2021/11/06 by Tan, Chee Han, Hohenegger, Christel, Osting, Braxton
#49R05 #76B45 #76M30 #FOS: Mathematics #FOS: Physical sciences #Fluid Dynamics (physics.flu-dyn) #Optimization and Control (math.OC) #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.2111.03766
In 1965, B. A. Troesch solved the isoperimetric sloshing problem of determining the container shape that maximizes the fundamental sloshing frequency among two classes of shallow containers: symmetric canals with a given free surface width and cross-sectional area, and radially symmetric containers with a given rim radius and volume [doi:10.1002/cpa.3160180124]. Here, we extend these results in two ways: (i) we consider surface tension effects on the fluid free surface, assuming a flat equilibrium free surface together with a pinned contact line, and (ii) we consider sinusoidal waves traveling along the canal with wavenumber α≥ 0 and spatial period 2π/α; two-dimensional sloshing corresponds to the case α= 0. Generalizing our recent variational characterization of fluid sloshing with surface tension to the case of a pinned contact line, we derive the pinned-edge linear shallow sloshing problem, which is an eigenvalue problem for a generalized Sturm-Liouville system. In the case without surface tension, we show that the optimal shallow canal is a rectangular canal for any α> 0. In the presence of surface tension, we solve for the maximizing cross-section explicitly for shallow canals with any given α≥ 0 and shallow radially symmetric containers with m azimuthal nodal lines, m = 0, 1. Our results reveal that the squared maximal sloshing frequency increases considerably as surface tension increases. Interestingly, both the optimal shallow canal for α= 0 and the optimal shallow radially symmetric container are not convex. As a consequence of our explicit solutions, we establish convergence of the maximizing cross-sections, as surface tension vanishes, to the maximizing cross-sections without surface tension.