P. D. Howell

The propagation of a liquid bolus along a liquid-lined flexible tube

We use lubrication theory and matched asymptotic expansions to model the quasi-steady propagation of a liquid plug or bolus through an elastic tube. In the limit of small capillary number, asymptotic expressions are found for the pressure drop across the bolus and the thickness of the liquid film left behind, as functions of the capillary number, the thickness of the liquid lining ahead of the bolus and the elastic characteristics of the tube wall. These results generalize the well-known theory for the low capillary number motion of a bubble through a rigid tube (Bretherton 1961). As in that theory, both the pressure drop across the bolus and the thickness of the film it leaves behind vary like the two-thirds power of the capillary number. In our generalized theory, the coefficients in the power laws depend on the elastic properties of the tube. For a given thickness of the liquid lining ahead of the bolus, we identify a critical imposed pressure drop above which the bolus will eventually rupture, and hence the tube will reopen. We find that generically a tube with smaller hoop tension or smaller longitudinal tension is easier to reopen. This flow regime is fundamental to reopening of pulmonary airways, which may become plugged through disease or by instilled/aspirated fluids.

Surface-tension-driven flow on a moving curved surface

The leading-order equations governing the flow of a thin viscous film over a moving curved substrate are derived using lubrication theory. Three possible distinguished limits are identified. In the first, the substrate is nearly flat and its curvature enters the lubrication equation for the film thickness as a body force. In the second, the substrate curvature is constant but an order of magnitude larger; this introduces an extra destabilising term to the equation. In the final regime, the radius of curvature of the substrate is comparable to the lengthscale of the film. The leading-order evolution equation for the thin film is then hyperbolic, and hence can be solved using the method of characteristics. The solution can develop finite-time singularities, which are regularised by surface tension over a short lengthscale. General inner solutions are found for the neighbourhoods of such singularities and matched with the solution of the outer hyperbolic problem. The theory is applied to two special cases: flow over a torus, which is the prototype for flow over a general curved tube, and flow on the inside of a flexible axisymmetric tube, a regime of interest in modelling pulmonary airways.

Curvature suppresses the Rayleigh-Taylor instability

The dynamics of a thin liquid film on the underside of a curved cylindrical substrate is studied. The evolution of the liquid layer is investigated as the film thickness and the radius of curvature of the substrate are varied. A dimensionless parameter (a modified Bond number) that incorporates both geometric parameters, gravity, and surface tension is identified, and allows the observations to be classified according to three different flow regimes: stable films, films with transient growth of perturbations followed by decay, and unstable films. Experiments and linear stability theory confirm that below a critical value of the Bond number curvature of the substrate suppresses the Rayleigh-Taylor instability.

The surface-tension-driven evolution of a two-dimensional annular viscous tube

We consider the evolution of an annular two-dimensional region occupied by viscous fluid driven by surface tension and applied pressure at the free surfaces. We assume that the thickness of the domain is small compared with its circumference so that it may be described as a thin viscous sheet whose ends are joined to form a closed loop. Analytical and numerical solutions of the resulting model are obtained and we show that it is well posed whether run forwards or backwards in time. This enables us to determine, in many cases explicitly, which initial shapes will evolve into a desired final shape. We also show how the application of an internal pressure may be used to control the evolution. This work is motivated by the production of non-axisymmetric capillary tubing via the Vello process. Molten glass is fed through a die and drawn off vertically, while the shape of the cross-section evolves under surface tension and any applied pressure as it flows downstream. Here the goal is to determine the die shape required to achieve a given desired final shape, typically square or rectangular. We conclude by discussing the role of our two-dimensional model in describing the three-dimensional tube-drawing process.

Control and optimization of solute transport in a thin porous tube

Predicting the distribution of solutes or particles in flows within porous-walled tubes is essential to inform the design of devices that rely on cross-flow filtration, such as those used in water purification, irrigation devices, flow field-flow fractionation, and hollow fibre bioreactors for tissue engineering applications. Motivated by these applications, a radially averaged model for fluid and solute transport in a tube with thin porous walls is derived by developing the classical ideas of Taylor dispersion. The model includes solute diffusion and advection via both radial and axial flow components, and the advection, diffusion and uptake coefficients in the averaged equation are explicitly derived. The effect of wall permeability, slip and pressure differentials upon the dispersive solute behaviour are investigated. The model is used to explore the control of solute transport across the membrane walls via the membrane permeability, and a parametric expression for the permeability required to generate a given solute distribution is derived. The theory is applied to the specific example of a hollow fibre membrane bioreactor, where a uniform delivery of nutrient across the membrane walls to the extra-capillary space is required to promote spatially uniform cell growth.

On the predictions and limitations of the Becker-Doring model for reaction kinetics in micellar surfactant solutions

We investigate the breakdown of a system of micellar aggregates in a surfactant solution following an order-one dilution. We derive a mathematical model based on the Becker-Döring system of equations, using realistic expressions for the reaction constants fit to results from Molecular Dynamics simulations. We exploit the largeness of typical aggregation numbers to derive a continuum model, substituting a large system of ordinary differential equations for a partial differential equation in two independent variables: time and aggregate size. Numerical solutions demonstrate that re-equilibration occurs in two distinct stages over well-separated timescales, in agreement with experiment and with previous theories. We conclude by exposing a limitation in the Becker-Döring theory for re-equilibration of surfactant solutions.

On the absence of marginal pinching in thin free films

This paper concerns the drainage of a thin liquid lamella into a Plateau border. Many models for draining soap films assume that their gas-liquid interfaces are effectively immobile. Such models predict a phenomenon known as “marginal pinching”, in which the film tends to pinch in the margin between the lamella and the Plateau border. We analyse the opposite extreme, in which the gas-liquid interfaces are assumed to be stress-free. We apply a nonlocal coordinate transformation that transforms the nonlinear governing equations into the linear heat equation. We thus obtain a travelling-wave solution and show that it is globally attractive. We therefore find that no marginal pinching occurs in this case: the film always approaches a monotonic profile at large times. This behaviour reflects a marked qualitative difference between a film with immobile interfaces and one with perfectly mobile interfaces. In the former case, suction of fluid into the Plateau border is localised, while in the latter it is instantaneously transmitted throughout the film.

Straining flow of a micellar surfactant solution

We present a mathematical model describing the distribution of monomer and micellar surfactant in a steady straining flow beneath a fixed free surface. The model includes adsorption of monomer surfactant at the surface and a single-step reaction whereby

A new pathway for the re-equilibration of micellar surfactant solutions

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Convection by a horizontal thermal gradient

monomer molecules combine to form each micelle. The equations are analysed asymptotically and numerically and the results are compared with experiments. Previous studies of such systems have often assumed equilibrium between the monomer and micellar phases, i.e. that the reaction rate is effectively infinite. Our analysis shows that such an approach inevitably fails under certain physical conditions and also cannot accurately match some experimental results. Our theory provides an improved fit with experiments and allows the reaction rates to be estimated.