Numerical investigation of the hydrodynamic drag of a circular cylinder coated with a thin layer of magnetic fluid

In order to reduce the drag of bodies in a viscous flow it has been proposed to apply to the surface exposed to the flow a layer of magnetic fluid, which can be retained by means of a magnetic field and thus act as a lubricant between the external flow and the body [1, 2]. In [1] the hydrodynamic drag of a current-carrying cylindrical conductor coated with a uniform layer of magnetic fluid was theoretically investigated at small Reynolds numbers. In order to simplify the equations of motion, the Oseen approximation was introduced for the free stream and the Stokes approximation for the magnetic fluid [3]. This approach has led to the finding of an exact analytic solution from which it follows that at Reynolds numbers Re 1 the drag of the cylinder can be considerably reduced if the viscosity of its magnetic-fluid coating is much less than the viscosity of the flow. The main purpose of the present study is to investigate, with reference to the same problem, how the magnetic-fluid coating affects the hydrodynamic drag at Reynolds numbers 1 Re 10²–10³, i.e., under separated flow conditions. In this case the simplifications associated with neglecting the nonlinear inertial terms in the Navier—Stokes equation are inadmissible, so that a solution can be obtained only by numerical methods.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 11–16, May–June, 1986.     

Cylinder in viscous incompressible fluid flow

We present a technique for calculating the temperature field in the vicinity of a cylinder in a viscous incompressible fluid flow under given conditions for the heat flux or the cylinder surface temperature. The Navier-Stokes equations and the energy equation for the steady heat transfer regime form the basis of the calculations. The numerical calculations are made for three flow regimes about the cylinder, corresponding to Reynolds numbers of 20, 40, and 80. The pressure distribution, voracity, and temperature distributions along the cylinder surface are found.It is known that for a Reynolds number R>1 the calculation of cylinder drag within the framework of the solution of the Oseen and Stokes equations yields a significant deviation from the experimental data. In 1933 Thom first solved this problem [1] on the basis of the Navier-Stokes equations. Subsequently several investigators [2, 3] studied the problem of viscous incompressible fluid flow past a cylinder.It has been established that a stable solution of the Navier-Stokes equations exists for R40 and that in this case the calculation results are in good agreement with the experimental data. According to [2], a stable solution also exists for R=44. The possibility of obtaining a steady solution for R>44 is suggested.Analysis of the results of [2] permits suggesting that the questions of constructing a difference scheme with a given order of approximation of the basic differential relations which will permit obtaining the sought solution over the entire range of variation of the problem parameters of interest are still worthy of attention.Calculation of the velocity field in the vicinity of a cylinder also makes possible the calculation of the cylinder temperature regime for given conditions for the heat flux or the temperature on its surface. However, we are familiar only with experience in the analytic solution of several questions of cylinder heat transfer with the surrounding fluid for large R within the framework of boundary layer theory [4].     

Diffusion flux to a distorted gas bubble at large Reynolds numbers

The diffusion flux to a distorted gas bubble situated in a uniform viscous incompressible fluid flow is determined for large Reynolds and Péclet numbers and finite Weber numbers. The bubble has the shape of an ellipsoid of revolution, oblate in the flow direction, making it possible to use the flow field derived by Moore [1] in the form of a two-term expansion with respect to the flow parameter =R^–1/2 (R is the Reynolds number; the zeroth term of the expansion corresponds to potential flow). The dependence of the diffusion flux onto the bubble surface on the Weber and Reynolds numbers is determined. The results of Winnikow [2] and Sy and Lightfoot [3] are thus generalized to the case of finite Weber numbers and a broader range of Reynolds numbers.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 70–76, July–August, 1976.     

