Sloshing of an ideal fluid in a horizontally forced rectangular tank

The motion of an ideal fluid in a rectangular tank is studied, under conditions in which the tank is subjected to horizontal sinusoidal periodic forcing. A novel technique is presented for solving the problem; it makes use of a Fourier-series representation in which the time-dependent coefficients are shown to obey a system of forced nonlinear ordinary differential equations. Time-periodic solutions are computed using further Fourier-series representations and Newton’s method to find the doubly subscripted arrays of coefficients. It is shown that the linearized solution describes the motion reasonably well, except near the regions of linearized resonance. A weakly nonlinear theory near resonance is presented, but is found to give a poor description of the motion. Extensive nonlinear results are shown which reveal intricate behaviour near resonance. A method is given for computing the stability of time-periodic solutions; it reveals that the solution branch corresponding to linearized theory is stable, but that additional unstable periodic solution branches may also be present. Further quasi-periodic and chaotic solutions are detected.