Vibrations of a stratified liquid in a basin of arbitrary shape

Kopachevskii, N. D.; Temnov, A. N.

doi:10.1016/0041-5553(86)90113-8

Cited by 4 publications

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“…Taking into consideration that the density does not vary with the time, that is,︀ = + ∇ 0 · ⃗ + ∇ · ⃗ = 0, linearizing (6), (7), we obtain…”

Section: Mathematical Formulation Of Problemmentioning

confidence: 99%

“…The problems on oscillations of a stratified liquid filling a bounded domain in a space have applications in the seiche theory, the theory of petroleum oscillations in tankers, in studying oscillations of cryogenic liquids in closed tanks. Not providing a detailed bibliography, we just mention monographs [1]- [6] and works [7], [8], where various aspects of oscillations of such system were studied. It is known that the presence of a vertical stratification of a liquid with respect to the density gives rise quite to interesting physical phenomena in such hydrosystems; these phenomena are related with the action of buoyancy forces.…”

Section: Introductionmentioning

confidence: 99%

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Small motions of ideal stratified liquid with a free surface totally covered by a crumbled ice

Kopachevskii¹,

Tsvetkov²

2018

Ufimskii Matematicheskii Zhurnal

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Let a rigid immovable vessel be partially filled with an ideal incompressible stratified fluid. We assume that in an equilibrium state the density of a fluid is a function of the vertical variable 3 , i.e., 0 = 0 ( 3 ). In this case the gravity field with constant acceleration ⃗ = − ⃗ 3 acts on the fluid, here > 0 and ⃗ 3 is unit vector of the vertical axis 3 , which is directed opposite to ⃗. Let Ω be the domain filled with a fluid in equilibrium state, be rigid wall of the vessel adherent to the fluid, Γ be a free surface completely covered with a crumbled ice. As the crumbled ice we mean that on the free surface, weighty particles of some substance float, and these particles do not interact or the interaction is negligible as the free surface oscillates. We should note that in foreign publications, such fluids are frequently called liquids with inertial free surfaces. The problem is studied on the base of an approach connected with application of so-called operator matrices theory. To this end, we introduce Hilbert spaces and some their subspaces, also auxiliary boundary value problems. The initial boundary value problem is reduced to the Cauchy problem for the differential second-order equation in Hilbert space. After a detailed study of the properties of the operator coefficients corresponding to the resulting system of equations, we prove a theorem on the strong solvability of the Cauchy problem obtained on a finite time interval. On this base, we find sufficient conditions for the existence of a strong (with respect to time variable) solution to the initial-boundary value problem describing the evolution of the hydrosystem. Small motions of ideal stratified liquid with a free surface totally covered by a crumbled ice.

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“…Taking into consideration that the density does not vary with the time, that is,︀ = + ∇ 0 · ⃗ + ∇ · ⃗ = 0, linearizing (6), (7), we obtain…”

Section: Mathematical Formulation Of Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Small motions of ideal stratified liquid with a free surface totally covered by a crumbled ice

Kopachevskii¹,

Tsvetkov²

2018

Ufimskii Matematicheskii Zhurnal

View full text Add to dashboard Cite

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“…Monographies [13], [14], and papers [15], [16], [17], [18], [19], [20] are dedicated to mathematical research of linear equations for small-amplitude internal waves. As for nonlinear Euler equations for incompressible fluid, correctness of many problems is not proved yet [21].…”

Section: Introductionmentioning

confidence: 99%

Study of internal wave breaking dependence on stratification

Kshevetskii,

Leble

2012

Preprint

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Mixing effect in a stratified fluid is considered and examined. Euler equations for incompressible fluid stratified by a gravity field are applied to state a mathematical problem and describe the effect. It is found out that a system of Euler equations is not enough for a formulation of correct generalized problem. Some complementary relations are suggested and justified. A numerical method is developed and applied for study of processes of vortex destruction and mixing progress in a stratified fluid. The dependence of vortex destruction on a stratification scale is investigated numerically and it is shown that the effect increases with the stratification scale. It is observed that the effect of vortex destruction is absent when the fluid density is constant. Some simple mathematical explanation of the effect is suggested.

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“…In the process, instead of the "method of auxiliary problems" that is customarily used in problems' of this type [2,3], we have used the method of "embedding" the boundary-value problem in an integral identity [4]. In our view, the applicability of this last method in the problem considered below is very problematical.…”

mentioning

confidence: 96%

Asymptotics of the spectrum and eigenvectors of a problem of fluid mechanics

Tsar'kov¹

1996

J Math Sci

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We consider small motions of an inhomogeneous incompressible ideal fluid filling a motionless vessel with elastic walls and subject to the action of a stationary potential force field. We obtain an operational formulation of the problem. We find new, very precise estimates of the eigenvalues and eigenvectors of this problem.In studying the small motions of a fluid filling a vessel with elastic walls, an operator (generalized) statement of the problem has been obtained. In the process, instead of the "method of auxiliary problems" that is customarily used in problems' of this type [2,3], we have used the method of "embedding" the boundary-value problem in an integral identity [4]. In our view, the applicability of this last method in the problem considered below is very problematical.An inhomogeneous incompressible ideal fluid with density p(x), x = (xl, x2, x3) in a state of rest fills a (not necessarily connected) region f~ in R 3 with a piecewise-smooth boundary. In f~ a stationary force field F = -V~o (~o being the potential) acts on the fluid, while 'the boundary 0f~ is subject to the reactive force of the wails of the motionless vessel. The functions p(x) and ~o(x) axe smooth up to the boundary, and p(x) is bounded away from zero. It is assumed that the system is in equilibrium, so that the relationholds in fl, where Po is the equilibrium pressure. Part of the boundary of fl is made up of absolutely rigid walls of the vessel, and its complement F consists of elastic elements of various kinds. There may be elastic connections between the different parts of F. In P we distinguish the part Fo "wetted" by the fluid from both sides (i.e., Fo is made up of the elastic partitions within the fluid) from the part F1 that separates the fluid from the "void." It is assumed that a stationary external pressure acts on Px. We note, finally, that a mass of density rn(x) is distributed over P. The elastic properties of P can be expressed by the operator D that connects the displacement of the surface P and the load that causes this displacement. Here we shall consider the displacement ( to be small in what foUows (the linear theory) and orthogonal to the equihbrium surface F; by "load" we mean a pressure with density q. We can now state our assumptions on the linear operator D. We consider the space L2(F) with inner product J (2) (r P where ~ is Lebesgue measure on F. We shall assume that there is a rigging H+ C La(F) C H_ of the space L2(F) by Hilbert spaces H+ and H_, where the imbeddings are dense and compact. As usual, the definition of the inner product (2) is extended so that (q, () makes sense for q e H_ and ( E H+, and H+ and H_ are in a dual relation with respect to this bilinear form that is consistent with the topology of the two spaces. The space H+ is the energy space [5] of the elastic system F. It is made up of all possible displacements ( of the surface F having finite potential energy, and the square of the norm [[(1t+ equals the potential energy of the system under displacement (. The space H_ is made up o...

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Vibrations of a stratified liquid in a basin of arbitrary shape

Cited by 4 publications

References 0 publications

Small motions of ideal stratified liquid with a free surface totally covered by a crumbled ice

Small motions of ideal stratified liquid with a free surface totally covered by a crumbled ice

Study of internal wave breaking dependence on stratification

Asymptotics of the spectrum and eigenvectors of a problem of fluid mechanics

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