The problem of a gas bubble in plane ideal fluid flow

We study the problem of two-dimensional fluid flow past a gas bubble adjacent to an infinite rectilinear solid wall.Two-dimensional ideal fluid flow past a gas bubble on whose boundary surface-tension forces act (or a gas bubble bounded by an elastic film) has been studied by several authors. Zhukovskii, who first studied jet flows with consideration of the capillary forces, constructed an exact solution of the problem of symmetric flow past a gas bubble in a rectilinear channel [1]. However, Zhukovskii's solution is not the general solution of the problem; in particular, we cannot obtain the flow past an isolated bubble from his solution. Slezkin [2] reduced the problem of symmetric flow of an infinite fluid stream past a bubble to the study of a nonlinear integral equation. The numerical solution of this problem has recently been found by Petrova [3]. McLeod [4] obtained an exact solution under the assumption that the gas pressure p₁ in the bubble equals the flow stagnation pressure p₀. Beyer [5] proved the existence of a solution to the problem of flow of a stream having a given velocity circulation provided p₁p₀.We examine the problem of two-dimensional ideal fluid flow past a gas bubble adjacent to an infinite rectilinear solid wall. The solution depends on the value of the contact angle . The existence of a solution is proved in some range of variation of the parameters, and a technique for finding this solution is given. The situation in which =1/2 is studied in detail.     

Interaction of two spherical particles in a steady flow of viscous fluid when the Reynolds number is not small

The problem of the interaction of two or more particles moving in a viscous incompressible fluid at small Reynolds numbers (Re 1) has been well studied. The linearity of the Stokes equations makes it possible to develop effective methods of solution of the problem for two and many particles [1]. If the Reynolds number is not small, the inertia forces in the Navier-Stokes equations cannot be ignored, and the problem becomes nonlinear, i.e., much more complicated. The present note is devoted to the problem of the interaction of two spherical particles in a steady uniform flow of a viscous incompressible fluid when the Reynolds number is not small. Asymptotic expressions are obtained for the interaction forces between the particles when the distances between them are large compared with their radius.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 142–144, May–June, 1983.     

Interaction of a jet exhausting from a body with an opposing supersonic stream of rarefied gas

The experimental investigation of supersonic flow past a sphere with a jet exhausting from the front point of the sphere into the flow at large [1] and moderate [2] Reynolds numbers Re has revealed an effect of shielding from the oncoming stream, this leading to a decrease in the drag coefficient of the sphere and of the energy flux to it. A numerical simulation of the flow has been made in the case of supersonic flow past a sphere with a sonic jet from a nozzle situated on the symmetry axis in the continuum regime [3]. In the present paper, this problem is investigated for flow of a rarefied gas on the basis of numerical solution of a model kinetic equation for a monatomic gas.     

Stability of flows of the Hamel type in channels with permeable walls

The stability of nonparallel flows of a viscous incompressible fluid in an expanding channel with permeable walls is studied. The fluid is supplied to the channel through the walls with a constant velocity v₀ and through the entrance cross section, where a Hamel velocity profile is assigned. The resulting flow in the channel depends on the ratio of flow rates of the mixing streams. This flow was studied through the solution of the Navier—Stokes equations by the finite-difference method. It is shown that for strong enough injection of fluid through the permeable walls and at a distance from the initial cross section of the channel the flow approaches the vortical flow of an ideal fluid studied in [1]. The steady-state solutions obtained were studied for stability in a linear approximation using a modified Orr—Sommerfeld equation in which the nonparallel nature of the flow and of the channel walls were taken into account. Such an approach to the study of the stability of nonparallel flows was used in [2] for self-similar Berman flow in a channel and in [3] for non-self-similar flows obtained through a numerical solution of the Navier—Stokes equations. The critical parameters *, R*, and Cr* at the point of loss of stability are presented as functions of the Reynolds number R₀, characterizing the injection of fluid through the walls, and the parameter , characterizing the type of Hamel flow. A comparison is made with the results of [4] on the stability of Hamel flows with R₀ = 0.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 125–129, November–December, 1977.The author thanks G.I. Petrov for a discussion of the results of the work at a seminar at the Institute of Mechanics of Moscow State University.     

