Numerical solution of a nonlinear problem of the motion of a plane hydrofoil beneath the free surface of an ideal incompressible fluid

A nonlinear problem of the motion of a hydrofoil of infinite span beneath the free surface of an ideal incompressible fluid with gravity is studied. The stream function is used as the dependent variable. Iterative algorithms for small and large Froude numbers based upon solving a linear boundary value problem in each step with subsequent updating of the shape of the free boundary are proposed. Typical predictions are given for a symmetric profile at different values of the submersion depth, the Froude number and the angle of attack. The free surface and streamlines shapes are shown. The dependence of the lift on the submersion depth for motion through fluid layers of different thickness is presented.Dnepropetrovsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 100–107, July–August, 1995.     

Viscous incompressible flow in a narrow channel with one free wall and fluid supply through a porous insert

The problem investigated relates the plane unsteady flow of a viscous incompressible fluid in a narrow channel one of whose walls is free and acted upon by a given load, while the other is rigidly fixed. The fluid enters the channel through a porous insert in the stationary wall. A model of the flow of a thin film of viscous incompressible fluid and Darcy's law for flow in a porous medium are used to find the distribution of fluid pressure and velocity in the channel and the porous insert in the two-dimensional formulation for fairly general boundary conditions in the case where the length of the porous insert exceeds the length of the free wall. In the particular case where the length of the porous insert is equal to the length of the free wall an exact stationary solution of the problem is obtained for a given value of the channel height. The stability of the equilibrium position of the free wall supported on a hydrodynamic fluid film is examined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 16–24, January–February, 1986.     

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.     

Peristaltic Flow of a Magnetohydrodynamic Johnson–Segalman Fluid

The problem of peristaltic transport of non-Newtonian fluid represented by the constitutive equation for a Johnson–Segalman fluid is analyzed for the case of a planar channel. The fluid is electrically conducting. The walls of the channel are electrically insulated and are transversely displaced by an infinite, harmonic travelling wave of long wavelength. The general solution of the non-linear equation resulting from the momentum equation is constructed for all values of Weissenberg number. The perturbation solution is also obtained. Some graphs are plotted for interesting physical parameters and discussed.     

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

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

Analytical Study of Fluid Flow and Heat Transfer during Forced Convection in a Composite Channel Partly Filled with a Brinkman–Forchheimer Porous Medium

In this paper, the problem of fully developed forced convection in a parallel-plate channel partly filled with a homogeneous porous material is considered. The porous material is attached to the walls of the channel, while the center of the channel is occupied by clear fluid. The flow in the porous material is described by a nonlinear Brinkman–Forchheimer-extended Darcy equation. Utilizing the boundary-layer approach, analytical solutions for the flow velocity, the temperature distribution, as well as for the Nusselt number are obtained. Dependence of the Nusselt number on several parameters of the problem is extensively investigated.     

Free surface asymptotics for thermocapillary flow at high marangoni numbers

Formal asymptotic expansions of the solution of the steady-state problem of incompressible flow in an unbounded region under the influence of a given temperature gradient along the free boundary are constructed for high Marangoni numbers. In the boundary layer near the free surface the flow satisfies a system of nonlinear equations for which in the neighborhood of the critical point self-similar solutions are found. Outside the boundary layer the slow flow approximately satisfies the equations of an inviscid fluid. A free surface equation, which when the temperature gradient vanishes determines the equilibrium free surface of the capillary fluid, is obtained. The surface of a gas bubble contiguous with a rigid wall and the shape of the capillary meniscus in the presence of nonuniform heating of the free boundary are calculated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 61–67, May–June, 1989.     

Nonlinear steady-state motion of an elastic plate floating on the surface of a liquid of infinite depth

The nonlinear problem of steady-state waves in an ideal fluid of infinite depth with a thin elastic plate floating on its surface is considered. The solution is found by a perturbation method. Three approximations are obtained. A case of branching of the solution is investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 119–123, March–April, 1987.     

Flow past a plate under the free surface of a weightless liquid

A study is made of the nonlinear problem of the flow without separation of a perfect weightless liquid past a plate near the free surface. This problem was first posed by Gurevich [1]. At present, there are only a general solution to the problem [2–4] and some numerical calculations [5], which have been made under definite restrictions and are inadequate for detailed information about the interaction between the free surface and the plate. In the present paper, a complete investigation of the problem is given. Convenient computational formulas are obtained together with asymptotic expansions of them, and detailed calculations are made for all depths of the plate.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 158–162, January–February, 1980.     

