Nonlinear interfacial waves in a circular cylindrical container subjected to a vertical excitation

• The two-layer ideal fluid interface waves were studied.
• Singular perturbation theory of two-time scale expansions was developed.
• A nonlinear amplitude equation with cubic and vertically excited terms was derived.
• Different surface patterns were obtained for different excited frequencies and amplitudes.

Singular perturbation theory of two-time scale expansions was developed in inviscid fluids to investigate the motion of single interface standing wave in a two-layer liquid-filled circular cylindrical vessel, which is subjected to a vertical periodical oscillation. It is assumed that the fluid in the circular cylindrical vessel is inviscid, incompressible and the motion is irrotational, a nonlinear amplitude equation including cubic nonlinear and vertically forced terms, was derived by the method of expansion of two-time scales without taking the influence of surface tension into account. By numerical computation, it is shown that different patterns of interface standing wave can be excited for different driving frequency and amplitude. We found that the interface wave mode become more and more complex as increasing of upper to lower layer density ratio γγ. The traits of the standing interface wave were proved theoretically. In addition, the dispersion relation and nonlinear amplitude equation obtained in this article can reduce to the known results for a single fluid when γ=0,h2→h1γ=0,h2→h1.