A. V. Kistovich

Classification of three-dimensional periodic fluid flows

Each physical factor—rotation, stratification, and compressibility of a fluid—is associated with a characteristic type of waves, which are usually analyzed independently [1]. However, elementary waves (acoustic, internal, gyroscopic) do not completely present the properties of periodic flows in the bulk of a real fluid, where all factors act simultaneously and hybrid waves with a complicated dispersion law exist. When studying waves, dissipation effects are considered as corrections ensuring flow attenuation [2]. However, in continua, dissipation factors determine the order of equations and the total number of elements of periodic flows, including waves and sets of boundary layers on rigid boundaries and free surfaces. The consistent inclusion of dissipation effects enables one to find self-consistent solutions of linearized problems of the generation of internal waves [3] without additional empirical parameters (force and mass sources [2]).

The structure of transient boundary flow along an inclined plane in a continuously stratified medium

Abstract An exact solution is found to the unsteady problem of flow formation along an inclined plane, compensating diffusive transfer in a stationary infinitely deep continuously stratified fluid whose density depends only on the concentration of dissolved matter (salinity).

Formation mechanism of a circumferential roller in a drying biofluid drop

Using the methods of theoretical thermohydromechanics, a model of protein roller formation at the periphery of a drying biofluid drop is constructed. As basic mechanisms of mass transfer-, temperature-, concentration-, and gravity-induced convections are considered, which produce a global toroidal flow with an ascending fluid filament at the center, dipping fluid along the free surface, and compensatory centripetal flow at the bottom. The shape of the protein roller is analyzed under the assumptions that the material deposition rate is proportional to the (i) material concentration and (ii) material flux.

Deformation of the free surface of a fluid in a cylindrical container by the attached compound vortex

Vortex flows in the fluid depth generate the pressurefields distorting the free surface [1]. Characteristic“dark spots” mark the depressions in the cores ofattached vortexes in river whirlpools. Floating piecesof ice in northern seas [2] and surface waves in straits[3] are collected into regular spiral branches. The distant measurements from sputniks [4] of the shape ofthe surface of the ocean are used for detection of largeocean vortexes and flows.Vortex flows in the cylindrical geometry are oftenused in applications (in the chambers of water passages, cyclones, separators, and other facilities). Thepressure drops in closed volumes can lead to the formation of gas and vapor–gas cavities, vibrations ofwhich form unsteady forces promoting destruction ofeven such large constructions as water passages ofhydroelectric power stations. The pressure distributionin the vortex, which is difficult to measure in complexunsteady flows [5], can be conveniently followed bythe shape of the free surface.The shape of the surface is easily determined if theideal incompressible fluid is rendered into the state ofthe solidstate rotation with angular velocity Ω in avertical cylindrical container with radius

Calculation of the structure of periodic flows in a continuously stratified fluid with allowance for diffusion

Interest in dissipation in inhomogeneous fluids isstimulated by the search for mechanisms of formationof the fine structure of both the ocean and the atmo-sphere and by the necessity of development of methodsfor calculating linear and nonlinear waves as well asconcomitant boundary layer flows. Singular compo-nents of the complete solutions are of increasing inter-est due to the use of microscopic electromechanicalsystems as well as micro- and nanotechnologies in con-trol systems of power-intensive devices. The internalstructure of boundary layers depends on the dimension-ality of the space of the problem. Two different bound-ary layers can exist in a three-dimensional periodic flowin a stratified viscous medium [1]. One of them is ananalogue of the periodic Stokes flow [2]. Its thicknessis determined by the wave frequency and kinematic vis-cosity of the medium. The thickness of the other inner-boundary layer, which is specific for a stratifiedmedium, depends on the Stokes scale and the geometryof the problem (slope of waves and radiating surfaces)[3]. Since the equations of motion are nonlinear, large-scale (regular) and small-scale (singular) elementsinteract with each other and with other flow compo-nents. Owing to this circumstance, the number of themechanisms of excitation of internal waves increases,and a range of conditions under which this effect occursextends [4].Inclusion of diffusion increases both the order of thegoverning system of equations and the number of formsof periodic motions, which complicates analysis of theproblem. In this work, the complete solution to the lin-earized problem of generation of three-dimensionalinternal waves with allowance for viscosity and diffu-sion has been constructed for the first time. A procedureof constructing solutions to multiscale singularly per-turbed equations depends on the ratio between the kine-matic viscosity ν and the salt diffusion coefficient D .For this reason, we analyze only the case of large valuesof the Schmidt number Sc = , which are typical forboth aqueous solutions of metal salts and seawater.We consider an exponentially stratified fluid whosedensity decreases with the height z aswhere Λ is the buoyancy scale, the direction of the z axis is opposite to the gravitational acceleration g , and N = = is the buoyancy frequency ( T

Fine structure of a conical beam of periodical internal waves in a stratified fluid

Stratified fluid flows caused by torsional or linear harmonic oscillations of a ring along the surface of an infinite vertical cylinder have been calculated by methods of the perturbation theory. The complete solutions of the linearized system of equations with sticking boundary conditions for velocity and impermeable boundary conditions for substance have been obtained taking into account viscosity and diffusion. Disturbances forming a conical beam of three-dimensional internal waves and families of small-scale components are identified. Formulas for calculating waves in media with different Schmidt numbers are described.

