, for the Fourier component with respect to one of the variables with allowance for all the saddle points we find an asymptotic form which parametrically depends on the second variable. The choice of the second variable makes it possible to prove analytically that in the absence of gravity the asymptotic behavior of the growing perturbations is determined by a single saddle point in the plane of that variable. In this way it is possible to justify the authors' previous conclusions [1] concerning the shape of the boundary L of the region D in the group velocity plane occupied by growing perturbations. In the presence of gravity the growth rates of perturbations corresponding to different group velocities are found numerically and the region D occupied by the growing perturbations is indicated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 23–30, March–April, 1985.     

Comparison of the RANS and PDF methods for air-particle flows

Two simulation methods, namely Reynolds-Averaged Navier–Stokes (RANS) equations, and Probability Distribution Function (PDF) are currently widely used for the modeling of multiphase flows. These two approaches are supplemented with appropriate closure equations that take into account all the pertinent forces and interaction effects on the solid particles, such as: particle–turbulence interactions; turbulence modulation; particle–particle interactions; particle–wall interactions; gravitation, drag and lift forces. The two methods have been used in order to simulate the turbulent particulate flow in upward pipes. The flow domain in all cases was a cylindrical pipe and the computations were carried for upward pipe flow. Monodisperse as well as polydisperse mixtures of particles have been considered. In general, the average velocity results obtained from the two methods are in close agreement, because the methods predict well the average velocity distribution of the carrier fluid as well as the solids. Thus, the differences in the average axial velocities predicted by the methods are not substantial. Differences in the turbulence intensity are more significant. A comparison of the numerical results obtained shows the relative importance of retaining the diffusion terms in both the axial and radial directions in the RANS method. Also the comparisons of the results show the relative effect of the lift forces in the distribution of solid particles.