Multiple-timescale asymptotic analysis of transient coating flows

New stability results for the widely studied paradigm "rotating cylinder coating flow" problem are found using a novel multiple-timescale asymptotic approach that is not only fully automated within an algebraic-manipulator platform, but also more widely applicable to diverse evolution equations, particularly those arising in thin-film flow on spatially periodic topographies. Hitherto undiscovered contributions to the capillary decay and gravitational drift in the Fourier modes comprising the coating-film thickness on the cylinder are found, the main discovery being the formal derivation of the functional form of a time-dependent decay rate that has previously been speculated only partially and heuristically. The new asymptotic approach admits analysis of the solution on a geometric progression of increasingly slow timescales, the slowest timescale being a priori dictated in the automated procedure. Theoretical results are in excellent agreement with those obtained from spectrally accurate numerical integrations of the evolution equation for the film thickness. The extent to which the predictions of prior related asymptotic studies are improved upon is quantified.