Thermal boundary conditions: an asymptotic analysis

Abstract

A generalized thermal boundary condition is derived for a material region, representing all thermal effects of an adjacent thin layer. The boundary condition is obtained by considering the equations of heat conduction in each region and performing an asymptotic expansion of the temperatures about the ratio of thermal conductivities. From the asymptotic theory, the generalized boundary condition arises naturally for the leading order problem. An example is given to demonstrate the utility of the effective boundary condition.

Appendices

Appendix A

In this appendix, we construct the Green’s function for the operator L defined in Sect. 5, and show that the spectrum of L is purely discrete and is as discussed in Sect. 5. First, recall that we are considering the space S of two component vectors u(x) and u ₁, where the first component is in the class of C ²(0,1) functions, and the second component is a real valued number. The reader will also recall that we are considering the operator L defined by

$$ LU=\left({{\begin{array}{*{20}l} {-u^{\prime \prime}(x)}\\ \frac{1}{\ell}{\left\{{u^{\prime}(1)+u(1)} \right\}}\\ \end{array}}} \right) $$

(7.1)

and that L acts on elements of S with the further property that u′(0) = 0and u ₁ = u(1). The Green’s function for this operator will be a two by two matrix defined by

$$ G(x,\xi,\lambda)=\left({{\begin{array}{*{20}l} {g_1 (x,\xi,\lambda)} \hfill & {g_3 (x,\lambda)}\\ {g_2 (\xi,\lambda)} \hfill & {g_4 (\lambda)}\\ \end{array}}} \right) $$

(7.2)

such that

$$ (L-\lambda^2G)=\left({{\begin{array}{*{20}l} {\delta (x-\xi)}& 0\\ 0 & 1\\ \end{array}}} \right) $$

(7.3)

along with the boundary conditions

$$ \frac{\partial g_1}{\partial x}(0,\xi,\lambda)=0 $$

(7.4)

$$ \frac{\partial g_3}{\partial x}(0,\lambda)=0 $$

(7.5)

$$ g_2 (\xi,\lambda)=g_1 (1,\xi,\lambda) $$

(7.6)

$$ g_4 (\lambda)=g_3 (1,\lambda) $$

(7.7)

which ensure that each column of G is in the domain of the operator L. The components of G may easily be solved for. Doing so, one finds that

$$ g_1 (x,\xi,\lambda)=\left\{{{\begin{array}{*{20}c} {-\left({\sin (\lambda \xi)/\lambda +K\cos (\lambda \xi)} \right)\cos (\lambda x)\quad \hbox{if}\,x < \xi}\\ {-\cos (\lambda \xi)\left({K\cos (\lambda x)+\sin (\lambda x)/\lambda} \right)\quad \hbox{if}\,x > \xi}\\ \end{array}}} \right. $$

(7.8)

$$ g_2 (\xi,\lambda)=-\cos (\lambda \xi)\left({K\cos (\lambda)+\sin (\lambda)/\lambda} \right) $$

(7.9)

where

$$ K=\frac{\sin (\lambda)(\lambda^2\ell -1)-\lambda \cos (\lambda)}{\lambda (\cos (\lambda)(1-\lambda^2\ell)-\lambda \sin (\lambda))} $$

(7.10)

and that

$$g_3 (x,\lambda)=\frac{\cos (\lambda x)}{\frac{1}{\ell} \left({\cos (\lambda)-\lambda \sin (\lambda)} \right)-\lambda^2\cos (\lambda)}$$

(7.11)

$$g_4 (x,\lambda)=\frac{\cos (\lambda)}{\frac{1}{\ell} \left({\cos (\lambda)-\lambda \sin (\lambda)} \right)-\lambda^2\cos (\lambda)}.$$

(7.12)

The spectrum of L may now be obtained by considering the integral of the Green’s function over a large circle in the complex λ plane. That is, we need to consider

$$ \frac{1}{2\pi i}\oint {Gd\lambda} $$

(7.13)

which may be evaluated by residues. It is clear that there are no branch cuts and that the poles of G are the solutions of

$$\tan \lambda =\frac{1-\lambda^2\ell}{\lambda}.$$

(7.14)

Hence, the spectrum of L is purely discrete as was to be shown.

Appendix B

In this appendix, we show that eigenvalues to the problem

$$ F^{\prime \prime}(x)+\lambda F(x)=0\quad \hbox{for}\,0 < x < 1 $$

(8.1)

$$ F^{\prime}(0)=0 $$

(8.2)

$$ F^{\prime}(1)+F(1)=\lambda \ell F(1) $$

(8.3)

are positive. To do this, we multiply Eq. (8.1) by F(x) and integrate the equation over the region 0 < x < 1. Integrating by parts and applying the boundary conditions (8.2), (8.3), we obtain

$$ \lambda \ell F(1)^2-F(1)^2-\int\limits_0^1 {F^{\prime}(x)^2dx} +\lambda \int\limits_0^1 {F(x)^2} dx=0 $$

(8.4)

and recall that ℓ is a fixed positive constant. Solving this equation for λ, we get

$$ \lambda =\frac{F(1)^2+\int\limits_0^1 {F^{\prime}(x)^2dx}}{\int\limits_0^1 {F(x)^2} dx+\ell F(1)^2}\geqslant 0. $$

(8.5)