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Analysis of a mathematical model related to Czochralski crystal growth. (English) Zbl 0969.35107

Summary: 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.

MSC:

35Q30 Navier-Stokes equations
35D05 Existence of generalized solutions of PDE (MSC2000)
35G30 Boundary value problems for nonlinear higher-order PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
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