On the equilibrium and stability of a capillary liquid with disconnected free surface in an open vessel (Q2516542)

A cylindrical vessel with arbitrary cross-section \(\Gamma\) is considered partially filled with an ideal incompressible liquid of density \(\rho\) and a total volume \(V\). The bottom of the vessel has a circular hole of radius \(r_0\). This hydrosystem is subjected to the action of a homogenous gravitational field directed vertically downward and characterized by the acceleration \(\overrightarrow{g}\). It is assumed that this system is under the conditions close to weightlessness and it is necessary to take into account the action of capillary (surface) forces. Let in the equilibrium state, the liquid occupy the domain \(\Omega= \Omega_0 \bigcup \Omega_1\), where \( \Omega_0 =\Gamma \times (0,h), 0 < h < h_0:=V/|\Gamma|\), is a subdomain corresponding to a part of the liquid inside the vessel and \(\Omega_1\) is a subdomain corresponding to a drop leaving the hole and suspended from the bottom of the vessel. By \(\Gamma_0\) it is denoted the upper free surface of the liquid and assumed that the wetting angle \(\delta\) on the contact boundary of the surface \(\Gamma_0\) with the solid wall \(S\) of the vessel is the right angle, i.e., \(\delta=\frac{\pi}{2}\). Thus, the surface \(\Gamma_0\) is horizontal and \(|\Gamma_0|=|\Gamma|\). The free surface of the drop is denoted by \(\Gamma_1\). It should be determined in the process of solution of this static problem together with the stability of capillary liquid partially occupies the vessel. The authors propose algorithms for the determination the free surface of the liquid and the boundary of the domain of its stability.