Two problems of convective diffusion to the surfaces of poorly streamlined bodies

Problems of diffusion to particles of nonspherical shape at large Peclet numbers have been analyzed in many papers (see [1–7], for example). The solution of the problem of mass exchange of an ellipsoidal bubble at low Reynolds numbers was obtained in [1] while the solution at high Reynolds numbers was obtained in [2, 3]. In [4] an expression is given for the diffusional flux to the surface of a solid ellipsoidal particle over which a uniform Stokes stream flows. Generalization to the case of particles of arbitrary shape was done in [5, 6], while generalization to any number of critical lines on the surface of the body was done in [7, 8]. The two-dimensional problem of steady convective diffusion to the surface of a body of arbitrary shape is analyzed in the approximation of a diffusional boundary layer (ADBL). The simple analytical expressions obtained are more suitable for practical calculations than those in [5-8], since they allow one to determine at once, in the same coordinate system in which the field of flow over the particle was analyzed, the value of the diffusional flux to its surface (from the corresponding hydrodynamic characteristics). The plane problem of the diffusion to an elliptical cylinder in a uniform Stokes stream is solved. The problems of the diffusion to a plate of finite dimensions (in the plane case) and a disk (in the axisymmetric case) whose planes are normal to the direction of the incident stream are considered. It is shown that, in contrast to the results known earlier (see [4, 6-15], for example), where the total diffusional flux was proportional to the cube root of the Peclet number, here it is proportional to the one-fourth power.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 104–109, November–December, 1978.The authors thank Yu. P. Gupalo, Yu. S. Ryazantsev, and Yu. A. Sergeev for a useful discussion.     

Optimization of body shape at small reynolds numbers

The problem of the optimization of the shape of a body in a stream of viscous liquid or gas was treated in [1–5]. The necessary conditions for a body to offer minimum resistance to the flow of a viscous gas past it were derived in [1], The necessary optimality conditions when the motion of the fluid is described by the approximate Stokes equations were derived in [2], The shape of a body of minimum resistance was found numerically in [3] in the Stokes approximation. The optimality conditions when the motion of the fluid is described by the Navier—Stokes equations were derived in [4, 5], and in [4] these conditions were extended to the case of a fluid whose motion is described in the boundary-layer approximation. The necessary optimality conditions when the motion of the fluid is described by the approximate Oseen equations were derived in [5] and an asymptotic analysis of the behavior of the optimum shape near the critical points was performed for arbitrary Reynolds numbers.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp, 87–93, January–February, 1978.     

Flows in the initial sections of channels with permeable walls

Calculations of two types of flows in the initial sections of channels with permeable walls are carried out on the basis of semiempirical turbulence theories during fluid injection only through the walls and during interaction of the external flow with the injected fluid. Experimental studies of the first type [1–3] show that at least within the limits of the lengths L/h<30 and L/a< 50 (2h is the distance between permeable walls of a flat channel anda is the tube radius) the velocity distributions in the laminar and turbulent flow regimes differ little and are nearly self-similar for solutions obtained in [4]. For sufficiently large Reynolds numbers, Re₀>100 (Re₀=v₀h/ or Re₀=v₀ a/, where v₀ is the injection velocity), and small fluid compressibility, the axial velocity component is described by the relations for ideal eddying motion: u=(/2)x× cos (y/2) in a flat channel and u=x cos (y²/2) in atube (the characteristic values for the coordinates are, respectively, h anda). Measurements indicate the existence of a segment of laminar flow; its length depends on the Reynolds number of the injection [3]. In the turbulent regime the maximum generation of turbulent energy occurs significantly farther from the wall than in parallel flow. Flows of the second type in tubes were studied in [5–7]. These studies disclosed that for Reynolds numbers of the flow at the entrance to the porous part of the tube Re=u₀ a/<3.10³ fluid injection with v₀/u₀>0.01 leads to suppression of turbu lence in the initial section of the tube. An analogous phenomenon was observed in the boundary layer with v₀/u₀>0.023 [8, 9]. Laminar-turbulent transition in flows with injection was explained in [10, 11] on the basis of hydrodynamic instability theory, taking into account the non-parallel character of these flows. The mechanisms for the development of turbulence and reverse transition in channels with permeable walls are not theoretically explained. Simple semiempirical turbulence theories apparently are insufficient for this purpose. In the present work results are given of calculations with two-parameter turbulence models proposed in [12, 13] for describing complex flows. Due to the sharp changes of turbulent energy along the channel length, a numerical solution of the complete system of equations of motion was carried out by the finite-difference method [14].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 43–48, September–October, 1976.     

