Analytical solutions of first-mode sloshing in new axisymmetric containers

• Three new families of containers for axisymmetric mode-one sloshing.
• Analytic solutions for depth-dependent frequencies.
• Comparison with planar analogue geometries.
• Limiting cases of isochronous containers determined.

A new family of axisymmetric containers denoted as DD and two new series of families of axisymmetric containers denoted AnAn and BnBn are found for first-mode sloshing. Each member depends on streamfunction separation constant ββ. The containers are of annular shape and the frequency distributions are obtained in analytical form. The DD family exists in the range 0.5≤β<∞0.5≤β<∞, involves logarithmic behavior, and exhibits a limiting shape in the form of an annular 90∘90∘ wedge at low depth. The AnAn container series which exist in the range 0≤β≤∞0≤β≤∞ vary from cylindrical annulii at β=0β=0 to axisymmetric isochronous shapes for β→∞β→∞. The BnBn container series exist in the range βn≤β≤∞βn≤β≤∞ are more interesting in that the shapes for the minimum values βnβn for existence of solutions in each family take the form of an annular 90∘∘ wedge at low depth and evolve to constant width at high depth. The shapes for all other values of β>βnβ>βn are of hyperbolic form at low depth and also evolve to constant width at high depth. The limiting geometries at each nn found as β→∞β→∞ correspond to axisymmetric isochronous containers.