Reports: AC9
46393-AC9 Harmonic Moments of Viscous Fingering Structures in Thin Oil Layers
We are studying the growth of air bubbles in an oil layer contained between two horizontal closely spaced circular plates. The bubble grows as the oil is removed from the perimeter of the plates. The multi-fingered patterns that develop as the oil is removed are similar to those formed by water when it replaces oil as it is extracted from an underground reservoir.
We have obtained the first laboratory results for the harmonic moments, which are integrals of integer powers of z = x + iy over the oil domain. We derived an expression that relates the time derivative of the harmonic moments to physically measurable properties --- viscosity, surface tension, and the spacing between the two plates that confine the oil layer. The time derivatives of the moments, determined directly by integrating over the contour encompassing an air bubble in an oil layer, are found to be in good accord with our theoretical prediction.
Our work demonstrates that the harmonic moment representation of interfaces is physically realizable and robust. The harmonic moments description should be applicable to a large body of phenomena where the velocity of an interface is proportional to the gradient of a scalar field. We submitted a manuscript on this work for publication in Physical Review E, and we received a strongly positive review. We made a revision in response to a suggestion of the reviewer, and the revised manuscript is now under review.
Now we are investigating the utility of phase field models to address fundamental open questions in the theory of viscous fingering. We are testing the phase field description of viscous fingering by comparing theoretical predictions with our laboratory observations for finger angles in rectangular, circular, and wedge shaped geometries. Our analyses and experiments are being conducted now by a graduate student who is working together with an undergraduate student and a postdoctoral research associate. The two previous graduate students have left, one after receiving his PhD and accepting a position in a laboratory supported by the U. S. Office of Naval Research; the other student accepted a position in a technology consulting firm.
