Thin-film flows involving deformable and porous interfaces

This thesis investigates thin-film flows involving deformable and porous interfaces, addressing both the fundamental fluid mechanics of free-surface flows prone to a novel class of instability and applications of related flows to biomedical transport. The first part of this thesis focuses on viscous gravity currents spreading over lubricated substrates. Such a free-surface flow involves two immiscible viscous fluids, the interface between them being deformable, with the upper-most surface in contact with the atmosphere. A theoretical framework is developed for such flows using the principles of lubrication theory. We find similarity solutions and perform asymptotic analyses to characterise various flow regimes and a stress singularity near the intrusion front. Building upon this foundation, a linear stability analysis reveals that such flows are prone to a new class of viscous fingering instabilities, arising from hydrostatic interactions between the two viscous fluids. Despite fundamental differences in the type of flow, this new class of fluid-mechanical instabilities curiously resembles a number of features typical of its closest predecessor: the well-known Saffman-Taylor instability, or what simply became known as viscous fingering. This challenges the perception that viscous fingering is limited to porous media, or Hele-Shaw cells, as was popularised in the decades of research since Saffman and Taylor in the 1950s. This thesis highlights how free-surface flows of fluids of unequal viscosity can exhibit similar fingering to that seen in porous media, widening the definition of what the fluid mechanical community perceives to be viscous fingering. We explore how this new class of instabilities depends on contrasts in the viscosity, density, and source flux, and what determines wavelength selection. Extending the problem to inclined substrates demonstrates that the onset and mechanism of instability are robust and not tied to geometric configuration. The second part of this thesis turns to biomedical transport in haemodialysis – a treatment option for patients affected by kidney failure. Such treatment, in itself, is a rich fluidmechanical problem, involving the flow of two viscous fluids (blood and a sterile solution, known as dialysate) in an artificial kidney known as a dialyser. Dialysers involve thousands of long and thin hollow fibres that facilitate the removal of toxins and excess fluid from the blood. On the scale of a single fibre, the length scales involved are such that both blood and dialysate behave as thin films of viscous fluid, separated by a semipermeable fibre membrane. We use lubrication theory to develop a consistent mathematical framework modelling the fluid flow and solute transport within a single fibre of a typical dialyser, characterising both diffusive and convective transport of toxins from the blood to the dialysate. By performing asymptotic analyses, we obtain analytical expressions for the clearance (characterising treatment efficiency) and recover classical results from the literature as special cases in various asymptotic limits. By incorporating fluid flow, our framework is faithful to the underlying hydrodynamics and provides a systematic foundation for improving dialyser design and exploring new treatment modalities.