Numerical analysis of solutocapillary Marangoni-induced interfacial waves

Spreading problems and solutocapillary waves are now routinely treated by semi-analytic lubrication theory leading to a 1D spatiotemporal system to be integrated numerically. In this review, such theories have been shown to be robust predictors of the pseudo-steady propagation at long times with only an initial transient period when the lubrication assumptions breakdown and the wave front is retarded due to bottom friction. Linear stability theory for bottom friction effects leads to 1D evolution equations that predict the scale of Marangoni stresses needed to excite waves and the solitary wave structure of their propagation. In general, applications which are sensitive to Marangoni effects naturally have high values of the Marangoni number (at least hundreds and potentially much higher in evaporation problems). Even when the Marangoni-induced effects are small amplitude, the gradients in stresses are such that numerical resolution requirements are steep. The idealization of interfacial dynamics to a domain with zero thickness (molecular effects) is computationally more demanding than the boundary layers induced in say high Reynolds number laminar flows. Therefore, specialized computational methods for treating open deformable interfaces with high transverse gradients are both required and are being successfully developed as reported here.