Axial creeping flow in the gap between a rigid cylinder and a concentric elastic tube

Abstract

We examine transient axial creeping flow in the annular gap between a rigid cylinder and a concentric elastic tube. The gap is initially filled with a thin fluid layer. We employ an elastic shell model and the lubrication approximation to obtain governing equations for the elastohydrodynamic interaction. At long axial length scales viscous forces are balanced by elastic tension, while at shorter length scales the viscous–elastic balance is achieved by means of an interplay between elastic bending, tension and shear stresses. Based on a viscous gravity current analogy in the tensile–viscous regime, we devise propagation laws for displacement flows which are induced by a variety of boundary conditions and examine different limits of the prewetting thickness. Next we focus on the moving elastohydrodynamic contact line at the edge of a penetrating film. A uniform matched asymptotic solution connecting the interior tension-based region with a boundary layer region near the propagation front is presented. Finally, a constructive example is shown in which isolated moving deformation patterns are created and superimposed to form a travelling wave displacement field. The presented interaction between viscosity and elasticity may be applied to fields such as soft robotics and micro-scale or larger swimmers by allowing for the time-dependent control of an axisymmetric compliant boundary.