Информационная система "Конференции"

Тезисы докладов

Численное решение дифференциальных и интегральных уравнений

Recently the contact-free method of thermogravity convection control by periodically changed heat field in a melt solution (heat field rotation method-HFRM) at crystal growth processes is actively and quite successfully developed and realized [1]. This approach allows to improve melt mixing in a medium of crystallization by the contact-free method through step-by-step switching of vertically aligned heaters around a growth crucible. HFRM mathematical model and its numerical algorithm has been proposed in this paper in order to make the subsequent analysis of how convective flows in a medium of crystallization depend on heat field parameters. The numerical modeling of crystal growth process is based on the simultaneous solution of three-dimensional non-stationary Navier-Stokes equations in Boussinesq approximation and heat transfer equation at the cyclic changed heat conditions on the boundary of calculated region. The assumption was made in considering a flat free surface of the liquid, favorable conditions for liquid adhesion on the crucible inner wall and crystal surface. Newton’s law describes the heat exchange between liquid in the crucible and the furnace environment except phase interface where the temperature remains constant. There were obtained the equation to estimate pressure and necessary boundary conditions. For model numerical solution we used a finite-difference algorithm where the approximation of system of equations was performed on an equidistant space grid. First at each time step, the temperature field in the liquid is calculated at from of the heat transfer equation. The next step was the determination of pressure. To solve the obtained system of algebraic equations we performed the expansion of the periodical grid function into a Fourier series at azimuthal coordinate [2]. The incomplete factorization iteration method was used to solve system of equations for each knot of angular coordinate [3]. The substitution of calculated pressure magnitudes into Navier-Stokes equations allows to obtain velocity distributions. The solution of algebraic equations system obtained by inexplicit approximation of Navier-Stokes and heat transfer equations is performed by Block Successive Over Relaxation iteration method. Several iterations should be performed to correlate the distribution of velocity and pressure in calculated region so as to increase the accuracy of calculations. The results of numerical calculations, namely, temperature distributions in the medium of crystallization, velocity field structures, trajectories of marker moving show the possibility of thermogravity convection control in the medium of crystallization using various heating regimes of crucible side walls in a growth furnace. These results agrees successfully with experimental dates. It shows the sufficiency of a mathematical statement of the problem and the efficiency of designed numerical algorithm.

Примечание. Тезисы докладов публикуются в авторской редакции

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