Shock-like free-surface perturbations in low-surface-tension, viscous, thin-film flow exterior to a rotating cylinder

Abstract

Continuing from the work of Hinch & Kelmanson (2003 Proc. R. Soc. Lond. A459, 1193–1213), the lubrication approximation is used to investigate the drift and decay of free–surface perturbations in the viscous flow exterior to a circular cylinder rotating about its horizontal axis in a vertical gravitational field. Non–dimensional parameters corresponding to gravity, γ = ρgbar{h}2/3ωμa, and surface tension, α = γh3/3ωμa4, are used to characterize the flow, where ω and a are respectively the angular velocity and radius of the cylinder, μ, ρ, σ and h are respectively the kinematic viscosity, density, surface tension and mean film thickness of the fluid, and g is the acceleration due to gravity. Within the parameter hierarchy γ2 << α << γ << 1, Hinch & Kelmanson (2003) discovered a complex interaction between rotation, gravity and surface tension, leading to a four–time–scale cascade over which drift and decay of free–surface perturbations occur. However, when α = o(γ2), the low–harmonic asymptotics of Hinch & Kelmanson (2003) cannot represent the shock–like solutions manifest in numerical simulations. Accordingly, the case of vanishingly small surface tension is investigated herein, and the resulting shock–like solutions are analysed. When the surface tension is identically zero, the resulting Hamiltonian problem may be solved explicitly via the method of characteristics, action–angle variables and strained–coordinate asymptotic expansions, which reveal a shock–formation time–scale of ω2μ3a3/3 g3h6. The strained (fast) time–scale which can be deduced a priori via action–angle variables, is consistent with that obtained via the independent asymptotic approach of Hinch & Kelmanson (2003), and the (slow) shock time–scale T = 30γ3t is derived and confirmed via spectral numerical integrations of the full lubrication approximation with vanishingly small, non–zero surface tension. With β = α/30γ3 << 1, a shock thickness of order O(β1/3) is discovered, and the leading–order transient in the surface elevation is found to satisfy a Kuramoto–Sivashinsky evolution equation, which is solved via multiple scales for the extreme cases β << 1 and β >> 1, and numerically otherwise. A universal scaling of the transient results is discovered which gives good agreement with the quasi–steady shock solution, even when the transient shock thickens in response to its decreasing amplitude. Depending upon critical values of α/γ2, β and γ, the transient solution is discovered to decay in one of only four possible sequences comprising one or more of T−1, T−½ and exp(−81αγ2t). Physical data indicate that all four decay sequences are observable in practice.