Stratified Liquid Research Articles

The cauchy-poisson waves in an inviscid rotating stratified liquid

Abstract Based upon the Boussinesq approximation, an initial-value investigation is made of the axisymmetric Cauchy-Poisson waves generated in an inviscid, rotating, stratified liquid of infinite depth by the initially prescribed free-surface elevation. The asymptotic analysis of the integral solution is carried out by the stationary-phase method to describe the solution for large time and large distance from the source of the disturbance. The asymptotic solution is found to consist of the classical free-surface gravity waves and the internal-inertial waves.

Energy characteristics of non-stationary internal waves in a vertical channel containing a stratified liquid

An explicit solution is obtained for the initial-boundary value problem of non-stationary internal waves in a vertical channel containing a stratified liquid generated by small vibrations of the bottom of the channel. The behaviour of the energy and the flow of energy of internal waves is studied as a function of time and the frequency of the force generating the vibrations. Asymptotic formulae for long times are obtained.

Fundamental solutions and Green's formulae for families of equations arising in the theory of the oscillations of a stratified viscous liquid

Fundamental solutions and Green's formulae for families of equations arising in the theory of the oscillations of a stratified viscous liquid

Diffraction by a strip parallel to the surface of a compressible stratified liquid

Diffraction by a strip parallel to the surface of a compressible stratified liquid

The oscillations of an oscillator near the interface between two liquids

The normal modes of a oscillator (an elastically attached solid) near the interface between two stable stratified liquids are studied. The liquids are assumed to be ideal and incompressible. The motion of the body causes internal waves to radiate along the interface. Numerical-analytical methods are developed to investigate small oscillations of the oscillator and the liquid, in a selfconsistent hydrodynamical model. Qualitative observations are made concerning the damping of the oscillations due to the dispersive properties of the medium, as well as the excitation and propagation of the internal waves. A study is made of the amplitude-frequency characteristic (AFC) of the oscillator incorporating the reaction of the waves; the AFC determines such global properties of the oscillating system as the selectivity, frequency variation, etc.

The cauchy‐poisson waves in an inviscid rotating stratified liquid

Based upon the Boussinesq approximation, an initial value investigation is made of the axisymmetric free surface flows generated in an inviscid rotating stratified liquid of infinite depth by the prescribed free surface disturbance. The asymptotic analysis of the integral solution is carried out by the stationary phase method to describe the solution for large time and large distance from the source of the disturbance. The asymptotic solution is found to consist of the classical free surface gravity waves and the internal‐inertial waves.

Solitons in a system of coupled Korteweg-de Vries equations

A system of two Korteweg-de Vries equations coupled by small linear and nonlinear terms, which is a model of a resonant interaction between two internal gravity-wave modes in a shallow stratified liquid, is considered. The present paper is a continuation of a preceding one, where only the linear coupling was dealt with. Various dynamical processes involving one-mode solitons are investigated. It is demonstrated that two solitons belonging to the different wave modes may form an oscillatory bound state (a bi-soliton) which provides an explanation for the numerical results of Gear & Grimshaw demonstrating leapfrogging motion of the two interacting solitons. In the framework of a perturbation theory based on the inverse scattering transform, the frequency of the bi-soliton's internal oscillations is found, and emission of radiation by a weakly excited bi-soliton is studied. Phase shifts and radiative energy losses accompnying a collision between two free solitons belonging to the different modes are calculated. In addition, a collision between a free soliton and a bi-soliton is considered, and it is demonstrated that the collision may result in break-up of the bi-soliton.

Spatial structure of a wake behind a sphere in a stratified liquid

Spatial structure of a wake behind a sphere in a stratified liquid

The theory of internal and surface waves in a stratified liquid

The initial-boundary value problem for the equation of a stratified liquid in the Boussinesq approximation is considered. This models the oscillations of a stratified liquid under an ice-field that has a free surface section. A solution is constructed that is expresed explicitly in terms of the eigenfunctions of some operator V, and its behaviour at long times is investigated.

Constructive control of the motion of oscillating systems with discrete and distributed parameters

The problem of controlling the motion of mechanical systems over an asymptotically large but fixed time interval is considered. The controlled objects may include elements with discrete parameters (material points, rigid bodies, weightless springs, etc.) and oscillating components with distributed parameters (strings, rods, elastic beams and shafts, membranes, plates, cavities with stratified liquid, etc.), whose frequency spectrum is denumerably infinite. The controlling actions, whether kinematic or dynamic in nature, are assumed to be concentrated with respect to the space variables. They may be movable, applied to the absolutely rigid parts of the system and / or fixed at the boundaries of the distributed elements (boundary control). This kind of control is of value in applications. On the assumption that techniques of mathematical physics /1, 2/ or the method of moments /3, 4/ yield an infinite-dimensional control problem for the Fourier coefficients of the solution relative to a set of basis functions for the boundary-value problem, an asymptotic approach is proposed for constructing approximate controls and the resulting scheme for the approximate solution of the problem is shown to be legitimate. A specific problem is examined as an illustration - rotation of an elastic rod in the plane by a torque applied at its end.

