On the decay and drift of free-surface perturbations in viscous thin-film flow exterior to a rotating cylinder

Free–surface viscous flow exterior to a circular cylinder rotating about its horizontal axis in a vertical gravitational field is considered. When the mean film thickness is small compared with the cylinder radius a, numerical simulations of the full Stokes equations reveal a surface–amplitude decay rate so slow that computational expense precludes investigation of large–time dynamics. However, numerical integrations of the simpler lubrication approximation are achievable to large times, and these reveal not only a slow decay to steady state, but also a gravity–induced phase lag, relative to the cylinder, in the wave modes in the free–surface elevation.A naive–expansion analysis reveals a complicated evolution in time with four different time–scales. Firstly, there is the fast process of rotating with the cylinder on a time–scale 1/ω, where ω is the angular velocity of the cylinder. Secondly, surface tension squeezes the free surface to a cylindrical shape on the time–scale μa4/s3, where μ is the dynamic viscosity of the fluid and s its surface tension. After this time, disturbances to the steady state take the form of an eccentricity of the cylindrical shape of the free surface that drifts in phase on the third time–scale of μ2a2/ω2g24, where ω is the density of the fluid and g the gravitational acceleration, and decays exponentially on the fourth and slowest time–scale of μ2μ3a6/μ2g2s7.The naive–expansion analysis thus suggests the drift rate and exponential decay rate in the fundamental mode to be proportional to 4 and 7 respectively. A rescaling is suggested wherein is the length–scale, whence a four–term, two–time–scale expansion for the film thickness yields explicit formulae for both the decay and drift rates, the former of which can be used in practical experiments to enforce the fastest possible decay to the steady state. An unusual delay in the resolution of secularity in the two–time–scale expansion is explained, as is the ability of the two–time–scale expansion to capture the four–time–scale physics. Results are presented for a selection of (non–dimensionalized) surface tension and gravity parameters, and excellent agreement is demonstrated between our asymptotics and the results of numerical simulations over varying time–scales. The convergence to the independently derived steady state is demonstrated, and a detailed explanation is presented of the influential physical mechanisms inherent in the multiple–scale expansion.