Blowing into a supersonic stream through a permeable surface

Distributed blowing of gas into a supersonic stream from flat surfaces using an inviscid flow model was studied in [1–9]. A characteristic feature of flows of this type is the influence of the conditions specified on the trailing edge of the body on the complete upstream flow field [3–5]. This occurs because the pressure gradient that arises on the flat surface is induced by a blowing layer whose thickness in turn depends on the pressure distribution on the surface. The assumption of a thin blowing layer makes it possible to ignore the transverse pressure gradient in the layer and describe the flow of the blown gas by the approximate thin-layer equations [1–5]. In addition, at moderate Mach numbers of the exterior stream the flow in the blowing layer can be assumed to be incompressible [3]. In [7, 8] a solution was found to the problem of strong blowing of gas into a supersonic stream from the surface of a flat plate when the blowing velocity is constant along the length of the plate. In the present paper, a different blowing law is considered, in accordance with which the flow rate of the blown gas depends on the difference between the pressures on the surface over which the flow occurs and in the reservoir from which the gas is supplied. As in [8, 9], the solution is obtained analytically in the form of universal formulas applicable for any pressure specified on the trailing edge of the plate.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 108–114, September–October, 1980.I thank V. A. Levin for suggesting the problem and assistance in the work.     

Numerical investigation of axisymmetric flows in the wake of blunt bodies in a supersonic stream of viscous gas

The axisymmetric flow in the near wake of spherically blunted cones exposed to a supersonic stream of viscous perfect heat-conducting gas is numerically investigated on the basis of the complete Navier-Stokes equations. The free-stream Mach numbers considered M = 2.3 and 4 were such that the gas can be assumed to be perfect, and the Reynolds numbers such that for these Mach numbers the flow in the wake is laminar but close to laminar-turbulent transition [1–4]. The flow structure in the near wake is described in detail and the effect of the Mach and Reynolds numbers on the base pressure, the total drag and the wake geometry is investigated. The results of calculating the flow in the wake of spherically blunted cones are compared with the experimental data [4].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 42–47, July–August, 1988.     

Small perturbation spectrum for plane-parallel flows of viscoelastic fluids

The investigation of the occurrence of a transition from the laminar to the turbulent flow regime in weak polymer solutions is of great practical interest. Experimental data indicate both an increase in flow stability and an occurrence of early turbulence [1]. Paper [2] explains the discrepancy in the experimental data for the numerical investigation of the first-mode symmetric perturbations, which are unstable for a Newtonian fluid. Paper [3] shows that other modes also become unstable in the case of the flow of a viscoelastic Maxwellian fluid in a channel. These features of the hydrodynamic stability of viscoelastic fluids indicate a significant rearrangement of the small perturbation spectrum. In the present paper, the perturbation spectrum for plane-parallel flows of viscoelastic Oldroyd and Maxwellian fluids is investigated at small Reynolds numbers, and at large and small wave numbers.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 164–167, November–December, 1984.     

Flow structure in motion of a spherical drop in a fluid medium at intermediate Reynolds numbers

The problem of flow of a viscous fluid around a spherical drop has been examined for the limiting case of small and large Reynolds numbers in several investigations (see [1–3], for instance; there is a detailed review of various approximate solutions in [4]). For the intermediate range of Reynolds numbers (approximately 1Re100), where numerical integration of the complete Navier-Stokes equations is necessary, there are solutions of special cases of the problem —flow of air around a solid sphere [5–7], a gas bubble [8, 9], and water drops [10]. The present paper deals with flow around a spherical drop at intermediate Reynolds numbers up to Re=200 for arbitrary values of the ratio of dynamic viscosities =₁/₂ inside and outside the drop. It is shown that a return flow can arise behind the drop in flow without separation. In such conditions the circulatory flow inside the drop breaks up. An approximate formula for the drag coefficient of the drop is given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 8–15, January–February, 1976.We thank L. A. Galin, G. I. Petrov, L. A. Chudov, and participants in the seminars led by them for useful discussions.     

