The Hamiltonian Structure of the Equations of Motion of a Liquid Drop Trapped Between Two Plates

Abstract

The equations of motion of an ideal incompressible liquid drop trapped between two parallel plates under the influence of surface tension and adhesion forces are studied. A main result of this paper is the proof that the equations of motion can be written in Hamiltonian form formula here Here [Dscr ] denotes a class of real-valued functions on the phase space [Nscr ] of the system and the Hamiltonian H ∈[Dscr ] is the energy function of the system. This allows the derivation of an equation for the (dynamic) contact angle, in which the free fluid surface meets the plates. The behaviour of the dynamic contact angle is a point of great controversy in the capillarity literature and the derivation confirms one of the existing models. In the second part of the paper, which can be read independently, existence and stability questions for rigidly rotating drops are dealt with. The existence of solutions to the equations of motion that describe rotationally symmetric drops which rotate rigidly between the plates with constant angular velocity is proved. These solutions can be regarded as relative equilibria of a mechanical system with symmetry. Using ideas of the energy-momentum method of Lewis, Marsden and Simo, a stability criterion for this kind of motion is provided. To derive this criterion, the second derivative of the so-called augmented energy functional at the relative equilibrium in directions which are transversal to the group orbit of this equilibrium is studied. The stability criterion is applied to rigidly rotating drops of cylindrical shape. These represent solutions to the equations of motion in the case that no adhesion forces act along the plates. The result extends previous work of Vogel and Lewis.