The work presented in this thesis focuses on gravity driven bilayer flow over a functional surface containing topography, with both liquids taken to be perfectly immiscible. Two such problems are considered and investigated systematically: (i) when the flow is confined between two rigid surfaces ("channel flow"); (ii) for the case of free-surface film flow down an inclined plane ("free-surface flow"). Both problems are underpinned by rigorous and comprehensive mathematical derivations, and the governing equation sets, resulting from application of the long-wave approximation, solved numerically using efficient and accurate finite difference algorithms programmed in C++. Such problems have received scant attention to-date.
The channel flow work begins by revisiting the problem investigated by Lenz and Kumar (2007) and Zhou and Kumar (2012), to explore bilayer flow for the particular case of one Newtonian liquid lying above another and confined by rigid surfaces aligned parallel to each other, the lower one containing a steep-sided topographical feature. The investigation carried out serves a number of important purposes, the first being to establish the validity of the modelling and numerical approaches adopted, with the mesh independent results obtained found to be in excellent agreement with earlier work. In addition, the depth-averaged equation set derived in the thesis enables solutions to be obtained when the Reynolds number is non-zero, in contrast to the work of others which achieved only partial success. Finally, the situation when the upper wall of the channel is allowed to move horizontally with a constant speed, inducing a shear flow, is investigated for the first time.
Bilayer free surface film flow over steep-sided topography, solutions to which have not been reported in the literature hitherto, is similarly investigated; comparisons having to be drawn for consistency and verification purposes with the case of single layer flow, Decré and Baret (2003), Gaskell et al. (2004), Veremieiev et al. (2010). Both zero and non-zero Reynolds number flow are considered and the governing equation sets and finite difference expressions re-derived to accommodate non-Newtonian behaviour, for the particular case of power-law liquids; it is found that for the latter case the associated depth-averaged equation set as formulated cannot be solved unless additional simplifications are adopted. In addition, for the case of Newtonian liquids, it is shown that the work can be extended to embody the more practical situation of three-dimensional bilayer film flow over topography. The mathematical model for this same film flow problem is extended to accommodate N layers, for the case when the Reynolds number is zero, with the derivation provided for completeness.
