A thin liquid film flowing down a vertical fiber exhibits complex and captivating interfacial dynamics of an unsteady flow, including the formation of droplets, irregular traveling waves, and a string of beads. The dynamics of traveling beads are particularly applicable in various fluid experiments due to their high surface-area-to-volume ratio, which enhances gas diffusion towards the liquid surface. Recent studies verified that when the flow undergoes regime transitions, the motion of the film thickness changes dramatically. The dynamics of the liquid film are influenced by the cylindrical geometry of the fiber and the presence of stabilizing and destabilizing forces, which pose additional challenges for numerical simulations. Many current numerical simulations of fiber coating dynamics, such as direct numerical simulation (DNS), rely on the full Navier-Stokes equations. Such numerical simulations incur significant computational costs, often requiring several days on a desktop computer. As a result of the high computational cost, it is not feasible to simulate regime transitions or extend the computational domain further downstream of the fluid.
In order to overcome these challenges, several reduced-order models using lubrication approximations have been developed. These models are much simpler than the full Navier-Stokes equations, yet they are capable of capturing droplet dynamics and transient patterns of the flow. However, these models have drawbacks in terms of their versatility, as one needs to modify the form of the equations depending on the assumptions made regarding boundary conditions and the scales of the problem. Depending on the boundary conditions at the solid-liquid interface, one may observe singularities, cusps, and non-classical shocks in finite time. If one were to consider a liquid film with a moderate thickness or a moderate Reynolds number, the resulting equation would exhibit significant differences.
In this dissertation, we consider a model for fiber coating at low Reynolds numbers with geometry such that the fluid thickness is larger than the fiber radius. We present a computationally efficient numerical method that can maintain the positivity of the film thickness as well as conserve the volume of the fluid in a coarse mesh setting. Our method allows simulations of regimes with isolated droplets and the Rayleigh-Plateau instability, commonly observed in laboratory experiments but particularly difficult to simulate. We create our positivity-preserving numerical method in the following way. First, we present a conserved and dissipative quantity for our continuous model. Next, we construct a continuous in time and discrete-in-space numerical method that satisfies the discrete equivalent of conservation of mass and an entropy estimate. We provide a proof of the positivity of our numerical method using a priori and a posteriori bounds and a proof of second-order consistency. Finally, we show that our method can be implemented efficiently using an adaptive time-stepping method to describe solutions that correspond quantitatively to laboratory experiments.
