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Axial Creeping Flow in the Gap between a Rigid Cylinder and a Concentric Elastic Tube

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. The study focuses on viscous-elastic time-scales for which the rate of solid deformation is of the same order-of-magnitude as the velocity of the fluid. We employ an elastic shell model and the lubrication approximation to obtain a forced nonlinear diffusion equation governing the viscous-elastic interaction. In the case of an advancing liquid front into a configuration with a negligible film layer (compared with the radial deformation of the elastic tube), the governing equation degenerates into a forced porous medium equation, for which several closed-form solutions are presented. In the case where the initial film layer is non-negligible, self-similarity is used to devise propagation laws for a pressure driven liquid front. When advancing external forces are applied on the tube, the formation of dipole structures is shown to dominate the initial stages of the induced flow and deformation regimes. These are variants of the dipole solution of the porous medium equation. Finally, since the rate of pressure propagation decreases with the height of the liquid film, we show that isolated moving deformation patterns can be created and superimposed to generate a moving wave-like deformation 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.