Analytical Methods for Heat Conduction in Composites and Porous Media
Analytical Methods for Heat Conduction in Composites and Porous Media
复制标题
复合材料和多孔介质中热传导的分析方法
DOI：
10.1002/9783527621408.ch5
复制
发表时间：
2008
影响因子：
0.8
通讯作者：
S. Rogosin
中科院分区：
文献类型：
--
作者：
V. Mityushev;E. Pesetskaya;S. Rogosin
本章的目的是描述应用于研究各种类型的复合材料和多孔介质中的稳态热传导的分析方法。我们提出了几个解析公式的有效（宏观）电导率张量推导出通过使用不同的方法的基础上，最近的结果偏微分方程理论和复分析。有效特征的研究最近已经成为一个独立的学科，有自己的哲学和机制。复合材料和多孔介质在几何形状和出现的物理问题类型上有所不同。对于复合材料，最受欢迎的是导电性，弹性，弹塑性和热弹性的问题（例如参考文献1）。[1-4]），但对于多孔介质，主要研究流体力学问题（如参考文献[1 - 4]）。[5-9]）。热传导研究的解析方法使我们能够部分统一复合材料和多孔介质的有效热性质理论。在本章中，考虑了孔隙中填充物（流体或气体）为静态时的纯稳态电导率问题。这些问题是多孔介质力学传热传质问题的基准[10]。本章的主要注意力将集中在对上述问题的分析或构造或封闭形式的解决方案上。对于这样的概念可以有不同的解释。对我们来说，得到一个解析解意味着找到一个公式，其中包含一个有限的集合的初等和特殊的功能，组成，积分，导数和甚至级数。此外，这样一个公式中的所有对象都必须有精确的含义（例如，积分和级数的收敛类型应该被描述）。最后，参数的域，以及所有的函数，积分等，必须明确确定。还将显示它们（或它们的交集，如果需要）是非空的。这种方法有点非传统。在经典著作中，人们认为级数不会形成封闭形式的解，但特殊函数可以。这导致了一些误解，因为
The goal of this chapter is to describe analytical methods applied to the study of steady heat conduction in various types of composites and porous media. We present several analytical formulas for the effective (macroscopic) conductivity tensor which are deduced by using different approaches based on the recent results in the theory of partial differential equations and complex analysis. The study of effective characteristics has recently become a separate subject with its own philosophy and machinery. Composites and porous media differ by geometry and by the type of physical problems that appear. For composites, the most popular are problems of conductivity, elasticity, elastoplasticity and thermoelasticity (eg Refs.[1–4]), but for porous media, problems of fluid mechanics are mostly studied (eg Refs.[5–9]).The analytical approach to the study of heat conduction allows us to unify partly the theory of the effective thermal properties in composite materials and porous media. In the present chapter, pure steady conductivity problems are considered when the filler of pores (fluid or gas) is static. Such problems are benchmarks of heat and mass transfer problems of the mechanics of porous media [10]. The main attention throughout this chapter will be paid to analytic or constructive, or closed form solutions to the above mentioned problems. Different interpretations can be given to such a notion. For us to get an analytical solution means to find the formula which contains a finite set of elementary and special functions, compositions, integrals, derivatives and even series. Besides, all objects in such a formula have to have a precise meaning (for instance, the type of the convergence of integrals and series should be described). Last, the domains of parameters, as well as all functions, integrals, etc., have to be explicitly determined. It will also be shown also that they (or their intersections, if necessary) are nonempty. This approach is slightly nontraditional. In classic books, it is supposed that series do not form closed form solutions, but special functions do. It leads to certain misunderstandings since not