Jan Franců

Modelling of the Czochralski flow

The Czochralski method of the industrial production of a silicon single crystal consists of pulling up the single crystal from the silicon melt. The flow of the melt during the production is called the Czochralski flow. The mathematical description of the flow consists of a coupled system of six P.D.E. in cylindrical coordinates containing Navier-Stokes equations (with the stream function), heat convection-conduction equations, convection-diffusion equation for oxygen impurity and an equation describing magnetic field effect.

On two-scale convergence and periodic unfolding

Abstract Two-scale convergence is an important tool in homogenization theory. The contribution deals with various primary and adjoint (based on unfolding) approaches to the two-scale convergence and their pro-and-con.

On Modeling of Bodies with Holes

Modeling of bodies with holes leads to boundary value problems for P. D. E. on perforated domains, i. e. multiply connected domains. The contribution aims to point out the problem of boundary conditions on the holes which depend on physical background of the problem.

Homogenization of Monotone Problems with Uncertain Coefficients

Abstract The homogenization problem for a nonlinear elliptic equation modelling some physical phenomena set in a periodically heterogeneous medium is studied. Contrary to the usual approach, the coefficients in the equation are supposed to be uncertain functions from a given set of admissible data satisfying suitable monotonicity and continuity conditions. The problem with uncertainties is treated by means of the worst scenario method.

Homogenization of heat equation with hysteresis

The contribution deals with heat equation in the form (c u + W[u])t = div(a . ∇u) + f, where the nonlinear functional operator W[u] is a Prandtl-Ishlinskii hysteresis operator of play type characterized by a distribution function η. The spatially dependent initial boundary value problem is studied. Proof of existence and uniqueness of the solution is omitted since the proof is a slightly modified proof by Brokate-Sprekels.The homogenization problem for this equation is studied. For e → 0, a sequence of problems of the above type with spatially e-periodic coefficients ce, ηe, ae is considered. The coefficients c*, η* and a* in the homogenized problem are identified and convergence of the corresponding solutions ue to u* is proved.

On Modelling of Czochralski Flow, the Case of Non Plane Free Surface

The flow of the melt during the industrial production of single crystal from melt by Czochralski method is called Czochralski flow. The mathematical description of the flow consists of a coupled system of six P.D.E. in cylindrical coordinates containing Navier-Stokes equations (with the stream function vorticity and swirl), heat convection-conduction equation, convection-diffusion equation for oxygen impurity and an equation describing magnetic field effect.