Abstract and Applied Analysis

Analysis of a mathematical model related to Czochralski crystal growth

Petr Knobloch and Lutz Tobiska

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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.

Article information

Abstr. Appl. Anal., Volume 3, Number 3-4 (1998), 319-342.

First available in Project Euclid: 8 April 2003

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


Knobloch, Petr; Tobiska, Lutz. Analysis of a mathematical model related to Czochralski crystal growth. Abstr. Appl. Anal. 3 (1998), no. 3-4, 319--342. doi:10.1155/S108533759800058X. https://projecteuclid.org/euclid.aaa/1049832729

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