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Solid spheres

Chase (1979), using some results from Landau and Lifshitz (1959), shows that, for well-separated spheres of radius R, the factor
   \begin{eqnarray}
q(\omega) = {{\rho_f}\over{\phi}}\left[1 + i\Delta(\omega)\right],
 \end{eqnarray} (34)
where
   \begin{eqnarray}
\Delta(\omega) = {{9}\over{4}}(\phi^{-1} - 1)
\left[1 + z - iz(1 + 2z/9)\right]z^{-2}
 \end{eqnarray} (35)
with $z = 2^{-1/2}\xi$. Here $\xi = (\omega R^2/\eta)^{1/2}$,and $\eta$ is the viscosity of the fluid, as usual. Comparing the main terms, we find that this formula shows
   \begin{eqnarray}
\alpha = 1 + {{1}\over{2}} (\phi^{-1} -1),
 \end{eqnarray} (36)
which is in complete agreement with (27), and
   \begin{eqnarray}
F(\xi) = 1 + (-i)^{1/2}\xi.
 \end{eqnarray} (37)
The result (37) is the first of several results showing that F behaves like $\omega^{1/2}$ for large $\omega$.


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Next: 2D duct Up: ANALYTICAL EXAMPLES OF THE Previous: ANALYTICAL EXAMPLES OF THE
Stanford Exploration Project
7/8/2003