Rimming flow of a weakly elastic fluid

Abstract

We present a theoretical description of the flow of a thin polymeric film on the inner wall of a rotating horizontal cylinder. We account for polymer elasticity with the quasi-linear Oldroyd-B constitutive relation. We apply several simplifications to derive analytical solutions to this otherwise analytically intractable problem. Because the film is thin and the Reynolds number is small, we can implement a lubrication approximation. Furthermore, if we consider weakly elastic polymers, we can expand perturbation series in the limit of a small Deborah number. The analytical approximation for the steady-state free surface shows qualitative agreement with previous numerical simulation. Numerical solutions of the approximate evolution equation demonstrate the destabilizing effect of elasticity on the film's perturbed steady-state profile. Keywords Rimming flow · Visco-elastic fluid · Approximate model List of symbols De Deborah number De = λ 1 Ω C B Inverse to the Bond number C B = δ 2 σ/r 0 μΩ e r , e θ Radial and azimuthal unit vectors g Gravity vector g = (−gsinθ, −gcosθ) h Thickness of the liquid layer h(θ, t) η 0 Characteristic thickness of the liquid layer h * = η 0 h K Ratio of solvent viscosity to solution viscosity, K = μ s /μ n Normal to the free surface p Pressure φ Mass flux through liquid layer φ (θ, t) = h 0 ν θ d R r Radial coordinate r 0 Radius of the cylinder R Non-dimensional radial coordinate t Time Communicated by Tim Phillips.