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.     

Penetrative convection in the isothermally incompressible fluid approximation

The onset of penetrative convection in an infinite horizontal fluid layer bounded by isothermal rigid or free nondeformable surfaces is numerically examined. It is assumed that the specific volume of the fluid depends quadratically on temperature and reaches a minimum inside the layer. The isothermally incompressible fluid convection model in which, as distinct from the Oberbeck-Boussinesq approximation, the thermal expansion is not assumed to be small is considered. Both the neutral stability curves of the conductive regime and the amplitudes of two-dimensional periodic and three-dimensional doubly-periodic convective flow are calculated. The results are compared with those previously obtained for the equations of penetrative convection in the Boussinesq approximation.Rostov-on-Don. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 40–52, March–April, 1996.     

Convective stability of a liquid in a spherical cavity

Three problems of convertive stability in a spherical liquid layer are considered. They differ with respect to the degree of freedom of the surface: a free, but nondeformable surface (a small droplet on a spider's web will serve as a model), a nonfree deformable surface (a droplet contaminated with an insoluble surface active agent), and a rigid unrestrained shell (dust-covered droplet). The Marangoni effects in the first two problems were investigated by Bratukhin [2] and are disregarded.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.3, pp. 24–28, May–June, 1992.     

Stability of a laminar boundary layer of a power-law no n-newtonian liquid

The stability of a laminar boundary layer of a power-law non-Newtonian fluid is studied. The validity of the Squire theorem on the possibility of reducing the flow stability problem for a power-law fluid relative to three-dimensional disturbances to a problem with two-dimensional disturbances is demonstrated. A numerical method of integrating the generalized Orr-Sommerfeld equation is constructed on the basis of previously proposed [1] transformations. Stability characteristics of the boundary layer on a longitudinally streamlined semiinfinite plate are considered.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 101–106, January–February, 1976.     

Unsteady flow over a rough bottom

The unsteady motion of an ideal incompressible fluid with a free surface, developing from a state of rest, is considered. The flow is assumed to be irrotational, continuous and two-dimensional; it may be the result either of an initial disturbance of the free boundary or of a given boundary pressure distribution. The rigid boundaries of the flow region are fixed, and the free surface does not cross them at any time during the motion. The fluid is located in a uniform gravity force field and there is no surface tension. A method which in the case of localized roughness of the bottom makes it possible to find the shape of the free surface at any moment of time with predetermined accuracy is proposed. The method involves reducing the initial linear problem to a Volterra integral equation of the second kind, the kernel of this equation being a nonlocal operator. This operator has a smoothing effect, which makes it possible to reduce the solution of the initial problem to the solution of an infinite, perfect lyregular system of Volterra integral equations for a denumerable set of auxiliary functions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 111–119, November–December, 1989.The author is grateful to I. V. Sturova and B. E. Protopopov for useful discussions and criticism.     

Effect of capillary wave radiation on the oscillations of a vibrator

The oscillations of a rigid body on an elastic tie (vibrator) in an ideal incompressible fluid with a free boundary, on which surface tension forces act, are considered. The linearized problem of hydrodynamics is solved approximately in the self-consistent formulation, the reaction forces exerted on the body by the fluid are calculated, and an integrodifferential equation of motion is obtained. Using asymptotic methods, the average characteristics determining the damping coefficient and the frequency shift of the oscillations of the vibrator are obtained with allowance for the effect of the capillary waves radiated by the vibrator. Qualitative effects depending on the parameters of the system are revealed. The authors' numerical simulation of the motion of the vibrator completely confirms the qualitative conclusions concerning the nature of the oscillations of a body in a fluid having surface tension.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 126–132, March–April, 1995.     

Free vibration of an elastic bottom plate of a partially fluid-filled cylindrical container with an internal body

An analytical method is developed to consider the free vibration of an elastic bottom plate of a partially fluid-filled cylindrical rigid container with an internal body. The internal body is a rigid cylindrical block that is concentrically and partially submerged inside the container. The developed method captured the analytical features of the velocity potential in a non-convex, continuous, and simply connected fluid domain including the interaction between the fluid and the structure. The interaction between the fluid and the bottom plate is included. The Galerkin method is used for matching the velocity potentials appropriate to two distinct fluid regions across the common horizontal boundary (artificial horizontal boundary). Then, the Rayleigh–Ritz method is also used to calculate the natural frequencies and modes of the bottom plate of the container. The results obtained for the problem without internal body are in close agreement with both experimental and numerical results available in the articles. A finite element analysis is also used to check the validity of the present method in the presence of the internal body. Furthermore, the influences of various variables such as fluid level, internal body radius, internal body length, and the number of nodal diameters and circles on the dynamic behaviour of the coupled system are investigated.     