Small vibrations of a sphere in a spherical volume of viscous fluid

Problems of the vibration of bodies in confined viscous fluids have been solved to determine the added masses and damping coefficients of rods [1–3] and floats [4–5]. The solutions of these problems, based on the use of simplifications of the boundary-layer method [4–6], are obtained analytically in general form and are in good agreement with the experimental data. However, in each specific case the possibility of using such solutions for given values of the fluid viscosity and vibration frequency must be justified either experimentally [2, 4, 5] or theoretically as, for example, in [1], where an analytic solution was obtained for concentric cylinders. The present paper offers a general solution of the problem of the small vibrations of a sphere in a spherical volume of fluid valid over a broad range of variation of the dimensionless kinematic viscosity. The limiting cases of this solution for both high and low viscosity are considered. The asymptotic expressions obtained are compared with calculations based on the analytic solution.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 29–34, March–April, 1986.     

Impact on a degenerate torus floating in a fluid of infinite depth

The vertical impact problem is considered for a degenerate torus, a solid obtained by rotation of a circle about its tangent, half-immersed in an ideal fluid of infinite depth. The impact is assumed to be such that the flow past the torus is attached (separationless impact). After odd continuation of the velocity potential across the free surface, the problem is divided into two subproblems. The first corresponds to the translational motion of the torus in an unbounded fluid along the axis of symmetry, and the second to the rotation of the torus. For a torus of general form the problem of translational motion along the axis of symmetry was considered in [1], in which a solution was constructed in toroidal coordinates in the form of a series in terms of Legendre functions, using the method of separation of variables, and an expression for the kinetic energy was found. In the present paper, the velocity potential is found in quadratures for the translational motion of the degenerate torus. For the rotational flow the problem is reduced to the solution of a Fredholm integral equation of the second kind. Degenerate bipolar coordinates are used. The numerical values of the apparent mass and the apparent moment of inertia are found. The separationless impact condition is derived.Rostov-on-Don. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 161–165, September–October, 1995.     

Asymptotic analysis of the unsteady problem of waves in a two-layer flow

The problem investigated is the unsteady problem of the internal waves generated in a two-layer flow by a certain periodic perturbation which leads to small deviations from the basic flow. A method of constructing an approximate solution uniformly valid throughout the region of variation of the variables and the parameters of the problem is indicated. It is confirmed that for large times and near-resonance parameters the motion of the fluid is described by the mixed problem for a cubic Schrödinger equation. Certain qualitative properties of the solution of this nonlinear problem are noted.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 82–90, November–December, 1987.The author is grateful to V. I. Bukreev and to I. V. Sturova for their interest in his work.     

Analytic investigation of a nonlinear problem of the effect of buoyancy on the evolution of an annular vortex

The evolution of a heavy axisymmetric vortex whose density differs from that of the surrounding irrotational ideal fluid is investigated analytically. If the vortex had no buoyancy (i.e., if the densities were identical), it would preserve its shape and velocity. An approximate analytic solution of the problem is obtained. This solution describes two types of evolution of the vortex shape corresponding to different values of the initial velocity and small buoyancy. The spectrum of a nonlinear wave developing on the vortex boundary is estimated.Nizhnii Novgorod. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 56–66, January–February, 1996.     

Symmetric cavitation flow past a wedge in the presence of a source on the wedge or in the flow

A solution is found to the problem of symmetric cavitation flow over a wedge by an ideal incompressible fluid (in accordance with Efros's scheme [1]) in the presence of a point source in the flow or on the wedge. Expressions are obtained for the forces exerted on the source and the wedge by the fluid, the conditions under which there is a negative resistance (thrust) are indicated, and the profiles of the free streamlines are constructed for different values of the flow parameters.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Shidkosti i Gaza, No. 6, pp. 137–141, November–December, 1979.We thank L. I. Sedov for his interest in the work.     

Some problems in the theory of plane jets of conducting fluid

Using the boundary-layer equations as a basis, the author considers the propagation of plane jets of conducting fluid in a transverse magnetic field (noninductive approximation).The propagation of plane jets of conducting fluid is considered in several studies [1–12]. In the first few studies jet flow in a nonuniform magnetic field is considered; here the field strength distribution along the jet axis was chosen in order to obtain self-similar solutions. The solution to such a problem given a constant conductivity of the medium is given in [1–3] for a free jet and in [4] for a semibounded jet; reference [5] contains a solution to the problem of a free jet allowing for the dependence of conductivity on temperature. References [6–8] attempt an exact solution to the problem of jet propagation in any magnetic field. An approximate solution to problems of this type can be obtained by using the integral method. References [9–10] contain the solution obtained by this method for a free jet propagating in a uniform magnetic field.The last study [10] also gives a comparison of the exact solution obtained in [3] with the solution obtained by the integral method using as an example the propagation of a jet in a nonuniform magnetic field. It is shown that for scale values of the jet velocity and thickness the integral method yields almost-exact values. In this study [10], the propagation of a free jet is considered allowing for conduction anisotropy. The solution to the problem of a free jet within the asymptotic boundary layer is obtained in [1] by applying the expansion method to the small magnetic-interaction parameter. With this method, the problem of a turbulent jet is considered in terms of the Prandtl scheme. The Boussinesq formula for the turbulent-viscosity coefficient is used in [12].This study considers the dynamic and thermal problems involved with a laminar free and semibounded jet within the asymptotic boundary layer, propagating in a magnetic field with any distribution. A system of ordinary differential equations and the integral condition are obtained from the initial partial differential equations. The solution of the derived equations is illustrated by the example of jet propagation in a uniform magnetic field. A similar solution is obtained for a turbulent free jet with the turbulent-exchange coefficient defined by the Prandtl scheme.     

