Displacement flows under elastic membranes. Part 1. Experiments and direct numerical simulations

Abstract

The injection of fluid into the narrow, liquid-filled gap between a rigid plate and an elastic membrane drives a displacement flow that is controlled by the competition between elastic and viscous forces. We study such flows using the canonical setup of an elasticwalled Hele-Shaw cell whose upper boundary is formed by an elastic sheet. We investigate both single-and two-phase displacement flows in which the localised injection of fluid at a constant flow rate is accommodated by the inflation of the sheet and the outward propagation of an axisymmetric front beyond which the cell remains approximately undeformed. We perform a direct comparison between quantitative experiments and numerical simulations of two theoretical models. The models couple the Föppl-von Kármán equations, which describe the deformation of the thin elastic membrane, to the equations describing the flow, which we model by (i) the Navier-Stokes equations or (ii) lubrication theory, respectively. We identify the dominant physical effects that control the system's behaviour and critically assess modelling assumptions that were made in previous studies. The insight gained from these studies is then used in Part 2 of this work where we formulate an improved lubrication model and develop an asymptotic description of the key phenomena.

Asymptotic regimes in elastohydrodynamic and stochastic leveling on a viscous film

“…For small systems, an analog to such a predicted transition is observed for a thin perturbed elastic plate resting on a nanoscopic fluid layer, where the restoring elastic bending force is opposed by van der Waals forces leading to an elastohydrodynamic touchdown [24] similar to capillary film dewetting [25]. To describe the dynamics theoretically, one can solve the full Navier-Stokes equation in the fluid phase using dynamic boundary conditions at the elastic interface given by the solution of the Föppl-von-Kármán equation [26], using e.g. the immersed-boundary method [27].…”

“…the immersed-boundary method [27]. Viscous flow in thin films can be described by the lubrication theory [28] that has been widely used to study different elastohydrodynamic flow phenomena [8,[22][23][24]26]. However, not much is known about how elastohydrodynamic flows are affected by the ratio between the geometric parameters that characterize the system as it undergoes large changes while the driving force remain the same.…”

Displacement flows under elastic membranes. Part 2. Analysis of interfacial effects

“…In Part 1 of this work (Pihler-Puzović et al 2015), we presented a model for the physical system that couples the Föppl-von-Kármán equations for the elastic sheet to the Navier-Stokes equations for the viscous liquid and includes a free surface that corresponds to the gas-liquid interface. The governing equations were solved numerically using the finite-element library oomph-lib (Heil & Hazel 2006).…”

Frequency response and resonance of a thin fluid film bounded by elastic sheets with application to mechanical filters

“…Interaction of fluid viscosity with solid elasticity in Hele-Shaw cells, or similar configurations, is relevant to various research subjects. Among these are viscous peeling problems (Hosoi & Mahadevan 2004;Lister et al 2013;Peng et al 2015;Pihler-Puzović et al 2015;Carlson & Mahadevan 2016), fluid driven crack propagation (Roper & Lister 2005;Spence et al 1987;Lai et al 2016), gravity currents on elastic substrates (Hewitt et al 2015;Howell et al 2016), and wrinkling of lubricated sheets (Huang & Suo 2002;Kodio et al 2016). At the inviscid flow limit, elastic-inertial fluid-structure-interaction have been shown to vary the solid structure resonance frequency.…”