Bubbly flow and its relation to conduction in composites

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Abstract
Following Wallis, the relation between non-viscous bubbly flow and conduction in composites is examined. The bubbles are treated as incompressible and correspond to non-conducting inclusions. A simple relation is found between the effective conductivity and the energy coefficient which is agreement with previous calculations. It is shown that the energy coefficient is frame dependent and, in the frame of zero volumetric flux, is equal to the virtual mass density. Zuber's virtual mass density corresponds to the conductivity of the Hashin–Shtrikman coated-sphere geometry. This connection is exploited to extend Zuber's result to ellipsoidal bubbles. The hyperbolicity of effective equations derived from a variational principle is analysed for various bubble configurations. Without bubble clustering the equations are ill-posed (unstable). However, when the bubbles group into ellipsoidal clusters the resulting effective equations are well-posed for a wide range of parameter values.

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