Investigation of supersonic flow around a blunt body with injection for high Reynolds numbers

The paper discusses the supersonic flow around a blunt smooth body by a stream of viscous gas with subsonic injection from the surface of the body. The effect of various injection cycles on the physical flow characteristics ahead of the body are studied in [1, 2]; the problem is considered in the approximation of a boundary layer. The nonuniform composition of the gas ahead of the body, chemical reactions between the various components, and the effect of radiation are taken into account. For a number of flow cycles, which are of practical importance, it will be of interest to consider higher approximations in powers of [=1/Re, see Eq. (1.1) below] in the shock layer ahead of the body and, in particular, to explain the action of the displacement effect and also the limits of applicability of the boundary-layer approximation assumed in [1, 2]. Extensive literature has been devoted to the asymptotics of the problem of flow around a blunt body of a viscous gas at high Reynolds numbers (see, for example, Van Dyke's book [3]). An investigation of the problem, based on the method of M. I. Vishik and L. A. Lyusternik, is contained in [4–6]. (The advantage of the use of Vishlik and Lyusternik's method in comparison with the method of internal and external expansion is discussed in [4].) The effect of injection on the flow has not been considered in the papers listed. In this paper, approximate solutions are constructed with an error of order and ² which take into account the effect of the injectionf on the flow . The approximate solutions are compiled from a more accurate nonviscous flow (external solution) and boundary-layer corrections. The boundary-layer corrections are constructed on a shock wave and a contact boundary in such a way that the solution would be continuous and quite smooth. For the external solution at the contact boundary, conditions are obtained which take into account the effect of viscosity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 69–77, January–February, 1974.     

A class of exact solutions of a nonlinear boundary-value problem for fluid flow in a rotating cylinder

The flow of an ideal incompressible weightless fluid that fills a rotating cylinder is investigated. The rotation axis of the cylinder is outside it and parallel to the cylinder generator, and the form of the cylinder section is determined in the process of solution of the problem. In the paper, a class of exact solutions of the problem is obtained in terms of elementary functions for different angular velocities of the cylinder. In these solutions, the flow field is formed by two straight vortex filaments parallel to the cylinder generator. The intensities of the vortex filaments are determined by the angular velocity . Investigations of ideal fluid flow in rotating vessels were begun already in the last century by Stokes and Zhukovskii [1]. The subject has been reviewed in monographs [2, 3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No, 1, pp. 71–75, January–February, 1984.     

Asymptotic model of the axisymmetric breakdown of a vortex filament in an incompressible fluid

The axisymmetric flow of an incompressible fluid is considered. An exact solution of the Euler equations corresponding to the breakdown of a straight vortex filament of intensity ₀ into a vortex filament of lesser intensity and a conical vortex surface is obtained. It is shown that beyond the breakdown point in the region bounded by the conical vortex surface reverse flows occur. An investigation of the problem with allowance for viscous effects at large Reynolds numbers makes it possible to establish a relation between the free parameters entering into the solution of the Euler equations. The results obtained are useful for investigating the problem of the breakdown of a swirled jet, whose solution has recently been receiving much attention [1, 2].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 47–52, November–December, 1989.     

