Thin Film Flow on Functional Surfaces: Stability and Morphology

Abstract

This thesis explores the stability and morphology of gravity induced film flow down smoothly corrugated rigid substrate, uniformly heated/cooled from below.
The problem of interest is complicated by the presence of a free-surface whose location is unknown .
This complication is overcome by reducing the governing equations of motion and energy to a manageable form within the framework of the well-known long-wave expansion, which exploits the disparity between the horizontal and vertical length scales in order to eliminate the depth-coordinate from the governing formulation.
Two methods for implementing a long-wave expansion are considered, with each leading to an asymptotic model of reduced dimensionality.
The first is a perturbation series of the fluid velocity and temperature with respect to a small parameter which represents the disparity between the horizontal and vertical length scales,
the second is a power series expansion with respect to the vertical coordinate in which the series truncation is correlated to the number of degrees of freedom with respect to the horizontal coordinate.


A key feature of the power series method is proof that, for any asymptotic model to be able to accurately resolve the thermodynamics beyond the trivial case of `a flat film flowing down a planar uniformly heated incline', the expansion of the fluid temperature must be quadratic to leading-order in the long-wave expansion.
The ensuing analysis reveals why heat transfer models based on the Nusselt linear temperature distribution fail to converge outside of the long-wave limit and details how asymptotic models can be extended to higher-order.
Superior predictions are obtained compared with earlier work and reinforced via a series of corresponding solutions to the full governing equations acquired using a purpose written finite element analogue, enabling comparisons of free-surface disturbance and temperature predictions to be made, as well as those of the streamline pattern and temperature contours inside the film.
In particular, the free-surface temperature is captured extremely well at moderate Prandtl numbers for film flow down smoothly corrugated substrate.


Investigation of the stability characteristics of gravity-driven film flow is opened with the classical problem of a thin film flowing down an inclined plate and its associated hydrodynamic stability as described by the Orr-Sommerfeld equation, which reveals the asymptotic methods are not able to fully capture the thermo-capillary effect in the heated/cooled case.
The stability problem is extended to film flow over non-planar substrate via Floquet theory, with the interaction between the substrate topography and thermo-capillarity investigated through a set of neutral stability curves.
Although no relevant experimental data is currently available for the heated film problem, existing numerical predictions and experimental data concerning the stability behaviour of isothermal film flow are taken as a reference point from which to explore the effect of both heating and cooling.