FLOWS WITH SURFACE TENSION 337(2) Complexity Measuring Frame. In <strong>the</strong> second choice,one defines a strictly positive, smooth functionf(a, t), whichmeasures <strong>the</strong> complexity of <strong>the</strong> curve. For example,[= 1 + x 2. Now, <strong>the</strong> product s~f is required to be equaleverywhere to its mean. That is,1 f2n Is~(o~, t) f(oq t) =~-~ j0 s~,(o~, t) f(og, t) do(1 L= 2-~ fo f(s, t)ds. (111)Thus, where f is large, s~ (<strong>the</strong> infinitesimal grid spacing) issmall, and vice versa. For general f, it is not possible toobtain an explicit expression for <strong>the</strong> tangential velocityneeded to dynamically enforce condition (111 ). However, itturns out that if f= C(K), <strong>the</strong>n an explicit expression isobtained. Previously, curvature constraints have been usedsuccessfully to statically remesh interfaces (see [9, 27], forexample)• However, <strong>the</strong> choice of T to dynamically enforce(111 ) is given byT(ct, t)= Cl~=0 • T(0, t)C(112)and <strong>the</strong> equal arclength frame again follows by settingC=I.More general reference frames (i.e., more general choicesfor f) may be useful, however. An important example where<strong>the</strong>y may be needed is in <strong>the</strong> case of anisotropic surfacetension [24, 1]. There, <strong>the</strong> surface tension coefficient isdirectionally dependent and so <strong>the</strong>re are preferred directionsof growth. It is possible to choose a reference framethat dynamically equidistributes grid points along <strong>the</strong> interfaceaccording to a measure of <strong>the</strong> anisotropic surfaceenergy• In this case, <strong>the</strong> appropriate tangential velocity isfound by solving a linear integro-differential equation• Thisis currently under study.ACKNOWLEDGMENTSWe thank Russ Caflisch, Anne Greenbaum, Leslie Greengard, JoeKeller, Mike Ward, and especially Dan Meiron and Ray Goldstein, forimportant and stimulating conservations. All three of <strong>the</strong> authors thank <strong>the</strong>hospitality and support of <strong>the</strong> Institute for Advanced Study where <strong>the</strong>yparticipated in <strong>the</strong> special year in fluid mechanics during <strong>the</strong> 1991 academicyear, where some of <strong>the</strong> initial work of this paper was done. There, <strong>the</strong>ywere supported by NSF Grant DMS-9100383 and by Department of <strong>the</strong>Air Force Grant F-49620-92-J-0023F. T.Y.H. acknowledges partial support<strong>from</strong> NSF Grant DMS-9003202 and Air Force Office of ScientificResearch Grant AFOSR-90-0090. J.S.L. acknowledges <strong>the</strong> support of <strong>the</strong>National Science Foundation through a Ma<strong>the</strong>matical Sciences PostdoctoralFellowship, <strong>the</strong> University of Minnesota Army High PerformanceComputing Research Center, <strong>the</strong> Minnesota Supercomputer Center, wheresome of <strong>the</strong> data analysis was performed, and <strong>the</strong> University of MinnesotaGeometry Center, where a movie was made of <strong>the</strong> pinching inertial vortexsheet. 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