Methods for numerical modeling of two-dimensional capillary surfaces

Abstract

Certain methods for numerical solving plane and axially symmetric problems on equilibrium shapes of a capillary surface are presented. The methods possess a high order of approximation on a nonuniform grid. They are easy to realize, fairly universal and suitable for constructing not only simply connected but also doubly connected and disconnected surfaces, including strongly curved ones. It is shown that the iterative algorithms constructed are absolutely stable at each iteration. The condition for convergence of iterations is obtained within the framework of a linear theory. To describe peak-shaped configurations of a magnetic fluid in a high magnetic field, an algorithm of generation of adaptive grid nodes in accordance with the surface curvature is proposed. The methods have been tested for the well-known problems of capillary hydrostatics on equilibrium shapes of a drop adjacent to the horizontal rotating plate under gravity, and of an isolated magnetic-fluid drop in a high uniform magnetic field. It has been established that they adequately respond to the physical phenomenon of a crisis of equilibrium shapes, i.e., they can be adopted to investigate the stability of equilibrium states of a capillary surface