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Squirt Transport

We next must obtain the coefficient $\gamma(\omega)$ in the mesoscopic transport law $-i\omega \zeta_{\rm int} = 
\gamma(\omega) (\overline{p}_{f1} - \overline{p}_{f2})$. Again, the approach is to first obtain the limiting behaviour at low and high frequencies and then to connect the two limits by a simple function.

The fluid response in phase 1 (the principal porespace) is governed by the Navier-Stokes equation $- \nabla p_{f1} + \eta \nabla^2 {\bf v}_1 
= -i \omega \rho_f {\bf v}_1$ and the compressibility law $K_f \nabla \cdot {\bf v}_1 
= i \omega p_{f1}$ where ${\bf v}_1$ is the local fluid velocity in the pores. Since for all frequencies of interest we have that $\omega \ll K_f/\eta$ (note that $K_f/\eta \approx 10^{12}$ s-1 for liquids and 1010 s-1 for gases), the fluid pressure in phase 1 is governed by the wave equation
\nabla^2 p_{f1} + \omega^2 \frac{\rho_f}{K_f} p_{f1} = 0\end{displaymath} (96)
and since the acoustic wavelength in the fluid is always much greater than the grain sizes, the fluid pressure in the principal porespace satisfies $p_{f1}({\bf r}) = \overline{p}_{f1}$ (a spatial constant) at all frequencies.

The focus, then, is on determining the flow and fluid pressure within the cracked grains (phase 2) that is governed by the local porous-continuum laws ${\bf Q}_2 = - (k_2/\eta) \nabla p_{f2}$ and
\frac{k_2}{\eta} \nabla p_{f2} + i \omega \frac{\alpha_2}{K_2^d B_2} 
p_{f2} = -i \omega \frac{\alpha_2}{K_2^d} p_{c2}\end{displaymath} (97)
where $p_{c2} = - K_2^d \nabla \cdot {\bf u}_2 + \alpha_2 p_{f2}$. This deformation and pressure change is excited by applying a uniform normal stress $-\Delta P {\bf n}$ to the surface of the averaging volume with the fluid pressure satisfying the boundary conditions ${\bf n} \cdot \nabla p_{f2}({\bf r}) = 0$ on $\partial E_2$ and $p_{f2}({\bf r}) = \overline{p}_{f1}$ on $\partial \Omega_{12}$.