The draining of a two-dimensional bubble

We consider a two-dimensional bubble at rest near the surface of a semi-infinite liquid layer. A lubrication analysis of the thin film above the bubble is matched to a capillary-static solution for the outer geometry. By analysing a transition region between the thinning viscous film and the capillary-static solution, we derive an effective boundary condition to be applied at the edge of the film. The result is a description of the drainage of liquid out of the film under gravity and surface tension. This drainage is ultimately responsible for rupture of the film and hence bursting of the bubble

Models for thin viscous sheets

Leading-order equations governing the dynamics of a two-dimensional thin viscous sheet are derived. The inclusion of inertia effects is found to result in an ill-posed model when the sheet is compressed, and the resulting paradox is resolved by rescaling the equations over new length- and timescales which depend on the Reynolds number of the flow and the aspect ratio of the sheet. Physically this implies a dominant lengthscale for transverse displacements during viscous buckling. The theory is generalised to give new models for fully three-dimensional sheets

Fluid mechanical modelling of the scroll compressor

This case-study concerns the flow of gas in a so-called Scroll Compressor. In this device a number of chambers of gas at different temperatures and pressures are separated by narrow channels through which leakage can occur. Using compressible lubrication theory, an estimate for the leakage rate is found in terms of the material properties of the gas and the geometry of the compressor. Thus a simple functional is obtained which allows the efficiency of different compressor designs to be compared. Next we derive a set of ordinary differential equations for the temperature and pressure in each chamber; the coupling between them arises from the leakage. The numerical solution of these equations allows a realistic simulation of a working compressor, and suggests some interesting possibilities for future designs. This problem arose at the 32nd European Study Group with Industry held in September 1998 at the Technical University of Denmark: the first ever to be held outside the United Kingdom. It was presented by Stig Helmer Jorgensen from DANFOSS, which is Denmark's largest industrial group and specialises in controls for refrigeration and heating. The Danish Study Group was a great success and is expected to be repeated annually henceforth. The feedback from DANFOSS has also been encouraging and hopefully this represents the start of a long-term collaboration

The evolution of a slender non-axisymmetric drop in an extensional flow

An asymptotic method for analysing slender non-axisymmetric drops, bubbles and jets in a general straining flow is developed. The method relies on the slenderness of the geometry to reduce the three-dimensional equations to a sequence of weakly coupled, quasi-two-dimensional Stokes flow problems for the cross-sectional evolution. Exact solution techniques for the flow outside a bubble in two-dimensional Stokes flow are generalised to solve for the transverse flow field, allowing large non-axisymmetric deformations to be described. A generalisation to the case where the interior contains a slightly viscous fluid is also presented. Our method is used to compute steady non-axisymmetric solution branches for inviscid bubbles and slightly viscous drops. We also present unsteady numerical solutions showing how the eccentricity of the cross-section adjusts to a non-axisymmetric external flow. Finally, we use our theory to investigate how the pinch-off of a jet of relatively inviscid fluid is affected by a two-dimensional straining cross-flow

The instability of a viscous sheet floating on an air cushion

The dynamics of a thin sheet of viscous liquid levitating on an air cushion is studied. Experimentally, it is observed that, after an initial settling stage, a local disturbance grows, eventually leading to the sheet blowing up like a viscous balloon. We derive a dynamical model for the levitating sheet and propose a mechanism for the onset of the instability. This instability is driven by the local drainage of the sheet due to a growing disturbance on its lower surface and is moderated by surface tension, the bending stiffness of the sheet and advection in the air layer. The balance between these effects determines the most unstable wavelength and this is illustrated by some numerical simulations

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

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

Intermittency in the transition to turbulence

It is commonly known that the intermittent transition from laminar to turbulent flow in pipes occurs because, at intermediate values of a prescribed pressure drop, a purely laminar flow offers too little resistance, but a fully turbulent one offers too much. We propose a phenomenological model of the flow, which is able to explain this in a quantitative way through a hysteretic transition between laminar and turbulent states, characterized by a disturbance amplitude variable that satisfies a natural type of evolution equation. The form of this equation is motivated by physical observations and derived by an averaging procedure, and we show that it naturally predicts disturbances having the characteristics of slugs and puffs. The model predicts oscillations similar to those which occur in intermittency in pipe flow, but it also predicts that stationary biphasic states can occur in sufficiently short pipes

Mathematical modelling of non-axisymmetric capillary tube drawing

This paper concerns the manufacture of non-axisymmetric capillary tubing via the Velloprocess, in which molten glass is fed through a die and drawn off vertically. The shapeof the cross-section evolves under surface tension as it flows downstream. The aim is to achieve a given desired final shape, typically square or rectangular, and our goal is to determine the required die shape. We use the result that, provided the tube is slowly varying in the axial direction, each cross-section evolves like a two-dimensional Stokes flow when expressed in suitably scaled Lagrangian coordinates. This allows us to use a previously derived model for the surface- tension-driven evolution of a thin two-dimensional viscous tube. We thus obtain, and solve analytically, equations governing the axial velocity, thickness and circumference of the tube, as well as its shape. The model is extended to include non-isothermal effects

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