D. S. Riley

Analytical and numerical studies of the stability of thin-film rimming flow subject to surface shear

Motivated by applications in rapidly rotating machinery, we have previously extended the lubrication model of the thin-film flow on the inside of a rotating circular cylinder to incorporate the effect of a constant shear applied to the free surface of the film and discovered a system rich in film profiles featuring shock structures. In this paper, we extend our model to include the effects of surface tension at leading order and take into account higher-order effects produced by gravity in order to resolve issues regarding existence, uniqueness and stability of such weak solutions to our lubrication model. We find, by analytical and numerical means, a set of feasible steady two-dimensional solutions that fit within a rational asymptotic framework. Having identified mathematically feasible solutions, we study their stability to infinitesimal two-dimensional disturbances. based on our findings, we conjecture which of the possible weak solutions are physically meaningful.

Melting of a sphere in hot fluid

Solid bodies immersed in hot fluids may melt. The molten material produced can then mix with, and be assimilated into, the fluid influencing its compositional and thermal states. Compositional convection of melt and thermal convection of cooled fluid around the solid determine the heat flux from the fluid to the solids surface. This, together with the thermal properties of the solid, controls the rate of melting. Experiments on melting wax spheres into water are described; these have shown how variations in the nature of melt flow round the sphere cause differing melting rates and hence the development of a distinctive melting morphology. Melting rates are calculated by a simple theoretical analysis which estimates melt layer thickness and the heat flux from the fluid. Melting rate predictions agree well with the experimental data. A geological application occurs when magma incorporates blocks of its surrounding wall rock. Relatively rapid melting rates are estimated, typically in the order of a half metre per day. Such fast rates indicate that this method of contamination may be an important influence on magmatic evolution in continental environments.

Extinction behaviour for two–dimensional inward-solidification problems

The problem of the inward solidification of a two–dimensional region of fluid is considered, it being assumed that the liquid is initially at its fusion temperature and that heat flows by conduction only. The resulting one–phase Stefan problem is reformulated using the Baiocchi transform and is examined using matched asymptotic expansions under the assumption that the Stefan number is large. Analysis on the first time–scale reveals that the liquid–solid free boundary becomes elliptic in shape at times just before complete freezing. However, as with the radially symmetric case considered previously, this analysis leads to an unphysical singularity in the final temperature distribution. A second time–scale therefore needs to be considered, and it is shown that the free boundary retains its shape until another non–uniformity is formed. Finally, a third (exponentially short) time–scale, which also describes the generic extinction behaviour for all Stefan numbers, is needed to resolve the non–uniformity. By matching between the last two time–scales we are able to determine a uniformly valid description of the temperature field and the location of the free boundary at times just before extinction. Recipes for computing the time it takes to completely freeze the body and the location at which the final freezing occurs are also derived.

TWO-DIMENSIONAL SOLIDIFICATION IN A CORNER

Analytical and numerical techniques are used to study the solidification of 1/2&pgr; and 3/2&pgr; wedges of liquid which are initially at their fusion temperature. An enthalpy method is used to obtain numerical solutions to these problems and the results are compared with asymptotic solutions for large and small Stefan numbers (the Stefan number being defined as the ratio of latent to sensible heats). The new solutions for small Stefan number are shown to provide surprisingly good approximations, especially for the 3/2&pgr; wedge. New results for heat transfer in a wedge (in the absence of a change of phase) are derived and applied in the asymptotic analysis, as are new conservation laws for the Stefan problem.

Substrate degradation in high-Rayleigh-number reactive convection

We study buoyancy-induced convection of a solute in an ideal two-dimensional fluid-saturated porous medium, where the solute undergoes a second-order reaction with a chemical substrate that is fixed in the underlying matrix. Numerical simulations at high Rayleigh number show how a flow is established in which a thin dynamic boundary layer beneath the solute source feeds slender vertical plumes beneath. We examine how the substrate is reactively degraded, at a rate enhanced by convective mixing. For the case when the substrate is abundant, we derive a reduced-order model describing the slow degradation of the substrate, which is formulated as a novel one-dimensional free-boundary problem. Numerical simulations and the reduced model reveal how, when the reaction is rapid compared to the convective time scale, the plumes propagate deep into the flow domain with reaction confined to a narrow region at their base. In contrast, slow reaction allows plumes to fill the domain before degradation of the substrate p...

Degenerate-diffusion models for the spreading of thin non-isothermal gravity currents

The gravitational spreading of a liquid with temperature-dependent viscosity is investigated. The aspect ratio of the layer of fluid is taken to be small, thus allowing significant simplifications to the equations governing the thermal and flow problems. The resulting equations are coupled through a dependency of the viscosity on temperature, three specific forms of which are considered. When the coupling is sufficiently strong, the flow is markedly different from the isothermal case and physically significant features seen in practice, such as a central plateau in the spreading profile, result.

Asymptotic solutions to the Stefan problem with a constant heat source at the moving boundary

In 1989, Huppert formulated mathematical models of the melting and/or freezing that occurs when a hot fluid begins to flow turbulently over a cold substrate. He solved the resulting Stefan problems numerically and presented certain asymptotic results. This note revisits one of his problems and adds to the mathematical analysis of its asymptotic structure. Generalizations and relationships with other results are also noted.

ASYMPTOTIC AND NUMERICAL SOLUTIONS FOR THE TWO-DIMENSIONAL SOLIDIFICATION OF A LIQUID HALF-SPACE

This paper concerns the two–dimensional inward solidification of a half–space filled with molten material at its fusion temperature. The temperature along half of the boundary surface is instantaneously lowered to a fixed temperature below that of fusion while the other half is either held at the fusion temperature or is insulated. New solutions are determined by asymptotic methods for small and large Stefan numbers; these are complemented by more general numerical solutions. The asymptotic and numerical results are compared and shown to be in good agreement. An error in a previous numerical study is also revealed and corrected.

Melt Spreading with Temperature-Dependent Viscosity

A hierarchy of mathematical models describing the non-isothermal spreading of a thin layer of melt with a temperature-dependent viscosity is described, the key parameters being the Peclet number and the rate of heat transfer through the melt’s surfaces. Extensive numerical and asymptotic results have been or will be presented elsewhere.