Operator Approach to Linear Problems of Hydrodynamics

Abstract. We investigate the classical eigenvalue problem that arises in hydrodynamics and is referred to as the sloshing problem. It describes free liquid oscillations in a liquid container W ⊂ R 3 . We study the case when W is an axially symmetric, convex, bounded domain satisfying the John condition. The Cartesian coordinates (x, y, z) are chosen so that the mean free surface of the liquid lies in (x, z)-plane and y-axis is directed upwards (y-axis is the axis of symmetry). Our first result states that the fundamental eigenvalue has multiplicity 2 and for each fundamental eigenfunction ϕ there is a change of x, z coordinates by a rotation around y-axis so that ϕ is odd in x-variable.The second result of the paper gives the following monotonicity property of the fundamental eigenfunction ϕ. If ϕ is odd in x-variable then it is strictly monotonic in x-variable. This property has the following hydrodynamical meaning. If liquid oscillates freely with fundamental frequency according to ϕ then the free surface elevation of liquid is increasing along each line parallel to x-axis during one period of time and decreasing during the other half period. The proof of the second result is based on the method developed by D. Jerison and N. Nadirashvili for the hot spots problem for Neumann Laplacian.

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“…. ) belong to H 1 (W ) and form a complete system in an appropriate Hilbert space (see, for example, [22]). …”

Section: Introductionmentioning

confidence: 99%

On high spots of the fundamental sloshing eigenfunctions in axially symmetric domains

Kulczycki

Kwaśnicki

2012

Proceedings of the London Mathematical Society

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“…This boundary-value problem (usually referred to as the sloshing problem) has been the subject of a great number of studies over more than two centuries (see Fox & Kuttler (1983) for a historical review). Advanced mathematical techniques based on the spectral theory of operators in a Hilbert space are developed for this problem; they can be found in the book by Kopachevsky & Krein (2001).…”

Section: Introductionmentioning

confidence: 99%

“…belong to the Sobolev space H 1 (W ) and form a complete system in an appropriate Hilbert space (e.g. Kopachevsky & Krein 2001).…”

Section: Introductionmentioning

confidence: 99%

“…A substantial body of work on problem (1.1)-(1.4) (a great part of which is numerical computation of eigenfrequencies) deals either with two-dimensional oscillations or with motions in containers having a vertical axis of symmetry (see Fox & Kuttler (1983) and references therein; references to papers published in Russian can be found in Kopachevsky & Krein (2001)). Here, we consider this problem for containers having another geometry, namely for horizontal cylinders of uniform cross section or troughs.…”

Section: Introductionmentioning

confidence: 99%

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On the ‘high spots’ of fundamental sloshing modes in a trough

Kulczycki

Kuznetsov

2010

Proc. R. Soc. A.

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We study an eigenvalue problem with a spectral parameter in a boundary condition. The problem describes sloshing of a heavy liquid in a container, which means that the unknowns are the frequencies and modes of the liquid's free oscillations. The question of 'high spots' (the points on the mean free surface, where its elevation attains the maximum and minimum values) is considered for fundamental sloshing modes in troughs of uniform cross section. For troughs, whose cross sections are such that the horizontal, top interval is the one-to-one orthogonal projection of the bottom, the following result is obtained: any fundamental eigenfunction attains its maximum and minimum values only on the boundary of the rectangular free surface of the trough.

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“…This and more general versions of the problem (it is usually referred to as the sloshing problem), has been the subject of a great number of studies over more than two centuries; see the paper [8] for a historical review and early results are described by Lamb in his Hydrodynamics [15]. Among recent works on this problem we mention the book [11] by Kopachevsky and Krein, and the papers [12], [13], and [14]. The latter two papers are, in particular, concerned with the question (related to the topic of our paper) when eigenvalues are simple.…”

Section: Introductionmentioning

confidence: 99%

‘High spots’ theorems for sloshing problems

Kulczycki

Kuznetsov

2009

Bulletin of the London Mathematical Society

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Abstract. We investigate several 2D and 3D cases of the classical eigenvalue problem that arises in hydrodynamics and is referred to as the sloshing problem. In particular, for a domain W ⊂ R 2 (canal's crosssection), where ∂W = F ∪ B and F (cross-section of the free surface of fluid) is an interval of the x-axis, whereas B (bottom's cross-section) is the graph of a negative function, the following result is proved. The fundamental eigenfunction u1 of the sloshing problem (the corresponding eigenvalue is simple) has monotonic traces on F and B; moreover, u1 attains its maximum and minimum values at the end-points of F .It is established that for the 2D (3D) ice-fishing problem with a single (circular) hole the function u 1 (both fundamental eigenfunctions) attains its maximum value at an inner point of F .A relationship between the high spots and hot spots theorems is considered.

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Operator Approach to Linear Problems of Hydrodynamics

Cited by 74 publications

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On high spots of the fundamental sloshing eigenfunctions in axially symmetric domains

On high spots of the fundamental sloshing eigenfunctions in axially symmetric domains

On the ‘high spots’ of fundamental sloshing modes in a trough

‘High spots’ theorems for sloshing problems

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