Integral model of the propagation of stationary potential waves in fluids

Much attention in fluid mechanics is traditionally devoted to the study of potential waves. For these waves, the constructive nature of the classical analytical models of fluid flows was originally demonstrated in [1] and definitions of the main properties of waves (in particular, the concepts of the group and phase velocities) were introduced [2, 3]. The important role played by nonlinear effects (which, together with dispersion, determine the disturbed surface shape) was established [4], and effective methods for the investigation of complex equations were developed [5]. The current interest in studying potential waves is related to the search for a mathematical description of the complicated pattern of the disturbed sea surface [7], including the waves of an anomalously large amplitude (rogue or freak waves), which cause considerable damage to seaside structures and ships. The problems of stationary wave propagation are conventionally analyzed using both differential [6] and integral methods. The first integral equation was obtained in 1921 by Nekrasov [8, 9] for determining the eikonal dependence of the slope of fluid particle velocities under the assumption of a symmetric shape of these waves. The properties of this equation were investigated in detail in many studies and used as a basis for the proof of the existence of stationary nonlinear waves [10, 11]. Later, a more complicated integral equation describing the fluid velocity in a wave was derived and approximately analyzed [12]. Recently, a system that includes an integral equation for the wave height and a differential equation for the velocity potential on the free surface was obtained by Ablowitz et al. [13] in solving the main problem of wave theory, namely, determining the disturbed liquid surface shape. In this study, we present a new nonlinear integral equation governing the displacements of a heavy fluid surface, which allows one to investigate the propagation of a wide class of perturbations on this surface. Let us consider the propagation of two-dimensional waves over the surface of a homogeneous, ideal incompressible fluid of depth h . The system under consideration includes the Euler equation, the continuity equation, and the potentiality condition together with the conventional boundary conditions imposed on the disturbed surface and the plane bottom [14]:

Free Oscillations of a Balanced Ball on the Horizon of Neutral Buoyancy in a Continuously Stratified Fluid

Interest in the investigation of free oscillations ofneutrally buoyant bodies in stratified fluids [1, 2],which arises from the practical needs of undersea navi-gation, has increased in the past few years owing to thefact that the number of neutrally buoyant probes drift-ing in the atmosphere and hydrosphere is progressivelyincreasing [3]. The work program involves regular sur-facing of submersible vehicles in order to read the dataand their submergence back to the working horizon. Inthe analysis of the measurement results, it is assumedthat the buoys do not disturb the structure of the strati-fied medium; the amplitude-frequency characteristicsof the process of stabilization of the buoy on the hori-zons of neutral buoyancy are determined in the approx-imation of an ideal fluid [1, 2, 4, 5]. The calculationresults are noticeably different from the data of labora-tory measurements [6] and seminatural measurements[7]. In this connection, it is interesting to analyze theprocess of stabilization of buoys in full detail with dueregard for the effects of buoyancy and dissipation. Thedevelopment of the methods of symbolic and numericalcalculation makes it possible to analyze more and morecomplex models of motion of bodies in an inhomoge-neous medium. In this paper, we construct and analyzean integro-differential model of oscillations of a ball ina continuously stratified viscous fluid.Consider an exponentially stratified fluid whosedensity ρ

Free convection from a point heat source in a stratified fluid

Abstract A non-stationary problem of free convection from a point heat source in a stratified fluid is considered. The system of equations is reduced to a single equation for a special scalar function which determinos the velocity field, and the temperature and salinity distribution. Relations are found connecting the spatial and temporal scales of the phenomenon with the parameters of the medium and the intensity of the heat source. The magnitude of the critical source intensity at which the fluid begins to move in a jet-flow mode is established. The structure of convective flows above the heat sources depends, in the stratified media, essentially on the nature of the stratification /1/ which may be caused by a change in the temperature of the medium /2, 3/ or its salinity /4–7/, and by the form of the heat source. When a temperature gradient exists within the medium, an ascending jet forms above the point source, mushrooming outwards near the horizon of the hydrostatic equilibrium. In the case of a fluid with salinity gradient, the jet is surrounded by a sheet of descending salty fluid, and a regular system of annular convective cells is formed around it /1/. The height of the stationary jet computed in /2, 3/ on the basis of conservative laws agrees with experiment. However, this approach does not enable the temperature and velocity distribution over the whole space to be found and does not enable the problem of determining the flow to be investigated. A stationary solution of the linearized convection equations /8/ does not correspond to detail to the observed flow pattern /1, 5–7/. In this connection the study of the non-linear, non-stationary convection equations is of interest. The purpose of this paper is to construct a non-linear, non-stationary free convection equation above a point heat source, and to analyse the scales of the resulting structure and the critical conditions under which the flow pattern changes.

Analytical models of stationary nonlinear gravitational waves

Euler’s equations with standard boundary conditions for the problem of potential surface waves of an arbitrary amplitude in a homogeneous liquid layer with a flat bottom are converted into the new system, including integral and differential equations for the of the potential and its time derivative near the surface. The basic formula of the theory of infinitesimal waves, paired Korteweg-de Vries (KdV) and Kadomtsev− Petviashvili (KP) equations, the envelope Zakharov−Shabat soliton follows from the system in limiting case. The resulting generalized equation, unlike traditional KdFand KP-equations is suitable for the description of waves on the surface of the initially quiescent fluid. A new exact solutions for gravity waves in a deep water, expressed in terms of complex Lambert’s functions are constructed.