平面双连通域稳态温度场的解析解

Analytical Solution of Steady-State Temperature Field in the Plane Double Connected Domain

目前对于平面双连通域稳态温度场的问题,解析解法主要有分离变量法、虚拟热源法、积分变换法和格林函数法等.这些方法仅适用于几何形状简单边界条件不复杂的情况.本研究采用复变函数方法,利用映射函数将物理平面上任意形状的双连通域映射为像平面上的轴对称圆环,然后将偏微分方程转化为常微分方程,利用传热学中第1类边界条件,在像平面内可以求出轴对称圆环的稳态温度场.然后再通过映射函数将结果返回物理平面,便得到物理平面上任意形状双连通域的稳态温度场.对于已知映射函数的偏心圆环和含一圆孔的半无限域的双连通域两个问题,通过本研究提出的方法给出了温度场的解析解.而对于一般的双连通域问题,本研究通过最优化技术给出了映射函数的求解方法,获得了复杂双连通域稳态温度场的解析解.

At present,the analytical solution for the steady-state temperature field of the plane double connected domain mainly includes the separation variable method,the virtual heat source method,the integral transformation method and the Green s function method.These methods are only applicable to the cases with simple geometric shapes and unobstructed boundary conditions.By means of the mapping function,this study used complex variable function method to map the double connected domain with arbitrary shape in the physical plane onto an axisymme- tric circular ring in the image plane,and then transformed the partial differential equations into ordinary differential equations.By the Dirichlet boundary condition in heat transfer,the steady-state temperature field of the axisymmetric circular ring can be obtained in the image plane.Then,returning the result to the physical plane by the mapping function,the steady-state temperature field of the double connected domain in the physical plane could be obtained.For the problems of the double connected domain with known the mapping function,such as the eccentric circular ring and a semi-infinite domain containing a circular hole,the proposed method provided an analytical solution of the steady-state temperature field.For the problem of the general double connected domain,this study gives the method of solving the mapping function through optimization techniques,and obtains the analytical solution of the steady-state temperature field of the complex double connected domain.