Abstract and Applied Analysis
Volume 3 (1998), Issue 3-4, Pages 319-342

Analysis of a mathematical model related to Czochralski crystal growth

Petr Knobloch1 and Lutz Tobiska2

1Institute of Numerical Mathematics, Faculty of Mathematics and Physics, Charles University, Malostranské námĕstí 25, Praha 1 11800, Czech Republic
2Institute of Analysis and Numerics, Otto von Guericke University Magdeburg, Postfach 4120, Magdeburg 39016, Germany

Received 16 June 1998

Copyright © 1998 Petr Knobloch and Lutz Tobiska. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper is devoted to the study of a stationary problem consisting of the Boussinesq approximation of the Navier–Stokes equations and two convection–diffusion equations for the temperature and concentration, respectively. The equations are considered in 3D and a velocity–pressure formulation of the Navier–Stokes equations is used. The problem is complicated by nonstandard boundary conditions for velocity on the liquid–gas interface where tangential surface forces proportional to surface gradients of temperature and concentration (Marangoni effect) and zero normal component of the velocity are assumed. The velocity field is coupled through this boundary condition and through the buoyancy term in the Navier–Stokes equations with both the temperature and concentration fields. In this paper a weak formulation of the problem is stated and the existence of a weak solution is proved. For small data, the uniqueness of the solution is established.