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46393-AC9
Harmonic Moments of Viscous Fingering Structures in Thin Oil Layers

Harry L. Swinney, University of Texas at Austin

 We are conducting experiments and analyses of the growth of an air bubble 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 oil is removed are similar to those formed by water when it replaces oil as it is extracted from an underground reservoir. We obtain high resolution digital movies of the developing patterns, and for each bubble pattern image we calculate its harmonic moments, which are integrals of integer powers of z = x + iy over the oil domain.

           Our work is the first to determine harmonic moments from laboratory measurements.  Harmonic moments are important because they form unique mathematical basis functions for describing the viscous fingering patterns that occur in many situations in nature and in petroleum and other industries.

           In contrast to a Fourier series or other representations, the harmonic moments are well behaved (that is, do not diverge) in the limit of zero interfacial surface tension. The harmonic moments are purely geometrical quantities, depending only on the shape of the pattern.  Yet our work demonstrates that measurements of the time evolution of the moments can be used to deduce the a physical property ---  surface tension --- if the viscosity of each fluid is known.  We are extracting a value of the surface tension from a harmonic moments analysis and are comparing the  result to the value of surface tension obtained by traditional techniques.

           Our work shows that the harmonic moments 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.

           Our experiments were conducted by a graduate student who has been awarded a PhD in Physics from the University of Texas at Austin.  He has accepted a staff position in an underwater acoustics research laboratory in Austin, where he will be able to apply directly many of the skills gained in his PhD research.            We are now preparing a manuscript on our results for publication, in collaboration with a theorist collaborator from Los Alamos National Laboratory and a former visitor in our laboratory who is now working in Moscow.  Also, in the coming year the principal investigator will present our results in seminars and conferences.

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