Gravity-driven continuous thin film flow over topography

This thesis is directed primarily at a systematic theoretical investigation of gravity-driven thin film flow over various topographical features, the effect of
inertia being of particular interest. The problem is solved using a hierarchy of models based, in order of decreasing complexity, on (i) the full Navier-Stokes system of equations; (ii) a depth averaged form of the latter; (iii) the lubrication equations. Every effort has been made to solve the underlying discrete equation set in each case efficiently using state-of-the-art solution strategies,
thus guaranteeing accurate and mesh-independent predictions. The solution of models (ii) and (iii) centres on the use of a multigrid methodology together with automatic, error controlled, time-stepping and the proper treatment of any associated nonlinear advective terms. A discrete analogue of model (i), for both two- and three-dimensional flows, is obtained using a finite element formulation with the free surface parametrised via the method of spines and the system solved using a parallel multifrontal method together with a memoryefficient
out-of-core storage approach.

A comprehensive set of results is presented for flow over both one- and twodimensional topography, generated using models (ii) and (iii); the predictions obtained are contrasted with each other and compared with existing related experimental data. The free-surface disturbance arising for the problems investigated is revealed to be influenced significantly by the presence of inertia
which leads to an increase in the magnitude and severity of the resulting capillary ridge, surge and trough formations present. A complementary exploration, using model (i) is undertaken which reveals the attendant internal flow
structure. It shows that two-dimensional flow over spanwise topography and three-dimensional flow over localised trench topography can lead to different internal, inertia dependent, flow topologies; findings that are consistent with previously reported results for the well-known lid-driven cavity problem.

Finally, the effect of a normal electric field on the free-surface disturbance generated by inertial thin film flow over topography is investigated using model (ii) coupled with a Fourier series separable solution of Laplace’s equation for the electric potential. Results for both two- and three-dimensional flow reveal that a significant electric field strength can be used to effectively planarise the free-surface capillary ridges and depressions that arise. The two-dimensional solutions obtained are consistent with those reported elsewhere for the case
when inertia is neglected and highlight the importance attached to choosing an appropriate means of embodying the latter. Furthermore, the novel results generated for three-dimensional flow demonstrate that as Reynolds number
increases, larger electric field strengths are required to planarise the associated free-surface disturbance.