Two-time scale dynamics of solutions to a rimming-flow equation

We study the dynamic behaviour of a thin viscous fluid film coating the inner wall of a rotating cylinder — a so-called rimming flow. The resulting partial differential equation:

ℎ_𝑡 + [ℎ + γℎ³(ℎ_θ + ℎ_θθθ) − δℎ³cos(θ)]_θ = 0,   for 𝑡 > 0, θ ∈ 𝕋,

describes the height ℎ > 0 of the fluid film. It is quasilinear, degenerate parabolic, and of fourth order. Three competing effects drive the dynamics of the interface — viscosity, surface tension, and gravity: If the surface-tension parameter γ > 0 is of order one and gravitational effects are neglected (δ = 0), then positive steady states of the PDE are characterised by positive constants, describing circles centered at the origin. These constant steady states are orbitally stable. The existence of positive steady states can further be guaranteed for small positive gravitational influence (δ > 0, small enough).

Finally, going to a rotating coordinate frame ξ = θ − 𝑡, we observe that for δ = 0 and fixed mass 𝑚 > 0, the corresponding positive travelling waves evolve only on a two-dimensional manifold ℳ(𝑚) of steady states, corresponding to circles not centred at the origin. If instead 0 < δ ≪ 1 is positive but small enough, solutions that stay bounded away from zero converge exponentially fast to a δ-neighbourhood of ℳ(𝑚). Close to ℳ(𝑚), we show existence of solutions on a large time scale 𝑡 ∼ 1/δ². Moreover, these solutions evolve on two distinct time scales: • On the large time scale 𝑡, they rotate at the speed of the cylinder around the origin. • On a slow time scale τ = δ²𝑡, the dynamics are governed by an ordinary differential equation in τ on ℳ(𝑚).