On the small vibrations of a section placed at the boundary of separation between two stratified liquids

The explicit solution of a boundary value problem for Sobolev's equation with a discontinuous coefficient is constructed. The problem describes the excitation of non-stationary internal waves in a two layer model of a stratified liquid. The behaviour of the solution at long times is investigated.

On the small vibrations of a stratified capillary liquid

An initial-boundary value problem is considered for the equation which describes the vibrations of an ideal, stratified liquid occupying the lower half space in the Boussinesq approximation. The Vaisala-Brunt frequency is assumed to be constant. The boundary condition on the planar boundary is a combination of the conditions on the solid cover and the free surface and, moreover, the latter contains a term which takes account of capillarity. A formulation of the problem is given, its solution is constructed and its behaviour at long times is investigated.

The problem of the vibrations of a rotating stratified liquid excited by a planar disc

The problem of the vibrations of a planar disc of arbitrary shape placed in a rotating stratified liquid is considered. An explicit solution is constructed for the problem under consideration and estimates of the solution when there is a finite perturbation are obtained. The problem of the existence of a steady state vibrational regime and its limiting amplitude is investigated.

The Cauchy-Poisson problem in an inviscid stratified liquid

Based upon the Boussinesq approximation, an initial value investigation is made of the axisymmetric free surface response of a nonrotating inviscid stratified liquid of finite or infinite depth to the initial displacement of the free surface. The asymptotic analysis of the integral solution is carried out by the stationary phase method to describe the solution for large time and distance from the origin of disturbance. It is shown that the asymptotic solution consists of the classical free surface gravity waves and the internal waves.

Mathematical and physical modelling of doublediffusive convection of aqueous solutions crystallizing at a vertical wall

Solidification of a double-diffusive liquid from a vertical wall in a rectangular cavity is investigated using both mathematical and physical modelling. A two-dimensional mathematical model is developed to simulate the process of solidification in a two-component liquid containing a denser solute. Observations on the solidification behaviour of aqueous sodium carbonate solutions are used to verify the results of the mathematical analysis. The theory and experiments provide a clear picture of the role of double-diffusion in producing vertical compositional and density stratification in an initially homogeneous liquid during solidification. The development of horizontally oriented convection cells in the stratified liquid is correlated with the magnitude of the destabilizing lateral temperature difference across the liquid region.

Electrohydrodynamic stability of two stratified power law liquids in couette flow

Consideration is given to the stability of the flow of two power law liquids under the influence of normal electric field between two infinite parallel planes when one of the planes moves with constant velocity in its own plane. It is found that the electric fields have a dramatic effect and can be chosen to stabilize or destabilize the flow. The effects of the power law parameters on the problem are examined.

The problem of the oscillations of a flat disc in a stratified liquid

The problem of the oscillations of an infinitely thin disc in a stratified liquid is considered, as well as the special features and difficulties encountered in studying problems of this type.

Small non-stationary vibrations of vertical plates in a channel with a stratified liquid

Three initial-boundary value problems for Sobolev's equation which describes the propagation of non-stationary internal waves in a channel with an exponentially stratified liquid have been studied. Waves are excited as a consequence of the small vibrations of one of the following three vertical structures set up in the channel: a barrier placed on the bottom of the channel, two barriers touching the walls of the channel and forming a diaphragm, and “baffles” which do not touch the walls. An explicit solution is constructed for each problem, its behaviour is studied at large times and the phenomena associated with the singularities of the velocity field of the fluid particles are investigated.

On the temperature field of a solar collector hot-water storage tank

The unsteady one-dimensional temperature field inside the hot-water in a storage tank of a thermosyphonic solar collector is investigated analytically. As a result of the low constant rates of the inflowing hot-water into the storage vertical tank via its top, it is conceived that heat between adjacent layers of the stratified hot liquid in the tank is dominantly transferred by conduction. The linearity of the temperature profile along the depth of the tank for each flow situation is heavily controlled by the inflow velocity and other restraining variables as can be perceived from the solution of the flow governing physical equation.

A problem related to the oscillations of a compressible stratified liquid

The problem of wave propagation in a compressible stratified liquid filling a two-dimensional semibounded channel is considered. An explicit solution of the wave equation is constructed and the behaviour of the solution at large time values is investigated.