Symmetric Singularity Formation in Lubrication-Type Equations for Interface Motion

Fourth order degenerate diiusion equations arise in a `lubrication approximation' of a thin lm or neck driven by surface tension. Numerical studies of the lubrication equation (LE) ht + (h n hxxx)x = 0 with various boundary conditions indicate that singularity formation in which h(x(t);t) ! 0 occurs for small enough n withànomalous' or`second type' scaling inconsistent with usual dimensional analysis. This paper considers locally symmetric or even singularities in the (LE) and in the modiied lubrication equation (MLE) ht + h n hxxxx = 0. Both equations have the property that entropy bounds forbid nite time singularities when n is suuciently large. Power series expansions for local symmetric similarity solutions are proposed for equation (LE) with n < 1 and (MLE) for all n 2 R. In the latter case, special boundary conditions that force singularity formation are required to produce singularities when n is large. Matching conditions at higher order terms in the expansion suggests a simple functional form for the time dependence of the solution. Computer simulations presented here resolve the self similarity in the onset of the singularity for approximately 30 decades in minx(h(x;t)). Measurements of the similarity shape and time dependences show excellent agreement with the theoretical prediction. One striking feature of the solution to (MLE) is a transition from a nite time singularity to an innnite time singularities n = 3=2. Also both equations (LE) and (MLE) exhibit symmetric singularitiesfor n < 0 with derivatives of order k vanishing at the singular point for all 2 < k < 4?2n. They also exhibit a blow up in derivatives of order greater than 4 ? 2n.