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

Kulczycki, Tadeusz; Kuznetsov, Nikolay

doi:10.1098/rspa.2010.0258

Cited by 14 publications

(11 citation statements)

References 10 publications

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“…Another simplification leading to manageable calculations involves two-dimensional containers, or long troughs with fluid sloshing only sideways. These were studied by Kulczycki-Kuznetsov [29] and Faltinsen-Timokha [14], among others.…”

Section: Mathematical Modelmentioning

confidence: 99%

The shape of the fundamental sloshing mode in axisymmetric containers

2015

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ABSTRACT. In the paper we numerically study positions of high spots (extrema) of the fundamental sloshing mode of a liquid in an axisymmetric tank. Our approach is based on a linear model reducing the problem to appropriate Steklov eigenvalue problem. We propose a numerical scheme for calculating sloshing modes and a novel method of making images of oscillating fluid. We also describe the relation of the high spot problem to the celebrated hot spots conjecture.

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Section: Mathematical Modelmentioning

confidence: 99%

The shape of the fundamental sloshing mode in axisymmetric containers

2015

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“…The second eigenvalue problem describes frequencies of sloshing of a fluid in the special case of a cylindrical "glass" with uniform cross-sections. (See [15] for a historical review and [5,20,26,27,28] for recent developments.) When we want to emphasize the dependence on the cylinder depth, we write the eigenvalues as λ j (L) and µ j (L).…”

Section: Sloshing Problemmentioning

confidence: 99%

Sharp spectral bounds on starlike domains

Laugesen¹,

Siudeja

2014

J. Spectr. Theory

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We prove sharp bounds on eigenvalues of the Laplacian that complement the Faber-Krahn and Luttinger inequalities. In particular, we prove that the ball maximizes the first eigenvalue and minimizes the spectral zeta function and heat trace. The normalization on the domain incorporates volume and a computable geometric factor that measures the deviation of the domain from roundness, in terms of moment of inertia and a support functional introduced by Pólya and Szegő.Additional functionals handled by our method include finite sums and products of eigenvalues. The results hold on convex and starlike domains, and for Dirichlet, Neumann or Robin boundary conditions.

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“…Roughly speaking, the question about high spots concerns monotonicity properties of fundamental sloshing eigenfunctions (see subsection 1.3 for a detailed description). Several results about the location of high spots were proved in [26] and [27]. One of them deals with the fundamental eigenfunction (it is unique up to a non-zero factor) of the two-dimensional sloshing problem in the case when the domain's top interval is the one-to-one orthogonal projection of the bottom.…”

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

Cited by 14 publications

References 10 publications

The shape of the fundamental sloshing mode in axisymmetric containers

The shape of the fundamental sloshing mode in axisymmetric containers

Sharp spectral bounds on starlike domains

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

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