Impact on a plate on the surface of a fluid

Questions of the interaction between solid and elastic structures with an ideal fluid which are associated with the initial stage of the impact and penetration of bodies in the fluid were considered in [1–4]. Results are presented below of an analysis of a central impact on a solid weightless plate which is on the surface of a compressible fluid. The impact velocity is much less than the speed of sound in the medium. Computations are performed by a finite-difference Lagrange method according to a program for plane motions of a continuous medium [5] by using a volume artificial viscosity of Neumann-Richtmayer type [6].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 143–145, May–June, 1978.     

Influence of an elastic plate on the motion of an inhomogenfous fluid

The influence of a thin elastic isotropic plate on the wave motion of an inhomogeneous fluid originating under the effect of external periodic perturbations is investigated. The fluid density increases constantly with depth. Analogous problems have been examined for an inhomogeneous fluid without a plate in [1, 2] and with a plate on the surface of a homogeneous fluid in [3–5].Sevastopol'. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 60–67, January–February, 1972.     

COUPLED VIBRATORY CHARACTERISTICS OF A RECTANGULAR CONTAINER BOTTOM PLATE

The natural frequencies of an elastic thin plate placed into a rectangular hole and connected to the rigid bottom slab of a rectangular container filled with fluid having a free surface are studied. The fluid is assumed to be incompressible, inviscid and irrotational, and the effect of surface waves is neglected. An analytical-Ritz method is developed to study the vibratory characteristics of the plate in contact with the fluid. First of all, the exact expression of the motion of the fluid is obtained, in which the unknown coefficients are determined by using the method of separation of variables and the method of Fourier series expansion. Then, the Ritz approach is used to obtain the frequency equation of the system. The vibrating beam functions are adopted as the admissible functions for the wet-mode expansion of the plate, and the added virtual mass incremental (AVMI) matrices are obtained for plates with arbitrary boundary conditions. Finally, a convergence study is carried out and some numerical results are given. The accuracy of AVMI factor solutions is discussed by comparing with the more accurate analytical-Ritz solutions presented in this paper. Furthermore, It is seen that the present method is also suitable for the vibration analysis of rectangular plates in contact with infinite fluid by taking the finite, but larger size fluid domain as an approximation in the computation.     

Bénard-Marangoni instability in a viscoelastic Jeffreys' fluid layer

Coupled buoyancy (Bénard) and thermocapillary (Marangoni) convection in a thin fluid layer of a viscoelastic fluid are studied. The viscoelastic fluid is modeled by Jeffreys' constitutive equation. The lower surface of the layer is in contact with a rigid heat-conducting plate while its upper surface is subject to a temperature-dependent surface tension. The critical temperature difference between both boundaries corresponding to the onset of convection is calculated. The role of the various viscometric coefficients is discussed. In the appendix it is shown that Jeffreys' constitutive relation is easily derived from thermodynamic considerations based on extended irreversible thermodynamics.     

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.     

A basis for a theory of nonlocal elastic filtration using the equations of elasticity

The proposed [1–3] nonlocal formulation of the hypothesis that the ground pressure is constant in nonstationary pressure filtration in a deep elastic stratum is derived from the equilibrium equations for the stratum-roof system. The roof is considered to be a flat plate [4] and the floor of the stratum is assumed to be rigid. An equation is established for the scale of the region of influence on the stress and pressure distributions at a point.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 82–86, July–August, 1971.The author wishes to thank V. N. Nikolaevskii for a number of useful observations.     

