The Elastic Landau-Levich Problem on a Slope

Abstract

The elastic analogue of the Landau-Levich dip-coating problem, in which a plate is withdrawn from a bath of fluid on whose surface lies a thin elastic sheet, is analysed for angle of withdrawal θ to the horizontal. The flow is controlled by the elasticity number, El, which is a measure of the relative importance of viscous and bending stresses, and θ. The leading order solution for small El is a steady profile in which the thickness of the film on the plate is found to vary as El^3/4 /(1 − cos θ)^5/8 . This prediction is confirmed in the limit θ « 1 by comparison with numerical simulation. Finally, the circumstances under which the assumption of a steady solution is no longer valid are discussed, and the time-dependent solution is described.