Fourth-order degenerate diffusion equations arise in a ‘lubrication approximation’ of a thin film or neck driven by surface tension. Numerical studies of the lubrication equation (LE) with various boundary conditions indicate that singularity formation in which occurs for small enough with ‘anomalous’ or ‘second type’ scaling inconsistent with usual dimensional analysis.
This paper considers locally symmetric or even singularities in the (LE) and in the modified lubrication equation (MLE) . Both equations have the property that entropy bounds forbid finite time singularities when is sufficiently large. Power series expansions for local symmetric similarity solutions are proposed for equation (LE) with and (MLE) for all . In the latter case, special boundary conditions that force singularity formation are required to produce singularities when 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 of approximately 30 decades in . 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 finite time singularity to an infinite time singularity . Also both equations (LE) and (MLE) exhibit symmetric singularities for with derivatives of order vanishing at the singular point for all . They also exhibit a blow up in derivatives of order greater than .