Surface Tension Driven Flow of a Droplet over a Uniform Prewet Surface: An Application of the Thin Film Equation

Supervisor: Dr Michael Dallaston

Understanding the behaviour of liquid droplets on solid substrates is crucial for various applications such as microelectronics, inkjet printing and coating technologies. Under the assumption that the length scale in the direction normal to the surface is significantly shorter than the rest, a reduction of the Navier-Stokes equations known as the thin-film equation can be used to describe the time evolution of a viscous liquid film as it interacts with a gaseous atmosphere and solid surface. We endeavour to model the symmetrical spreading, driven by surface tension, of a thin viscous droplet over a uniform prewet substrate via numerical methods. We propose both implicit and explicit numerical schemes for solving the thin-film equation for comparison purposes. The finalised method is successful in numerically solving the thin-film equation and the time evolution of a liquid droplet over a flat substrate with a precursor film (prewet surface) has been plotted. The implicit numerical scheme allows for higher accuracy regarding the spatial mesh and can facilitate improvements in computational efficiency. Furthermore, the resulting progression of droplet shape over time is consistent with current research surrounding droplet flow geometry. This numerical model showcases the applicability of partial differential equations such as the thin-film equation for modelling liquid droplet dynamics. The resulting visualisation of droplet geometry assists in understanding the complex small-scale behaviours of liquid films on solid surfaces as a result of surface tension forces.