Inertial effects on thin-film wave structures with imposed surface shear

This thesis provides a depth-averaged analytical and numerical approach to the mathematical simulation of thin-film flow on a flat inclined plane relevant to gravity-driven flows subject to high surface shear. Motivated by modelling thin-film structures within an industrial context, wave structures are investigated for flows with moderate inertial effects and small film depth aspect ratio e. Approximations are made assuming a Reynolds number, Re ~O (1/e) and a depth-averaged approach used to simplify the governing Navier-Stokes equations. A classical, parallel, Stokes flow is expected in the absence of any wave disturbance based on a local quadratic profile; in this work a generalised approach, which includes inertial effects, is solved. Flow structures are identified and compared with studies for Stokes flow in the limit of negligible inertial effects. Both two-tier and three-tier wave disturbances are constructed to study film profile evolution subject to shear at the free surface. An evaluation of film profiles is given from a paramet- ric study for wave disturbances with increasing film Reynolds number. An evaluation of standing wave and transient film profiles is undertaken which identifies new profiles not previously predicted when inertial effects are neglected. A revised integral boundary layer model incorporating a more general cubic velocity profile is also introduced, to better capture fluid re- circulation associated with a capillary region, and is developed to provide a better understanding of the internal flow dynamics within the thin-film layer. Notably, the wavelength and amplitude of the capillary ripples are analysed. The effect of the boundary conditions between the fluid and the plane is undertaken to simulate slip properties of various substrates over which the fluid may flow. A Navier slip condition is proposed at this boundary and its effect on the wave structure is examined both with and without the inclusion of inertia. The corresponding film dynamics are analysed with increased slip at the fluid-plane boundary and the effect on the wave structures formed are discussed. In a subsequent chapter solitary wave structures are investigated through a study of gravity-driven flow structures as associated with an oscillating inlet. The effects of increasing the film Reynolds number of these flows is evaluated together with an investigation of the stability characteristics relevant to inlet frequency and inertial effects. The effect of surface shear on solitary waves is examined, both as a stabilising and a destabilising factor on perturbations introduced at the inlet. A final section provides an overview of the outcomes from this study