Reiterated homogenization applied to heat conduction in heterogeneous media with multiple spatial scales and perfect thermal contact between the phases

Abstract

Several types of heterogeneous media with multiple spatial scales presently offer good potential to improve upon more traditional materials used in heat transfer and other engineering applications. An example might be a large scale, otherwise homogeneous medium filled with dispersed small-scale particles that form aggregate structures at an intermediate scale. In this paper, the strong-form Fourier heat conduction problem in such media is formulated using the method of reiterated homogenization. The constituent phases are assumed to have a perfect thermal contact at the interface. The ratio of two successive length scales of the medium is a constant small parameter $\varepsilon$. The method is an up-scaling procedure that writes the temperature field as an asymptotic multiple-scale expansion in powers of the small parameter $\varepsilon$. The technique leads to two pairs of local and homogenized problems, linked by effective coefficients. In this manner the phenomenon description at the smallest scale is seen to affect the medium macroscale, or effective, behavior, which is the main interest in engineering. To facilitate the physical understanding of the derived sub-problems, an analytical solution is obtained for the heat conduction problem in a laminated binary composite. The present formulation shall serve as a basis for future efforts to numerically compute effective properties of heterogeneous media with multiple spatial scales.

Appendix

The expressions (25) and (26) for the boundary conditions of the first local problem and (46) and (47) for those of the second local problem are now deduced.

Perfect thermal contact-derived boundary conditions It results from Eqs. (3) and (7) up to $\varepsilon ^{4}$ that

$$\begin{aligned} \left[ \!\left[ T_0 + \varepsilon T_1 + \varepsilon ^2 T_2 + \varepsilon ^3 T_3 + \varepsilon ^4 T_4 \right] \!\right] _{\Gamma _s} = 0, \quad \mathrm{on} \; \Gamma _s = \Gamma _Y \cup \Gamma _Z . \end{aligned}$$

(89)

Equating to zero the terms corresponding to equal powers of $\varepsilon$,

$$\begin{aligned} \text{ from } \, \varepsilon ^{0}, \quad \left[ \!\left[ T_0 \right] \!\right] _{\Gamma _s}&= 0, \end{aligned}$$

(90)

$$\begin{aligned} \text{ from } \, \varepsilon ^{1}, \quad \left[ \!\left[ T_1 \right] \!\right] _{\Gamma _s}&= 0, \end{aligned}$$

(91)

$$\begin{aligned} \text{ from } \, \varepsilon ^{2}, \quad \left[ \!\left[ T_2 \right] \!\right] _{\Gamma _s}&= 0. \end{aligned}$$

(92)

From Eqs. (35) and (90), and the condition that the $T_i$, $i=0,1,2,\ldots$, are differentiable functions, one concludes that $T\mathbf{(x)}$ and its derivatives are continuous across $\Gamma _s$.

Combining Eqs. (43) and (91) results in

$$\begin{aligned} \left[ \!\left[ -\chi ^j\mathbf{(y)} \dfrac{\partial {T}}{\partial {x_j}} + \tilde{\tilde{T}}_1\mathbf{(x)} \right] \!\right] _{\Gamma _s} = 0, \end{aligned}$$

(93)

which is satisfied upon requiring that

$$\begin{aligned} \left[ \!\left[ \chi ^j \right] \!\right] _{\Gamma _Y} = 0, \quad \mathrm{on}\; \Gamma _{Y} . \quad \quad \quad (47) \end{aligned}$$

Now combining Eqs. (44) and (92) results in

$$\begin{aligned} \left[ \!\left[ -\chi _{y}^\ell (\mathbf{z}) \left( \dfrac{\partial {T}}{\partial {x_\ell }} - \dfrac{\partial {\chi ^j}}{\partial {y_\ell }} \dfrac{\partial {T}}{\partial {x_j}} \right) + \tilde{T}_2(\mathbf{x},\mathbf{y}) \right] \!\right] _{\Gamma _s} = 0, \end{aligned}$$

(94)

which is satisfied upon requiring that $\tilde{T}_2$ be continuous across $\Gamma _Y$ and

$$\begin{aligned} \left[ \!\left[ \chi _{y}^j \right] \!\right] _{\Gamma _Z} = 0, \quad \mathrm{on}\; \Gamma _{Z} . \quad \quad \quad (26) \end{aligned}$$

Heat flux-derived boundary conditions It results from Eqs. (2), (7) up to $\varepsilon ^{4}$, and (9) that, on $\Gamma _s = \Gamma _Y \cup \Gamma _Z$,

$$\begin{aligned}&\left[\left[ - k_{i j} \left( \varepsilon ^{-2} \frac{\partial T_0}{\partial z_j} + \varepsilon ^{-1} \left( \frac{\partial T_0}{\partial y_j} + \frac{\partial T_1}{\partial z_j} \right) + \varepsilon ^{0} \left( \frac{\partial T_0}{\partial x_j} + \frac{\partial T_1}{\partial y_j} + \frac{\partial T_2}{\partial z_j} \right) \right. \right.\right.\nonumber \\&\quad +\varepsilon ^{1} \left( \frac{\partial T_1}{\partial x_j} + \frac{\partial T_2}{\partial y_j} + \frac{\partial T_3}{\partial z_j} \right) + \varepsilon ^{2} \left( \frac{\partial T_2}{\partial x_j} + \frac{\partial T_3}{\partial y_j} + \frac{\partial T_4}{\partial z_j} \right) \nonumber \\&\quad +\left.\left.\left. \varepsilon ^{3} \left( \frac{\partial T_3}{\partial x_j} + \frac{\partial T_4}{\partial y_j} \right) + \varepsilon ^{4} \frac{\partial T_4}{\partial x_j} \right) \right]\right] _{\Gamma _s} n_i = 0. \end{aligned}$$

