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1998 Analysis of a mathematical model related to Czochralski crystal growth
Petr Knobloch, Lutz Tobiska
Abstr. Appl. Anal. 3(3-4): 319-342 (1998). DOI: 10.1155/S108533759800058X


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.


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Petr Knobloch. Lutz Tobiska. "Analysis of a mathematical model related to Czochralski crystal growth." Abstr. Appl. Anal. 3 (3-4) 319 - 342, 1998.


Published: 1998
First available in Project Euclid: 8 April 2003

zbMATH: 0969.35107
MathSciNet: MR1749414
Digital Object Identifier: 10.1155/S108533759800058X

Primary: 35D05 , 35G30
Secondary: 35Q30 , 76D05 , 76Rxx

Keywords: Boussinesq aproximation , Czochralski method , Navier–Stokes equations , nonstandard boundary conditions , weak solvability

Rights: Copyright © 1998 Hindawi

Vol.3 • No. 3-4 • 1998
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