On the Bifurcation and Stability of Rigidly Rotating Inviscid Liquid Bridges

Abstract

We consider a mathematical model that describes the motion of an ideal fluid of finite volume that forms a bridge between two fixed parallel plates. Most importantly, this model includes capillarity effects at the plates and surface tension at the free surface of the liquid bridge. We point out that the liquid can stick to the plates due to the inner pressure even in the absence of adhesion forces. We use both the Hamiltonian structure and the symmetry group of this model to perform a bifurcation and stability analysis for relative equilibrium solutions. Starting from rigidly rotating, circularly cylindrical fluid bridges, which exist for arbitrary values of the angular velocity and vanishing adhesion forces, we find various symmetry-breaking bifurcations and prove corresponding stability results. Either the angular velocity or the angular momentum can be used as a bifurcation parameter. This analysis reduces to find critical points and corresponding definiteness properties of a potential function involving the respective bifurcation parameter.