On uniformly rotating fluid drops trapped between two parallel plates

Kruse, H. P., J. E. Marsden and J. Scheurle

Lect. in Appl. Math., AMS, 29, (1993), 307-317

Abstract:

This contribution is about the dynamics of a liquid bridge between two fixed parallel plates. We consider a mathematical model and present some results from the doctoral thesis [10] of the first author. He showed that there is a Poisson bracket and a corresponding Hamiltonian, so that the model equations are in Hamiltonian form. The result generalizes previous results of Lewis et al. on the dynamics of free boundary problems for "free" liquid drops to the case of a drop between two parallel plates, including, especially the effect of capillarity and the angle of contact between the plates and the free fluid surface. Also, we prove the existence of special solutions which represent uniformly rotating fluid ridges, and we present specific stability conditions for these solutions. These results extend work of Concus and Finn [2] and Vogel [18],[19] on static capillarity problems (see also Finn [5]). Using the Hamiltonian structure of the model equations and symmetries of the solutions, the stability conditions can be derived in a systematic way. The ideas that are described will be useful for other situations involving capillarity and free boundary problems as well.