Convective diffusion in a periodic array of spheres for small reynolds numbers

It is shown that in flow past a system of spheres of radius a situated at the nodes of a cubic lattice with the period b in the direction of one of the principal translations of the lattice under the condition (a/b) · · P^1/31 (P is the Péclet number, P1), the concentration of dissolved material absorbed by the sphere surfaces diminishes logarithmically at distances large compared with b, but small compared with L=Pb²/4a. At distances considerably larger than L, the decrease is described by an exponential law which coincides with the law of concentration decrease at distances much larger than b in the case of a spatially homogeneous distribution of the spheres. We consider the flow of an incompressible fluid with the velocity U past a system of spheres of radius a. We assume that the Reynolds number R=Ua/ (where , the kinematic viscosity coefficient, is much larger than unity). Dissolved in the fluid is a material of concentration c which is absorbed by the sphere surfaces. The diffusion coefficient D is assumed to be sufficiently small for the Péclet number P=Ua/D to be much larger than unity. The spheres are situated at the nodes of a cubic lattice with the period b. As will be shown below, it is necessary that P(a/b)³1. Under these assumptions the concentration varies in a thin (of the order aP^–1/3) diffusion layer near the surface of each sphere. A diffusion wake is formed behind each sphere. The transverse dimensions of this wake for a sufficiently widely spaced lattice (aP^1/3 b) exceed the effective thickness of the diffusion boundary layer, which enables us to reduce the problem of concentration absorption on the surface of the system of spheres to the problem considered by Levich [1] concerning the convective diffusion of a material of constant constant concentration flowing past a single sphere.Hasimoto [2] considers the solution of the Stokes equation describing the motion of a viscous fluid past an array of spheres situated at the nodes of a cubic lattice. However, he does not give an expression for the velocity field applicable near the surface of some single sphere which is necessary to the solution of the diffusion problem.In the method of Lamb [3] (§336) and Burgers [4], in dealing with the flow of a viscous stream past a single sphere, one considers the equation of motion in space, including the interior of the sphere, and not just the solution of the equation in the space outside the sphere with boundary conditions at the sphere surface. At the center of the sphere one places a concentrated force and a system of multipoles whose magnitude is chosen in such a way as to ensure fulfillment of the required boundary conditions.This idea of introducing an effective potential is used in [2] to find the velocity field of a fluid flowing past an array of spheres. We propose a treatment of the effective potential method somewhat different from that of [2].The authors are grateful to V. G. Levich and V. S. Krylov for their comments.     

Permeable plate at angle of attack in steady flow of a viscous fluid

An approximate solution is presented for the problem of the resistance of a permeable plate of widthl at an angle of attack in a steady plane flow of an incompressible viscous fluid for the case of both small and very large Reynolds numbers with different permeability laws. The results obtained in the case of large Reynolds numbers are compared with the corresponding results for flow past plane rod grids.     

Channel flow of an anisotropically conducting medium in a zone of entry into a magnetic field

A considerable number of papers are devoted to the problem of the deformation of a plane flow of a conducting liquid moving through a channel |x| < , 0 y h=const in a zone of entry into a magnetic field B=(0, 0, B. (x)), where (x) is the Heaviside unit function((x)=0 for x < 0 and (x)=i for x < 0). Apparently the first paper in this direction was that of Shercliff [1, 2] in which the asymptotic (for x .o- )profile of a perturbed velocity was. determined for a flow of an isotropic conducting liquid in a channel with nonconducting walls. The flow considered by Shemliff takes place in magnetohydrodynarnic flowmeters. Complete calculation of the perturbed flow of an isotropie conducting liquid in the channel of a magnetohydrodynamic generator is carried out in [3]. Asymptotic velocity profiles in the channel of a magnetohydrodynamic generator, with ideally segmented electrodes and the flow of an anisotropically conducting medium along them, were found in [4]. General formulas for the calculation of the asymptotic velocity profile, from the known distribution of the perturbing forces along the channel, are presented in [5]. In [6, 7] the Green function is proposed for the solution of the equation for the stream function of the perturbed flow. Finally, in [8], the solution for the perturbed flow of an anisotropically conducting liquid in a channel with continuous electrodes is described by means of the Green function, and the asymptotic profiles of the velocity are calculated.In this paper the flow of anauisotropically conducting liquid is determined in a channel with ideally segmented electrodes. The solution is set up with the aid of the Fourier method. The resulting series, in which the slowly converging part can be related to the asymptotic profile [4] calculated from the solution of an ordinary differential equation, make it possible to determine the velocity field rapidly. A detailed deformation pattern of the velocity profile is set up. Certain general properties of the flow in a zone of entry into a magnetic field are noted; with the aid of these it is possible to discover the error in the calculations [8].