On the evolution of non-axisymmetric viscous fibres with surface tension, inertia and gravity

We consider the free boundary problem for the evolution of a nearly straight slender fibre of viscous fluid. The motion is driven by prescribing the velocity of the ends of the fibre, and the free surface evolves under the action of surface tension, inertia and gravity. The three-dimensional Navier-Stokes equations and free-surface boundary conditions are analysed asymptotically, using the fact that the inverse aspect ratio, defined to be the ratio between a typical fibre radius and the initial fibre length, is small. This first part of the paper follows earlier work on the stretching of a slender viscous fibre with negligible surface tension effects. The inclusion of surface tension seriously complicates the problem for the evolution of the shape of the cross-section. We adapt ideas applied previously to two-dimensional Stokes flow to show that the shape of the cross-section can be described by means of a conformal map which depends on time and distance along the fibre axis. We give some examples of suitable relevant maps and present numerical solutions of the resulting equations. We also use analytic methods to examine the coupling between stretching and the evolution of the cross-section shape

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 $n$ 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

Mathematical modelling of the overflowing cylinder experiment

The overflowing cylinder (OFC) is an experimental apparatus designed to generate a controlled straining flow at a free surface, whose dynamic properties may then be investigated. Surfactant solution is pumped up slowly through a vertical cylinder. On reaching the top, the liquid forms a flat free surface which expands radially before overflowing down the side of the cylinder. The velocity, surface tension and surfactant concentration on the expanding free surface are measured using a variety of non-invasive techniques. A mathematical model for the OFC has been previously derived by Breward, Darton, Howell and Ockendon and shown to give satisfactory agreement with experimental results. However, a puzzling indeterminacy in the model renders it unable to predict one scalar parameter (e.g. the surfactant concentration at the centre of the cylinder), which must be therefore be taken from the experiments. In this paper we analyse the OFC model asymptotically and numerically. We show that solutions typically develop one of two possible singularities. In the first, the surface concentration of surfactant reaches zero a finite distance from the cylinder axis, while the surface velocity tends to infinity there. In the second, the surfactant concentration is exponentially large and a stagnation point forms just inside the rim of the cylinder. We propose a criterion for selecting the free parameter, based on the elimination of both singularities, and show that it leads to good agreement with experimental results

The drainage of a foam lamella

We present a mathematical model for the drainage of a surfactant-stabilised foam lamella, including capillary, Marangoni and viscous effects and allowing for diffusion, advection and adsorption of the surfactant molecules. We use the slender geometry of a lamella to formulate the model in the thin-film limit and perform an asymptotic decomposition of the liquid domain into a capillary-static Plateau border, a time-dependent thin film and a transition region between the two. By solving a quasi-steady boundary-value problem in the transition region, we obtain the flux of liquid from the lamella into the Plateau border and thus are able to determine the rate at which the lamella drains. Our method is illustrated initially in the surfactant-free case. Numerical results are presented for three particular parameter regimes of interest when surfactant is present. Both monotonic profiles and those exhibiting a dimple near the Plateau border are found, the latter having been previously observed in experiments. The velocity field may be uniform across the lamella or of parabolic Poiseuille type, with fluid either driven out along the centre-line and back along the free surfaces or vice versa. We find that diffusion may be negligible for a typical real surfactant, although this does not lead to a reduction in order because of the inherently diffusive nature of the fluid-surfactant interaction. Finally, we obtain the surprising result that the flux of liquid from the lamella into the Plateau border increases as the lamella thins, approaching infinity at a finite lamella thickness

Mathematical modelling of curtain coating

We present a simple mathematical model for the fluid flow in the curtain coating process, exploiting the small aspect ratio, and examine the model in the large-Reynolds-number limit of industrial interest. We show that the fluid is in free fall except for a region close to the substrate, but find that the model can not describe the turning of the curtain onto the substrate. We find that the inclusion of a viscous bending moment close to the substrate allows the curtain to “turn the corner”

Boundary conditions for free surface inlet and outlet\ud problems

We investigate and compare the boundary conditions that are to be applied to free surface problems involving inlet and outlets of Newtonian fluid, typically found in coating processes. The flux of fluid is a priori known at an inlet, but unknown at an outlet, where it is governed by the local behaviour near the film-forming meniscus. In the limit of vanishing capillary number Ca it is well-known that the flux scales with Ca2/3, but this classical result is nonuniform as the contact angle approaches . By examining this limit we find a solution that is uniformly valid for all contact angles. Furthermore, by considering the far-field behaviour of the free surface we show that there exists a critical capillary number above which the problem at an inlet becomes over-determined. The implications of this result for the modelling of coating flows are discussed

Mathematical modelling of elastoplasticity at high stress

This paper describes a simple mathematical model for one-dimensional elastoplastic wave propagation in a metal in the regime where the applied stress greatly exceeds the yield stress. Attention is focussed on the increasing ductility that occurs in the over-driven limit when the plastic wave speed approaches the elastic wave speed. Our model predicts that a plastic compression wave is unable to travel faster than the elastic wave speed, and instead splits into a compressive elastoplastic shock followed by a plastic expansion wave