Shendeleva M L
School of EEIE, South Bank University, 103 Borough Road, London SE1 0AA, United Kingdom.
Phys Rev E Stat Nonlin Soft Matter Phys. 2001 Sep;64(3 Pt 2):036612. doi: 10.1103/PhysRevE.64.036612. Epub 2001 Aug 29.
An instantaneous line heat source located in the medium consisting of two half-spaces with different thermal properties is considered. Green's functions for the temperature field are derived using the Laplace and Fourier transforms in time and space and their inverting by the Cagniard-de Hoop technique known in elastodynamics. The characteristic feature of the proposed approach consists in the application of the Cagniard-de Hoop method to the transient heat conduction problem. The idea is suggested by the fact that the Laplace transform in time reduces the heat conduction equation to a Helmholtz equation, as for the wave propagation. Derived solutions exhibit some wave properties. First, the temperature field is decomposed into the source field and the reflected field in one half-space and the transmitted field in the other. Second, the laws of reflection and refraction can be deduced for the rays of the temperature field. In this connection the ray concept is briefly discussed. It is shown that the rays, introduced in such a way that they are consistent with Snell's law do not represent the directions of heat flux in the medium. Numerical computations of the temperature field as well as diagrams of rays and streamlines of the temperature field are presented.
考虑位于由具有不同热性质的两个半空间组成的介质中的瞬时线热源。利用时间和空间的拉普拉斯变换和傅里叶变换以及弹性动力学中已知的卡尼亚尔 - 德胡普技术对其进行反演，推导了温度场的格林函数。所提出方法的特点在于将卡尼亚尔 - 德胡普方法应用于瞬态热传导问题。这个想法是由以下事实提出的：时间上的拉普拉斯变换将热传导方程简化为亥姆霍兹方程，就像波传播一样。导出的解表现出一些波动特性。首先，温度场在一个半空间中分解为源场和反射场，在另一个半空间中分解为透射场。其次，可以推导出温度场光线的反射和折射定律。就此简要讨论了光线概念。结果表明，以与斯涅尔定律一致的方式引入的光线并不代表介质中热通量的方向。给出了温度场的数值计算以及温度场的光线和流线图。
