PhD in Applied Mathematics
Asymptotic and numerical studies of nonlinear
evolution equations of viscous thin–film flows
Christian M Groh
The Univeristy of Leeds
School of Mathematics
September 2010
Examination: 09. December 2010
Supervised by Prof. Mark A Kelmanson
Science is built up of facts, as a house is built of stones;
but an accumulation of facts is no more a science than a heap of
stones is a house.
— Henri Poincaré (1905)
Abstract
This thesis considers weakly nonlinear spatio–temporal evolution equations of the type arising in the thin–film modelling of free–surface viscous flows, with particular reference to a general class of problems, industrially motivated in part, in which the spatial variation is periodic. First, the emphasis is on the development of two solution methods, numerical and asymptotic, for such evolution equations; the two disparate methodologies then admit mutually supportive results. Second, armed with the validated techniques, the emphasis is on confidently identifying new physical phenomena inherent in classical and well–studied problems, in particular the celebrated "Moffatt coating–flow problem" (and increasingly sophisticated variations thereon) of a thin annulus of viscous liquid adhering to the exterior surface of a rotating horizontal circular cylinder.
The numerical methods are distinctive because they utilise the inherent spatial periodicity to solve the evolution equations via Fourier–spectral discretisations that are not only accurate to machine precision, but also considerably more efficient than the (fourth–order) finite–difference schemes employed in prior related studies. In preparation for the investigation of new physical phenomena, a suite of numerical post–processing tools is also developed for the determination of properties such as wave–mode decay and drift rates.
The asymptotic methods are distinctive on two fronts. First, they consider in detail the correct a priori scalings required to render the asymptotic expansions uniformly valid; this reveals, for the first time, explicit restrictions on the admissible parameters governing the competing physical effects of film thickness, gravity, surface tension and inertia. Second, the solution techniques — specifically, multiple–timescale methods — are implemented in a fully automated fashion using an algebraic manipulator, here in Maple; this admits the generation of solutions to an asymptotic order exceeding those of previous studies by several orders of magnitude. A fully automated asymptotic procedure is similarly developed and implemented for linear analyses about the steady state.
Finally, the mutually supportive numerical and asymptotic methods are applied to increasingly sophisticated models of the aforementioned coating–flow problem, the latter method yielding explicit formulae for all new results. This reveals several hitherto–undiscovered transient dynamics of the well–studied Pukhnachev problem (the Moffat problem augmented by capillary effects); in particular, a new cascade–like structure describing the surface–tension–induced annihilation of free–surface harmonics is discovered. Pukhnachev's problem is then augmented by inertial effects, and new results on their influence on flow stability, in particular, the existence of a new large–time limit–cycle behaviour, is revealed.