High-frequency limit of $\gamma(\omega)$

It has already been commented that in the extreme high-frequency limit where each patch behaves as if it were sealed to flow ($\dot{\zeta}_{\rm int}=0$), the theory of Hill (1963) applies. Hill demonstrated, among other things, that when each isotropic patch has the same shear modulus, the volumetric deformation within each patch is a spatial constant. The fluid pressure response in this limit $p_{fi}^\infty$ is thus a uniform spatial constant throughout each phase except in a vanishingly small neighborhood of the interface $\partial \Omega_{12}$ where equilibration is attempting to take place. The small amount of fluid-pressure penetration that is occuring across $\partial \Omega_{12}$ can be locally modeled as a one-dimensional process normal to the interface.

Using the coordinate x to measure linear distance normal to the interface (and into phase 1), one has that equation (56) is satisfied by

$$\begin{eqnarray} p_{f1} &=& p_{f1}^{\infty} + C_1 e^{i\sqrt{i\omega/D_1} \, x } \\ p_{f2} &=& p_{f2}^{\infty} + C_2 e^{-i\sqrt{i\omega/D_1} \, x }\end{eqnarray}$$ (72) (73)

where the diffusivities are defined $D_i = k K B_i/(\eta_i \alpha)$.The constants C_i are found from the continuity conditions (57) to be

$$\begin{eqnarray} C_1 &=& \frac{-1}{1 + \sqrt{\eta_2 B_2/(\eta_1 B_1)}} (p_{f1}^\... ... \sqrt{\eta_2 B_2/(\eta_1 B_1)}} (p_{f1}^\infty - p_{f2}^\infty).\end{eqnarray}$$ (74) (75)

Although not actually needed here, we have that $p_{fi}^\infty = B_i p_{ci}$where the uniform confining pressure of each patch is given by equations (40) and (41) so that the fluid pressure difference between the phases goes as

$$\frac{p_{f1}^\infty - p_{f2}^\infty}{\Delta P} = \frac{B_1 - B_2}{1- \beta (B_1/v_1 + B_2/v_2)}.$$ (76)

This equation is exactly the difference between equations (50) and (51). Because the penetration distance $\sqrt{D_i/\omega}$ vanishes at high-frequencies, we may state that to leading order in the high-frequency limit, $\overline{p}_{f1} - \overline{p}_{f2} = p_{f1}^\infty - p_{f2}^\infty$.

To obtain the high-frequency limit of the transport coefficient $\gamma(\omega)$, we use the definition (53) of the internal transport (note that $-{\bf n} \cdot \nabla p_{f1} = \partial p_{f1}/\partial x$)

$$-i \omega \zeta_{\rm int} = \frac{1}{V} \frac{k}{\eta_1} \i... ...partial \Omega_{12}} \frac{\partial p_{f1}} {\partial x} \, dS$$ (77)

along with equations (72) and (74). The result is

$$\lim_{\omega \rightarrow \infty} \gamma(\omega) = i^{3/2} ... .../(\eta_1 B_1 K)}}{ 1 + \sqrt{\eta_2 B_2/(\eta_1 B_1)}}\right).$$ (78)

Here, S is again the area of $\partial \Omega_{12}$ contained within a volume V of the patchy composite.