控制熔体温度三维稳定态方程的解析解

Exact Solution of Temperature-Governing Equations for Three-Dimension Steady State Crystal Growth

在周期性边界条件及均匀流场的作用下,从数学角度求解三维稳态晶体生长温度控制方程的解析解,首先将其解展开为关于的Fourier级数,然后引入微分算子,利用复数的性质及一些特殊的数学方法求解特征根,从而求得控制方程的解析解,并确定解中各系数的关系.理论结果表明沿晶体生长方向温度分布具有周期性振荡衰减性质.

At the presence of periodic boundary condition and uniform flow field,the three-dimensional steady mathematical model of crystal growth in the process of solidification of metal melt has been analyzed.In order to obtain the analytic solution of the three-dimensional governing equation of stable crystal growth,the unknown function is expanded into Fourier series with the variable x,y,then introduces differential operator and solves the characteristic roots by using the properties of complex number and several special mathematical methods to obtain the analytic solution of the governing equation and the relations between the coefficient in its solution. The theoretical result shows that the temperature distribution in the direction of crystal growth is exponentially damped oscillation.