Fingering instabilities in gravity and surfactant-related thin film flows

Thin-film fluid flows are central to a number of biological, industrial and chemical applications and processes. Thin films driven by external forces are highly susceptible to instabilities leading to the break-up of the film into fingering-type patterns. These fingering-type patterns are usually undesirable as they lead to imperfections and dry spots. This behaviour has motivated theoreticians to try to understand the behaviour of the flow and the mechanisms by which these instabilities occur.This work focuses on modelling the dynamics of a thin viscous droplet spreading down an inclined pre-wetted plane due to gravity and surfactant-related effects. We use high resolution numerical simulations combined with analytical solutions to describe the influence of different competing physical effects on the spreading behaviour. We also obtain the spreading and thinning rates for the droplet based on fluid and surfactant conservation arguments where a power-law behaviour is not assumed a priori. In particular, a quasi-steady similarity solution is obtained for one-dimensional flow at the leading edge of the droplet. A linear stability analysis shows that this base state is linearly unstable to long-wavelength perturbations in the transverse direction. This suggests that the onset of the fingering instabilities originate from this region. The influence of surfactant, particularly, the Marangoni effect is shown to increase the growth rate and band of unstable wavenumbers compared to gravity-driven spreading alone. Moreover, with the addition of insoluble surfactant it is shown that this region is linearly unstable for all inclination angles. This is in contrast to gravity-driven spreading where it has been shown that there is a threshold angle below which this region is linearly stable. Stability criteria are obtained in the small wave number limit. For gravity-driven spreading, capillary effects are shown to be responsible for the instability in this limit as reported by previous studies. The Marangoni effect is shown to be behind this instability at small inclination angles and in the small wave number limit when surfactant effects are included. Two-dimensional simulations undertaken here support the linear stability results and are useful in exploring the nonlinear stability of the flow