Detached flow over a step with the formation of a turbulent wake

A method of calculating the plane turbulent layer behind a step interacting with a free potential flow of incompressible fluid is developed. The method includes consideration of the initial boundary layer and injection (or suction) in the isobaric bottom region. Friction on the wall behind the step is neglected, which corresponds to symmetric quasisteady flow behind the straight edge of a plate. The inviscid flow is represented by the Keldysh-Sedov integral equations; the flow in the wake with a one-parameter velocity profile is represented by three first-order differential equations—the equations of momentum for the wake and motion along its axis and the equation of interaction (through the displacement thickness) of the viscous flow with the external potential flow. The turbulent friction in the wake is given, accurate to the single empirical constant, by the Prandtl equation. The different flow regions — on the plate behind the step, the isobaric bottom region, and the wake region — are joined with the aid of the quasi-one-dimensional momentum equation for viscous flow. The momentum equation for the flow as a whole serves as the closure condition. The obtained integrodifferential system of equations is approximated by a system of nonlinear finite-difference equations, whose solution is obtained on a computer by minimization of the sum of the squares of the discrepancies. The results of the calculations agree satisfactorily with experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 17–25, May–June, 1977.We are grateful to V. I. Kuptsov for consultation and help in programming and to Z. A. Donskova who assisted in the calculations and preparation of the paper.     

HYDROELASTIC VIBRATION OF A CIRCULAR CONTAINER BOTTOM PLATE USING THE GALERKIN METHOD

The free vibration of a flexible thin plate placed into a circular hole and elastically connected to the rigid bottom slab of a circular cylindrical container filled with fluid having a free surface is studied. The liquid is assumed to be incompressible, inviscid and irrotational. The effect of the free surface wave is also taken into account in the analysis. First of all, the exact expression of velocity potential of the liquid movement is derived by a combination of the superposition method and the method of separation of variables. With the help of the Fourier–Bessel series expansion, part of the unknown coefficients in the solution is determined by the consistency condition between the liquid movement and the plate vibration, in the form of integrals associated with the dynamic deflection of the plate. Then, the Galerkin method is applied to derive the eigenfrequency equation of the fluid–plate interaction. Finally, the effects of various parameters and the free surface wave on eigenfrequencies of the fluid–plate system are discussed. As a consequence, the accuracy of the nondimensional added virtual mass incremental (NAVMI) factor solution has also been evaluated by comparing with the more accurate Galerkin solution. It is shown that the proposed method is also applicable to the vibration analysis of circular plates in contact with an infinite liquid by only taking a finite but larger size of liquid to replace the infinite liquid in the computation.     

Exact solution of the problem of convective heat exchange for an elliptic cylinder and a plate in a fluid with small Prandtl number

The heat exchange problem is solved for an elliptic cylinder and a plate in an incompressible fluid flow with small Prandtl number Pr1. For flow along a plate, the values of the Nusselt number Nu obtained by solving the complete energy equation and the heat boundary layer equation are compared.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 26–31, May–June, 1996.     

Boundary conditions on a rigid surface in a two-phase flow

A study is made of the boundary conditions on a rigid surface in a two-component disperse flow. Appropriate boundary conditions are obtained for the kinetic equation and macroscopic equations of a pseudogas of solid particles proposed in [1–3]. The reasons for the occurrence of bubbles in two-phase systems are discussed. On the basis of the similitude parameters of the kinetic equation of the pseudogas, disperse systems are classified generally on the basis of the concentration of solid particles and their diameters.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 46–51, May–June, 1980.I thank V. P. Myasnikov for suggesting the problem and for a helpful discussion.     

Diffraction of a sound wave by a nonmoving plate

The problem of determining the velocity field excited by a sound wave impinging on a plate at rest is analyzed as an initial- and boundary-value problem with a movable boundary for the two-dimensional wave equation. The latter problem is solved by the formulation and inversion of integral equations of the Volterra type. The solution is obtained in closed form for any angle of inclination of the incident wave relative to the plate surface and is represented by recursion relations allowing for the influence of any number of diffracted waves generated in succession at the plate boundary.Moscow. Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 2, pp. 123–130, March–April, 1972.     

Slow motion of a permeable sphere in a viscous fluid

Flow of a viscous fluid past a permeable sphere is investigated in the Stokes approximation. An example of such a flow is flow past a perforated or meshed spherical surface. The elements of the sphere contain rigid impermeable sections and openings through which the fluid can flow. The interaction of the sphere with the flow is described by two drag coefficients, which established the connection between the flow velocity of the fluid at the sphere and the stress tensor on it. The dependence of the flow pattern and also the drag and flow rate of the fluid on these coefficients is investigated. In special cases, the obtained solution describes flow past solid and liquid spheres.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 165–167, September–October, 1982.     

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.