(95)

Equating to zero the terms corresponding to equal powers of $\varepsilon$,

$$\begin{aligned} \text{ from } \, \varepsilon ^{-2}, \quad \left[ \!\left[ - k_{i j} \frac{\partial T_0}{\partial z_j} \right] \!\right] _{\Gamma _s} n_i&= 0, \end{aligned}$$

(96)

$$\begin{aligned} \text{ from } \, \varepsilon ^{-1}, \quad \left[ \!\left[ - k_{i j} \left( \frac{\partial T_0}{\partial y_j} + \frac{\partial T_1}{\partial z_j} \right) \right] \!\right] _{\Gamma _s} n_i&= 0, \end{aligned}$$

(97)

$$\begin{aligned} \text{ from } \; \varepsilon ^{0} , \quad \left[ \!\left[ - k_{i j} \left( \frac{\partial T_0}{\partial x_j} + \frac{\partial T_1}{\partial y_j} + \frac{\partial T_2}{\partial z_j} \right) \right] \!\right] _{\Gamma _s} n_i&= 0. \end{aligned}$$

(98)

From Eqs. (35) and (36) it follows that Eqs. (96) and (97) are identically verified.

Combining Eqs. (35), (36), (38), and (98) results in

$$\begin{aligned} \left[ \!\left[ - k_{i j} \left( \frac{\partial T}{\partial x_j} + \frac{\partial \tilde{T}_1}{\partial y_j} - \frac{\partial \chi ^\ell _y(\mathbf{z})}{\partial z_j} \left( \frac{\partial T}{\partial x_\ell } + \frac{\partial \tilde{T}_1}{\partial y_\ell } \right) + \frac{\partial \tilde{T}_2}{\partial z_j} \right) \right] \!\right] _{\Gamma _s} n_i = 0, \end{aligned}$$

(99)

or

$$\begin{aligned} \left[ \!\left[ - \left( k_{i \ell } - k_{i j} \frac{\partial \chi ^\ell _y}{\partial z_j} \right) \left( \frac{\partial T}{\partial x_\ell } + \frac{\partial \tilde{T}_1}{\partial y_\ell } \right) \right] \!\right] _{\Gamma _s} n_i = 0. \end{aligned}$$

(100)

Recognizing in Eq. (100) that T, $\tilde{T}_1$, and their derivatives are continuous across $\Gamma _s$, Eq. (98) is satisfied on the portion $\Gamma _Z$ upon requiring that

$$\begin{aligned} \left[ \!\left[ - \left( k_{i j} - k_{i \ell } \frac{\partial \chi ^j_y}{\partial z_\ell } \right) \right] \!\right] _{\Gamma _Z} n_i = 0, \quad \mathrm{on}\; \Gamma _{Z} . \qquad \qquad (25) \end{aligned}$$

Now the combination of Eqs. (35), (43), (44), and (98) can be written as

$$\begin{aligned} \left[ \!\left[ \frac{\partial T}{\partial x_j} \left( - k_{i j} + k_{i \ell } \frac{\partial \chi ^j}{\partial y_\ell } + k_{i \ell } \frac{\partial \chi ^j_y}{\partial z_\ell } - k_{i \ell } \frac{\partial \chi ^m_y}{\partial z_\ell } \frac{\partial \chi ^j}{\partial y_m} \right) \right] \!\right] _{\Gamma _s} n_i = 0. \end{aligned}$$

(101)

Given continuity of T and its derivatives across $\Gamma _s$, Eq. (101) yields, on application of the average operator over Z,

$$\begin{aligned} \left[ \!\left[ \left\langle - \left( k_{i j} - k_{i \ell } \frac{\partial \chi ^j_y}{\partial z_\ell } \right) + \left( k_{i \ell } - k_{i m} \frac{\partial \chi ^\ell _y}{\partial z_m} \right) \frac{\partial \chi ^j}{\partial y_\ell } \right\rangle _{\small {Z}} \right] \!\right] _{\Gamma _s} n_i = 0. \end{aligned}$$

(102)

Finally, Eq. (98) is also satisfied on the portion $\Gamma _Y$ upon requiring that

$$\begin{aligned} \left[ \!\left[ - \left( k^{1}_{i j} - k^{1}_{i \ell } \frac{\partial \chi ^j}{\partial y_\ell } \right) \right] \!\right] _{\Gamma _Y} n_i = 0, \quad \mathrm{on}\; \Gamma _{Y} . \end{aligned}$$

(103)