Mass transfer in multicomponent systems occurs through convection and diffusion. For a viscous Newtonian flow, convection may be modelled using the Navier–Stokes equations, whereas the diffusion of multiple species within a common phase may be described by the generalised Onsager–Stefan–Maxwell equations. In this talk we present a novel finite element formulation which fully couples convection and diffusion with these equations. In the regime of vanishing Reynolds number, we use the principles of linear irreversible dynamics to formulate a saddle point system which leads to a stable formulation and a convergent discretisation. The wide scope of applications for this novel numerical method is illustrated by considering transport of oxygen through the lungs, gas separation processes, mixing of water and methanol and salt transport in electrolytes.

Stably density stratified shear flows arise widely in geophysical settings. Instabilities of these flows occur on scales that are too small to be directly resolved in numerical simulations, e.g., of the oceans and atmosphere, yet drive diabatic mixing events that often exert a controlling influence on much larger-scale processes. In the limit of strong stratification, the flows are characterized by the emergence of highly anisotropic layer-like structures with much larger horizontal than vertical scales. Owing to their relative horizontal motion, these structures are susceptible to stratified shear instabilities that drive spectrally non-local energy transfers. To efficiently describe the dynamics of this ``layered anisotropic stratified turbulence'' regime, a multiple-scales asymptotic analysis of the non-rotating Boussinesq equations is performed. The resulting asymptotically-reduced equations are shown to have a generalized quasi-linear (GQL) form that captures the essential physics of strongly stratified shear turbulence. The model is used to investigate the mixing efficiency of certain exact coherent states (ECS) arising in strongly stratified Kolmogorov flow. The ECS are computed using a new methodology for numerically integrating slow--fast GQL systems that obviates the need to explicitly resolve the fast dynamics associated with the stratified shear instabilities by exploiting an emergent marginal stability constraint.

Throughout a product development project, many decisions must be made. These include whether to start, stop, continue, or re-direct a project based on the learnings of the project team. Some of these decisions are related to the risk of achieving certain product performance attributes and they are often based on experimental observations in the laboratory or in field applications of early prototypes. Sometimes, these observations provide sufficient insight but often a significant uncertainty remains. Mathematical simulation can provide deeper insight into the mechanisms, may indicate limiting parameters and transport steps, and allows exploration of novel prototypes without actually making them. This talk will illustrate how Mathematics have been used to inform project development projects and their guiding decisions at WL Gore by describing examples from three very different applications.

We introduce a classification tool for mathematical theories based on Robinson's notion of model companionship; roughly the idea is to attach to a mathematical theory $T$ those signatures $L$ such that $T$ as axiomatized in $L$ admits a model companion. We also introduce a slight strengthening of model companionship (absolute model companionship - AMC) which characterize those model companionable $L$-theories $T$ whose model companion is axiomatized by the $\Pi_2$-sentences for $L$ which are consistent with the universal theory of any $L$-model of $T$.

We use the above to analyze set theory, and we show that the above classification tools can be used to extract (surprising?) information on the continuum problem.

In the first part of this talk, we will (briefly) derive the original calculation by Hawking in 1974 to determine the radiation given off by a black hole, giving the result in the form of an integral of a classical solution to the linear wave equation.
In the second part of the talk, we will take this integral as a starting point, and rigorously calculate the radiation given off by a forming spherically symmetric, charged black hole. We will then show that for late times in its formation, the radiation given off approaches the limit predicted by Hawking, including the extremal case. We will also calculate a bound on the rate at which this limit is approached.