Removing the Stiffness from Interfacial Flows with Surface Tension

318 HOU, LOWENGRUB, AND SHELLEYThe source of stiffness is made clear by the linear motion.The stability constraint for an explicit time integrationmethod applied to Eq. (29) has the form~It ff[rl~] + S1 )ffEq=~]. (34)S~ S aA "frozen coefficient" analysis of Eq. (34) reveals the stricterstability constraint~It < C. (g=h)3/S, (35)where g,=min, s,. Therefore, the stability constraint isdetermined by the minimum grid spacing in arclength,which is strongly time dependent. Our experience is that theLagrangian motion of the points can lead to "pointclustering" and hence to very stiff systems, even for flows inwhich the interface is smooth and the surface tension issmall.For inertial vortex sheets, the growth rate for perturbationsabout the flat equilibrium is given byS~2 = V~k2_~ ikl 3. (36)Again, the surface tension controls a high wave-numberinstability. The instability here is the Kelvin-Helmholtzinstability and it is due to the shearing motion across theinterface. As can be seen from the growth rate, the surfacetension is a dispersive regularization in contrast to theHele-Shaw case where it is dissipative. Again, by linearizingaround the time dependent inertial vortex sheet F=(x(0~, t), y(~, t)) with strength 7(~, t), Beale et al. [ 10, 12]find the dominant behavior for r/, again the normal componentof the perturbation, to be),2 Stltt= --~S J~O[t/~ct~t]'a'=4 ?/~t"3I- ~-~-3 2s= (37)The perturbation in y is eliminated, to leading order, byusing two time derivatives on ~/. A frozen coefficient analysisleads to the dynamic stability constraintWhile the time step constraint (38) is less restrictive thanthat for Hele-Shaw flows, our experience is that pointclustering, through the Lagrangian point motion, still leadsto prohibitively stiff systems.4. THE SMALL SCALE BEHAVIOR OF UThe stiffness of an explicit method occurs because theevolution at small length scales is controlled by a high orderterm introduced by the curvature. The normal velocity Ufrom the Birkhoff-Rott integral contains the physically relevantpart of the velocity field, and the curvature appears init through the vortex sheet strength. In this section, thesmall scale behavior of U is analyzed and precisely determinedin terms of the vortex sheet strength.First, the normal velocity U is given in a convenient form.Let the complex position of the interface be given byz(e, t) = x(e, t) + iy(~, t). Then, the normal velocity is givenbyU(o~,t)=_llm{Z ~ +oo y(o~',t) }P.v. f tl .(39)This quantity is clearly related to the Hilbert transform ofthe vortex sheet strength 7 over the curved interface F. Ouroriginal intuition was that the small-scale behavior of thisexpression could be found by simply retaining one term inthe expansion of the denominator in the Birkhoff-Rottintegral to yield a Hilbert transform over a flat interface,z = ~t, with a variable coefficient prefactor. This intuition isset rigorously in the following way. The kernel in theBirkhoff-Rott integral is rewritten as1 1mz(o~,t)-z(og, t) z,.(o~-o~')[ 1+ z(oq t)- z(og, t)(40)Note that the bracketed term has a removable singularity at= e', provided that z is a smooth function ofcq the interfacedoes not self-intersect, and s~ > 0. The Birkhoff-Rottintegral can then be rewritten asf+~ 7(0~', t)P.V. -~ z(cq t) -z(~', t) d0(n ~ff[7]=_ +f+~Z~ oo7(~',t)g(ot, o~',t) d o~',(41)At < C. (gah)3/2/S. (38) where g is the term in the brackets of Eq. (40). Thus, the