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. 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 nonnegligible, 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.

Asymptotic regimes in elastohydrodynamic and stochastic leveling on a viscous film

“…magmatic intrusion in the Earth's crust [1,2], to fracturing and crack formation in glaciers [3], to pumping in the digestive and arterial systems [4][5][6], or the construction of 2D crystals for electronic engineering [7]. Elastohydrodynamic flows have been studied in model geometries in order to understand their generic features and the inherent coupling between the driving force from the elastic deformations of the material and the viscous friction force resisting motion [8][9][10][11][12][13][14][15][16].…”

Transient gas flow in elastic microchannels

“…In this case we define the total stiffness as k = (p − p 0 )/d = (k −1 u + k −1 l ) −1 (where k u and k l are the stiffness of the upper and lower surfaces, respectively). For axial flow in the thin annular gap between an elastic shell and an inner rigid cylinder, the stiffness k is given by k = Ew t /r 2 t (Elbaz & Gat 2016), where E is Young's modulus, w t is the shell thickness and r t is the radius of the cylinder. In addition, based on Gervais et al (2006), for rectangular microchannels k can be approximated by k = E/w c c 1 , where w c is the width of the channel and c 1 is an order-one proportionality constant.…”

“…with an X F = O(T 1/5 ) spread rate, typical of the early-time propagation of the incompressible peeling problem (e.g. Elbaz & Gat 2016), and a validity range of…”

“…This can also be done analytically by matching the ZKB solutions, which are valid in the bulk outside the front, to an exponential solution near the front which decays in the far field. The process was addressed by the authors in a related problem (see Elbaz & Gat 2016).…”

“…While previous studies modelled the adhesive as a Hookean elastic material, McEwan & Taylor (1966) examined the opposite limit of a Newtonian viscous fluid, which enabled calculation of the peeling speed as a function of the applied tension. Other works involving viscous peeling dynamics include Hosoi & Mahadevan (2004), who examined the peeling and levitation of an elastic sheet over a thin viscous film, and Lister, Peng & Neufeld (2013), who studied axisymmetric viscous peeling of an elastic sheet from a flat rigid surface by injection of fluid between the surface and the sheet (additional relevant works include Hodges & Jensen (2002), Hewitt, Balmforth & De Bruyn (2015), Elbaz & Gat (2016), Thorey & Michaut (2016), Young & Stone (2017)).…”