We examine transient axial creeping flow in the annular gap between a rigidcylinder and a concentric elastic tube. The gap is initially filled with a thinfluid layer. The study focuses on viscous-elastic time-scales for which therate of solid deformation is of the same order-of-magnitude as the velocity ofthe fluid. We employ an elastic shell model and the lubrication approximationto obtain a forced nonlinear diffusion equation governing the viscous-elasticinteraction. In the case of an advancing liquid front into a configuration witha negligible film layer (compared with the radial deformation of the elastictube), the governing equation degenerates into a forced porous medium equation,for which several closed-form solutions are presented. In the case where theinitial film layer is non-negligible, self-similarity is used to devisepropagation laws for a pressure driven liquid front. When advancing externalforces are applied on the tube, the formation of dipole structures is shown todominate the initial stages of the induced flow and deformation regimes. Theseare variants of the dipole solution of the porous medium equation. Finally,since the rate of pressure propagation decreases with the height of the liquidfilm, we show that isolated moving deformation patterns can be created andsuperimposed to generate a moving wave-like deformation field. The presentedinteraction between viscosity and elasticity may be applied to fields such assoft-robotics and micro-scale or larger swimmers by allowing for thetime-dependent control of an axisymmetric compliant boundary.
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