B. R. Duffy

TRAVELLING SOLITARY WAVE SOLUTIONS TO A COMPOUND KDV-BURGERS EQUATION

Abstract Explicit travelling solitary wave solutions to a compound KdV-Burgers equation are obtained by using an automated method. A two-dimensional generalization is discussed.

The Jacobi elliptic-function method for finding periodic-wave solutions to nonlinear evolution equations

The sn- and cn-function methods for finding nonsingular periodic-wave solutions to nonlinear evolution equations are described in a form suitable for automation, where sn and cn are the elliptic Jacobi snoidal and cnoidal functions, respectively. Some new solutions are presented.

The strong influence of substrate conductivity on droplet evaporation

We report the results of physical experiments that demonstrate the strong influence of the thermal conductivity of the substrate on the evaporation of a pinned droplet. We show that this behaviour can be captured by a mathematical model including the variation of the saturation concentration with temperature, and hence coupling the problems for the vapour concentration in the atmosphere and the temperature in the liquid and the substrate. Furthermore, we show that including two ad hoc improvements to the model, namely a Newtons law of cooling on the unwetted surface of the substrate and the buoyancy of water vapour in the atmosphere, give excellent quantitative agreement for all of the combinations of liquid and substrate considered.

TRAVELLING SOLITARY WAVE SOLUTIONS TO A SEVENTH-ORDER GENERALIZED KDV EQUATION

Abstract A new explicit travelling solitary wave solution to a seventh-order generalized KdV equation is obtained. Its two-dimensional generalization is discussed.

On the effect of the atmosphere on the evaporation of sessile droplets of water

An experimental and theoretical study of the effect of the atmosphere on the evaporation of pinned sessile droplets of water is described. The experimental work investigated the evaporation rates of sessile droplets in atmospheres of three different ambient gases (namely, helium, nitrogen, and carbon dioxide) at reduced pressure (from 40 to 1000 mbars) using four different substrates (namely, aluminum, titanium, Macor, and polytetrafluoroethylene) with a wide range of thermal conductivities. Reducing the atmospheric pressure increases the diffusion coefficient of water vapor in the atmosphere and hence increases the evaporation rate. Changing the ambient gas also alters the diffusion coefficient and hence also affects the evaporation rate. A mathematical model that takes into account the effect of the atmospheric pressure and the nature of the ambient gas on the diffusion of water vapor in the atmosphere and the thermal conductivity of the substrate is developed, and its predictions are found to be in enc...

Curing simulation of thermoset composites

Abstract This paper deals with the modelling and simulation of resin flow, heat transfer and the curing of multilayer thermoset composite laminates during processing in an autoclave. Darcys Law and Stokes’ slow-flow equations are used for the flow model and, for approximately isothermal flows, a similarity solution is developed. This permits the decoupling of the velocity and thermal fields. A two-dimensional convection–diffusion heat equation with an internal heat generation term is then solved numerically, together with the equation for the rate of cure, using a finite difference scheme on a moving grid. The simulations are performed with varying composite thicknesses, and a comparison of numerical results with known experimental data confirms the approximate validity of the model.

A numerical technique for solving unsteady non-Newtonian free surface flows

Abstract A numerical method has been developed for solving two-dimensional generalized Newtonian fluid flow with multiple free surfaces. It is an extension of the GENSMAC code which solves the time-dependent Navier-Stokes equations for the primitive variables of velocity and pressure in an arbitrary domain. Like GENSMAC, it is a finite-difference technique, based on staggered grids, using (virtual) marker particles as a means of flow visualization. The code has been employed to solve three time-dependent problems: extrudate die swell, viscous jet buckling, and injection moulding in complex cavities. Both Newtonian and non-Newtonian results are displayed.

The rate of spreading in spin coating

In this paper we reconsider the fundamental problem of the centrifugally driven spreading of a thin drop of Newtonian fluid on a uniform solid substrate rotating with constant angular speed when surface-tension and moving-contact-line effects are significant. We discuss analytical solutions to a number of problems in the case of no surface tension and in the asymptotic limit of weak surface tension, as well as numerical solutions in the case of weak but finite surface tension, and compare their predictions for the evolution of the radius of the drop (prior to the onset of instability) with the experimental results of Fraysse & Homsy (1994) and Spaid & Homsy (1997). In particular, we provide a detailed analytical description of the no-surface-tension and weak-surface-tension asymptotic solutions. We demonstrate that, while the asymptotic solutions do indeed capture many of the qualitative features of the experimental results, quantitative agreement for the evolution of the radius of the drop prior to the onset of instability is possible only when weak but finite surface-tension effects are included. Furthermore, we also show that both a fixed- and a specific variable-contact-angle condition (or ‘Tanner law’) are capable of reproducing the experimental results well.

On the gravity-driven draining of a rivulet of viscous fluid down a slowly varying substrate with variation transverse to the direction of flow

In this paper we use a lubrication approximation to investigate the locally unidirectional gravity-driven draining of a thin rivulet of Newtonian fluid down a slowly varying substrate. The work generalizes the recent study by Duffy and Moffatt [Chem. Eng. J. 60, 141 (1995)] of gravity-driven draining down a locally planar substrate to include the effects of substrate variation transverse to the direction of flow. Asymptotic and numerical results are obtained for several simple convex and concave transverse substrate profiles. In all the cases investigated these results show a number of common features. In particular, they show that a single stable slowly varying rivulet running continuously from the top to the bottom of a large horizontal circular cylinder is possible only if the transverse substrate profile is a sufficiently shallow trough. If the profile is a deeper trough then no such rivulet is possible near the bottom of the cylinder, while if the profile is a ridge then no such rivulet is possible n...

On the lifetimes of evaporating droplets with related initial and receding contact angles

A physically credible relationship based on the unbalanced Young force between the initial and receding contact angles of an evaporating droplet is proposed and used to give a complete description of the lifetime of a droplet evaporating in an idealised stick-slide mode. In particular, it is shown that the dependence of the lifetime on the initial contact angle is qualitatively different from that when the relationship between the initial and receding contact angles is not taken into account.