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

Abstract

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.

Analytically approximate natural sloshing modes for a spherical tank shape

“…The higher modes cannot be directly excited by a harmonic horizontal tank forcing. For higher liquid depths, 1 < h < 2, our analysis demonstrates that (i) an artificial normal liquid flow through the 'dry' tank surface (caused by the analytical continuation) becomes far from zero, (ii) the 'high spots', the points on the mean free surface where its elevation attains the maximum and minimum values, are not at the wall, which is in agreement with the assertion proved by Kulczycki & Kwaśnicki (2012).…”

The shape of the fundamental sloshing mode in axisymmetric containers

“…The above reduction was applied by many authors (e.g. [30,20,38]). Note that there is no boundary condition on R, but if m > 0 we need continuity of the three dimensional solution, while for m = 0 we need continuous derivative (three dimensional solutions are harmonic, hence smooth).…”

The Legacy of Vladimir Andreevich Steklov

“…On the other hand (see [34], Proposition 1.3), if the angle between B and F is bigger than π 2 and smaller than π then ϕ 1 (x, y, 0), ϕ 2 (x, y, 0) attain their extrema inside F as is shown in Figure 8, top.…”

“…In [34] (see Theorems 1.1 and 1.2), the following is proved. If W is a convex body of revolution confined to the cylinder {(x, y, z) : (x, y, 0) ∈ F, z ∈ R} (this condition was introduced by F. John in 1950), then three assertions hold: (i) ν 1 = ν 2 ; (ii) the corresponding eigenfunctions ϕ 1 and ϕ 2 are antisymmetric; (iii) the high spots of these modes are attained on ∂F .…”

“…This reduces the risk of spilling from a snifter. Thus, the position of a high spot depends on the container's shape as is schematically shown in Figure 3 for a coffee cup (a) and a snifter (b); theorems guaranteeing these kinds of behavior are proved in [34].…”