Solution of the problem of outflow of fluid from a half-space through a partially permeable wall

An exact solution is obtained to the problem of outflow of a perfect incompressible fluid from a half-space through an opening, occupied by a permeable plate. It is shown that the flow rate Q of the fluid in the case of outflow through a permeable plate can exceed the flow rate Q₀ of the fluid in the case of jet outflow through a free space.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 174–176, September–October, 1985.     

Heat flow in a channel with discontinuity of the temperature on the wall

We study the temperature field in the flow of a viscous fluid in a circular tube when there is an abrupt change in the boundary condition for the temperature on the walls at a section of the channel. Following the classical studies [1, 2], this problem has often been considered (for example, in [3, 4, 5]) under different assumptions about the type of flow, the form of the boundary conditions, and the values of the Péclet number. The solutions hitherto obtained are frequently cumbersome and do not exhaust all situations of physical interest. In the present paper, we find the solution to the problem for the case of Poiseuille flow, boundary conditions of the first kind for the temperature, and arbitrary values of the Péclet number. We establish an expression that determines the Nusselt number at different sections of the channel. The results of calculations based on the obtained formulas are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Shidkosti i Gaza, No. 5, pp. 194–198, September–October, 1979.     

Three-dimensional problem of exponentially stratified flow over bottom roughness in the case of finite depth

The three-dimensional problem of the flow of an exponentially stratified fluid of finite depth over bottom roughness is considered in the rigid roof approximation and in the presence of a free surface. In the rigid roof approximation the solution is obtained in the form of a Fourier series in the vertical Lagrangian coordinate, and the series coefficients are expressed in terms of single integrals outside a horizontal strip whose sides are parallel to the flow axis and tangential to the projection of the support of the function describing the bottom roughness. This makes it possible to investigate the near field in regions not considered in [1, 2]. The presence of a small parameter in the boundary condition at the free surface makes it possible to find, in the first approximation, the wave motions and nonwave disturbances at the free surface in the near and far fields. In the near field the width of the wave zone is of the order of the flow depth, expands with distance from the bottom and is broadest at the free surface. As distinct from the annular disturbances within the fluid, the pattern of the nonwave disturbances at the free surface depends on the polar angle. The law of similarity for the diverging waves at the free surface is also obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 101–111, May–June, 1990.The authors are grateful to É. V. Teodorovich for discussing the formulation of the problem.     

Secondary flows in a layer with a free surface

The loss of stability of a plane-parallel incompressible viscous heat-conducting fluid flow in a horizontal layer subject to a longitudinal temperature gradient is considered. The lower surface of the layer is assumed to be rigid, while the upper one is free with a surface tension coefficient depending linearly on temperature. Both boundaries are assumed to be thermally-insulated. The critical value of the temperature gradient as a function of other relevant parameters is determined by analyzing the spectrum of the linearized problem. Secondary flows arising after the onset of instability are determined from an analysis of the full nonlinear problem using the expansion of the solution in a power series in terms of a supercritical state parameter in the vicinity of the bifurcation point. Three types of secondary flows are studied: plane two-dimensional waves propagating along the temperature gradient; plane waves travelling at a certain angle to the gradient; and three-dimensional waves propagating along the gradient. A numerical method of problem solution, based on the polynomial approximation, is described.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 85–98, September–October, 1994.     

Impact of a circular disk and washer on a fluid in a cylindrical vessel

The solution of the problem of the axisymmetric motion of an ideal incompressible fluid in a cylindrical vessel of finite depth is obtained for small vibrations of a flexible circular disk and washer (disk with centered hole) on the surface of the fluid. On the basis of this solution the virtual mass is determined as a function of the dimensions of the vessel, disk and washer for the special case of a rigid nondeformable disk and washer.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 103–111, January–February, 1995.