2-D free surface modeling of a liquid contained in an axisymmetric vessel

The emphasis of this research is to determine the free surface shape of a liquid contained in an axisymmetric vessel under the influence of gravitational and centrifugal forces in a microgravity environment. The numerical scheme developed for determining the free surface shape is based on a graphical method developed by Slobozhanin and Tyupstov[7]. The differential equations used in calculating the free surface are the governing equations for the equilibrium free surface line, nondimensionalized in terms of the Bond number and Rotation number, and two equations based on the physical constraints of the problem. The difficulty in the straight forward integration of the equations to obtain the free surface coordinates is due to two unknowns q and r_A. q is an unknown which arises from the development of the governing equations and r_A is the coordinate of the boundary condition satisfied at the vessel wall. The numerical scheme uses a quasi-Newton's method (see [3]) to determine q r_A and to some specified tolerance. After q r_A and are found the governing equations can be integrated to obtain the coordinates of the free surface line. The results obtained by the numerical method are in agreement with the results obtained by the graphical method, and in qualitative agreement with results obtained by Concus [4] and the results given in [2].