Mathematical modelling of heat transfer during electron-beam autocrucible melting by means of the steady-state Stefan problem

A two-dimensional axisymmetric mathematical model of electron-beam autocrucible melting is developed and examined. Here, the hypothesis is used that forced convective heat transfer in the melt may be modelled with the help of the coefficient of effective thermal conductivity, _E. A simplified approach is used in which _E is assumed to be known. In another approach the value of _E depends on a prescribed value of a mean melt-stirring velocity and a mean liquid-pool radius which is determined in the course of solving of the problem. With the help of the Kirchoff transformation and a Green function we may reduce the problem to a nonlinear Hammerstein integral equation. Here, a dependence of the thermal-conductivity coefficient on the temperature, _S(T), at the cooled surfaces is disregarded and constant (mean) values of _S are utilized. In order to solve the problem in the case where this dependence _S(T) is taken into account, an axuiliary Green-function method is proposed which also permits to take into account a change of the heat-exchange coefficients on the autocrucible. This reduces the problem to a system of three integral Hammerstein equations. Numerical solutions of the nonlinear integral equations are obtained with the help of a variational (projective-net) method for the case of circular scanning of an electron beam over the heated surface. The computational results are well consistent with experimental data.     

长方体内多量散射介质光子密度波扩散方程研究

进一步改进了已有物理模型,从理论上解决了三维有限体积内光子密度波扩散方程的求解问题,得到了长方体边界条件下的格林函数的解析解,给出了实验可测量光通量与待测物吸收系数改变量之间可进行数值计算的表达式。本物理模型更加接近实际情况,解决了已有模型与实际情况偏差较大的矛盾,为进行反演计算提高成像质量提供了理论保证。     

多散射介质中光子密度波扩散方程的二阶求解

对于生物医学光子学领域中的新技术-扩散光子密度波技术进行了理论上的研究与扩展,给出了长为a,宽为b,高为无限大的矩形边界条件下光子密度波扩散方程的二阶微扰解析解.由于实际测到的物理量多为光通量,演算出了与入射面相对应面处的光子通量和介质中的吸收系数改变量之间的关系.并利用人体前臂肌肉组织的典型光学参数,给出了二阶微扰解的模拟计算结果,并与一阶微扰计算结果进行了比较.结果表明,二阶微扰解比原有的一阶微扰解在计算精度上有一定的提高,尤其是在异常体附近精度更是有较大的提高.所得二阶结果不仅为利用扩散方程进行生物组织成像问题提供了更加精确的理论基础,而且对于图像质量尤其是分辨异物的能力的提高具有促进作用.     

半无限空间中光子密度波扩散方程的求解

改进了原无限大模型，求解了无限大空间中光子密度波的扩散方程，找出了相应条件下的Green函数，演算出了与入射面相对应的光子通量和介质中的吸收系数改变量之间的关系，所用模型更接近于实验研究情况，所得结论能直接用于多散射介质中成像问题的实验研究。即通过对光子通量的检测，反演出吸收系数改变量的分布，从而实现用光子密度波理论在多散射介质中的成像问题的解决。     

水平气井不稳定渗流数学模型的格林函数求解方法

水平井技术作为提高单井产量的有效手段,能数倍增加储层泄气面积,而目前水平气井的渗流问题研究较少。在气体不稳定渗流基本微分方程的基础上,采用Green函数法,通过积分叠加原理获得在箱式边界条件下水平井不稳定渗流微分方程的拟压力解,为水平气井产能提供了理论依据。