This is the second edition of the book first published in 1992 and presenting the possible benefit of the use of Green’s functions in the effective resolution of heat conduction problems. The book is divided in twelve chapters which describe the notions of Green’s function and its use for the resolution of heat conduction problems in different geometries and for different boundary conditions imposed on the boundary of the bodies. The book ends with eleven appendices which present the expressions of Green functions in different geometries and other associated tools.
Chapter 1 starts with a short historical introduction on Green’s works and with a short presentation of the interest of Green’s functions in the current situation. Then, the authors write the balance equations which lead to the linear heat equation written as \(\rho c\frac{\partial T}{\partial t} =-\nabla \cdot q+g(x,t)\), where \(q\) is the heat flux \(-k\nabla T\) and \( \nabla \cdot q\) is its divergence, \(g(x,t)\) correspond to the source terms, \( k\) (resp., \(\alpha =k/\rho c\)) is the thermal conductivity (resp., diffusivity) of the homogeneous body. The authors also write this heat equation in cylindrical or spherical coordinates. They present a general boundary condition which may be imposed and written as \(k_{i}\frac{\partial T }{\partial n_{i}}+h_{i}T=f_{i}\), where \(n_{i}\) is the unit outer normal vector. An initial condition is finally added \(T(.,0)=F\). In the case of a 1D body, the steady heat equation reduces to the second-order differential equation \(\frac{d^{2}T}{dx^{2}}+\frac{g}{k}=0\). The authors solve this equation directly or computing first the Green function. In the transient case, the authors compute the Green function using Laplace transform, and they present the solution of the heat equation using this Green function in an infinite, or semi-infinite body and in the case of a flat plate. The first chapter ends with some insights in the study of heterogeneous bodies.
In the short Chapter 2, the authors give a classification of the geometries, of the possible boundary conditions, of the structure of the initial date and of the source terms which can be considered. They refer to this classification in the further chapters of the book.
In Chapter 3, the authors derive the Green’s function solution equation for transient 1D heat conduction problems. They start with the problem \(\frac{1 }{\alpha }\frac{\partial T}{\partial t}=\frac{\partial ^{2}T}{\partial x^{2}} +\frac{1}{k}g\), with the boundary condition \(k_{i}\frac{\partial T}{\partial n_{i}}\mid _{x_{i}}+h_{i}T\mid _{x_{i}}=f_{i}\), \(i=1,2\), and with the initial condition \(T(\cdot,0)=F\). They write the corresponding equation for the Green function \(G\) and then the solution of the heat problem in terms of \(G\). They then consider a rectangular domain with various boundary conditions on the different pieces of the boundary. The chapter ends with an analysis of the 3D case.
Chapter 4 contains the presentation of methods which lead to the expression of transient Green’s functions: the method of images for finite bodies, the Laplace transform, the method of separation of variables and the method of eigenfunctions expansions for other bodies. In each case, the authors discuss many examples where these methods can be applied. The chapter ends with some computations for the Green functions in the steady case. These computations are often based on series expansions which lead to the problem of their convergence.
Chapter 5 explores some efficient tools which can be used in the computations of an approximation of the Green function when the exact expression is given as a series. The authors indicate in some examples the number of terms which have to be considered in order to get a good approximation of the Green function.
In Chapter 6, the authors compute the solution of transient heat conduction problems using the Green functions they have built in Chapter 4. They start with 1D space problems, first considering finite or semi-infinite bodies, then flat plates. They end with 2D finite and rectangular domains or semi-infinite bodies. In these different situations, the authors compute the solution for different boundary conditions.
Chapter 7 and Chapter 8 are devoted to the computation of the solutions of transient heat conduction problems in cylindrical or spherical coordinates, respectively. In Chapter 7, the authors compute the solution of different kinds of heat conduction problems using Green’s functions. They start with the case of an infinite cylinder and go on with a long solid cylinder, or a hollow cylinder. They also consider an infinite body with a circular hole and thin shells. In order to obtain the expressions of the solutions, they use some of the techniques which have have already been presented in the preceding chapter, and especially the method of seperation of variables. In Chapter 8, the authors consider the case of infinite bodies, of solid or hollow spheres and of the exteriors of spheres. Each of these two chapters end with considerations on steady heat conduction problems for these bodies.
In Chapter 9, the authors consider the case of steady-periodic conduction for transient problems. The heating terms are here supposed to be periodic. Hence the solution of the heat conduction problem has to be found periodic. The authors draw some computations using complex variables. They present many examples in the 1D space case, in cyclindrical or spherical coordinates. They also consider layered bodies and they end with 2D or 3D domains.
Chapters 10 and 11 are devoted to the computations of Green’s functions for bodies which are not as “simple” as plates, cylinders, spheres, etc., and to the modifications which have to be introduced in the methods presented in the previous chapters. The authors have indeed in mind bodies whose normal vector on the boundary is not always parallel to a coordinate axis or which are not homogeneous. They thus consider the diffusion equation written in the form \(\nabla \cdot (k(r)\nabla T)+g(r,t)-m(r)^{2}T=\rho (r)c_{p}(r)u(r) \frac{\partial T}{\partial u}\), where the position vector \(r\) occurs in the different terms. They first modify the Green function solution method presented in Chapter 3 in this case, and they prove that they have to solve linear systems when they look for solutions expressed in terms of an appropriate basis functions. This is what they call a Galerkin-based method. Some examples are presented which illustrate this Galerkin-based method. The authors then consider 1D nonhomogeneous bodies, and the chapter ends with a short consideration on the case of fins and on the fin effect. In Chapter 11, several situations are studied where the Galerkin-based method can be applied. The authors consider different types of boundary conditions and different kinds of bodies, such as heterogeneous materials. They also apply this method to the description of fluid flow in ducts.
In the final Chapter 12, the authors move to considerations on numerical methods. They start from the application of Duhamel’s theorem in order to express the solution of a linear heat conduction problem and which leads to integral equations. The authors explain how the unsteady surface element formulation may be used in order to solve these Duhamel’s integral equations. Once again, the chapter gives many examples where this method can be used.
Each chapter ends with exercises whose solutions are partially indicated at the end of the book. Each chapter also ends with a list of references. Because of the many notes and tables throughout the chapters and of the different appendices, the book is self-contained and presents a comprehensive and useful material for engineers who have to solve heat conduction problems in different situations. They will get here the expressions of the Green function or that of the solution of the heat conduction problems for different bodies and under different kinds of boundary conditions. They will also learn how to compute the Green function and the solution, using the efficient methods which are exposed and applied in many examples.