Liquid film dynamics in horizontal and tilted tubes: dry spots and sliding drops

Using a model derived from lubrication theory, we consider the evolution of a thin viscous film coating the interior or exterior of a cylindrical tube. The flow is driven by surface tension and gravity and the liquid is assumed to wet the cylinder perfectly. When the tube is horizontal, we use large-time simulations to describe the bifurcation structure of the capillary equilibria appearing at low Bond number. We identify a new film configuration in which an isolated dry patch appears at the top of the tube and demonstrate hysteresis in the transition between rivulets and annular collars as the tube length is varied. For a tube tilted to the vertical, we show how a long initially uniform rivulet can break up first into isolated drops and then annular collars, which subsequently merge. We also show that the speed at which a localized drop moves down the base of a tilted tube is non-monotonic in tilt angle.

A bistable reaction-diffusion system in a stretching flow

We examine the evolution of a bistable reaction in a one-dimensional stretching flow, as a model for chaotic advection. We derive two reduced systems of ordinary differential equations (ODEs) for the dynamics of the governing advection-reaction-diffusion partial differential equations (PDE), for pulse-like and for plateau-like solutions, based on a non-perturbative approach. This reduction allows us to study the dynamics in two cases: first, close to a saddle-node bifurcation at which a pair of nontrivial steady states are born as the dimensionless reaction rate (Damkoehler number) is increased, and, second, for large Damkoehler number, far away from the bifurcation. The main aim is to investigate the initial-value problem and to determine when an initial condition subject to chaotic stirring will decay to zero and when it will give rise to a nonzero final state. Comparisons with full PDE simulations show that the reduced pulse model accurately predicts the threshold amplitude for a pulse initial condition to give rise to a nontrivial final steady state, and that the reduced plateau model gives an accurate picture of the dynamics of the system at large Damkoehler number. Published in Physica D (2006)

Sloshing and slamming oscillations in collapsible channel flow

We consider laminar high-Reynolds-number flow through a finite-length planar channel, where a portion of one wall is replaced by a thin massless elastic membrane that is held under longitudinal tension T and subject to an external pressure distribution. The flow is driven by a fixed pressure drop along the full length of the channel. We investigate the global stability of two-dimensional Poiseuille flow using a method of matched local eigenfunction expansions, which is compared to direct numerical simulations. We trace the neutral stability curve of the primary oscillatory instability of the system, illustrating a transition from high-frequency ‘sloshing’ oscillations at high T to vigorous ‘slamming’ motion at low T . Small-amplitude sloshing at high T can be captured using a low-order eigenmode truncation involving four surface-based modes in the compliant segment of the channel coupled to Womersley flow in the rigid segments. At lower tensions, we show that hydrodynamic modes contribute increasingly to the global instability and we demonstrate a change in the mechanism of energy transfer from the mean flow, with viscous effects being destabilising. Simulations of finite-amplitude oscillations at low T reveal a generic slamming motion, in which the the flexible membrane is drawn close to the opposite rigid wall before rapidly recovering. A simple model is used to demonstrate how fluid inertia in the downstream rigid channel segment, coupled to membrane curvature downstream of the moving constriction, together control slamming dynamics.