Numerical solution for equation of liquid drop on inclined plane

Abstract

The problem for the equation of the liquid drop on the inclined plane […] is considered. Here […], D - circle area, V - volume of the liquid in the drop, A 9x; 9y} - the unknown constant (Lagrange multiplier). H(x,y,u) - is a given function [1] with the specific expression depending on the angle of the plane on which the drop lies is inclined. The finite difference method [..] for the solution of the problem (1) is proposed. The operator A(u) could be presented in several different forms: in the rectangular system of the (x,y} variables (the curving contour), in polar system of coordinates, as the system of finite elements or as transformation of the curving linear contour system to the rectangular one [1]. The iterative method for the solution of the finite difference system (2) is proposed. In previous publication of the authors [2] the two-stage iterative methods for the symmetrical drop were proposed. The solution was searched by using the bisection method to obtain the parameter A and then the iterative method to solve the equation with the nonlinear operator A(u) was applied. In this paper the one-stage iterative method instead of two-stage iterative one is proposed maintaining the condition l{u) = V (volume preservation) at each iteration. The convergence of the iterative method is considered. The compatibility of the iterative method with the specific integral nonlocal condition is indicated. The numerical experiment was carried out for the symmetrical drop.