The applied, numerical and asymptotic analysis of acoustic, electromagnetic and elastic scattering by smooth scatterers (e.g. a cylinder or a sphere) is a classical topic in applied mathematics. However, many real-world applications involve highly non-smooth scatterers with geometric structure on multiple length scales. Examples include acoustic scattering by trees and other vegetation in the modelling of urban noise propagation, electromagnetic scattering by snowflakes and ice crystal aggregates in climate modelling and weather prediction, and elastic scattering by cracks and other interfaces in seismic imaging and hydrocarbon exploration. In such situations it may be more appropriate to model the scatterer not by a smooth surface but by a fractal, a geometric object with self-similarity properties and detail on every length scale. Well-known examples include the Cantor set, Sierpinski triangle and the Koch snowflake. In this talk I will give an overview of our recent research into acoustic scattering by such fractal structures. So far our work has focussed on establishing well-posedness of the scattering problem and integral equation reformulations of it, and developing and analysing numerical methods for obtaining approximate solutions. However, there remain interesting open questions about the high frequency (short wavelength) asymptotic behaviour of solutions, and whether the self-similarity of the scatterer can be exploited to derive more efficient approximation techniques.

Many microorganisms must navigate strange biological environments whose physics are unique and counter-intuitive, with wide-ranging consequences for evolutionary biology and human health. Mucus, for instance, behaves like both a fluid and an elastic solid. This can affect locomotion dramatically, which can be highly beneficial (e.g. for mammalian spermatozoa swimming through cervical fluid) or extremely problematic (e.g. the Lyme disease spirochete B. burgdorferi swimming through the extracellular matrix of human skin). Mathematical modeling and numerical simulations continue to provide new fundamental insights about the biological world in and around us and point toward new possibilities in biomedical engineering. These complex fluid phenomena can either enhance or retard a microorganism's swimming speed, and can even change the direction of swimming, depending on the body geometry and the properties of the fluid. We will discuss analytical and numerical insights into swimming through model viscoelastic (Oldroyd-B) and liquid-crystalline (Ericksen-Leslie) fluids, with a special focus on the important and in some cases dominant roles played by the presence of nearby boundaries.

Peel Ports operate a number of locks that allow ships to enter and leave the port. The lock gates comprise a single caisson structure which blocks the waterway when closed and retracts into the dockside as the gate opens. Build up of silt ahead of the opening lock gate can prevent it from fully opening or requiring excessive power to move. If the lock is not able to fully open, ships are unable to enter the port, leading to significant operational impacts for the whole port. Peel ports are interested in understanding, and mitigating, this silt build up. 

In this talk I will discuss our recent work on two problems. The first problem concerns with capillary rise between rough structures, a fundamental wetting phenomenon that is functionalised in biological organisms and prevalent in geological or man-made materials. Predicting the liquid rise height is more complex than currently considered in the literature because it is necessary to couple two wetting phenomena: capillary rise and hemiwicking. Experiments, simulations and analytic theory demonstrate how this coupling challenges our conventional understanding and intuitions of wetting and roughness. For example, the critical contact angle for hemiwicking becomes separation-dependent so that hemiwicking can vanish for even highly wetting liquids. The rise heights for perfectly wetting liquids can also be different in smooth and rough systems. The second problem concerns with droplets (or condensates) formed via a liquid-liquid phase separation process in biological cells. Despite the widespread importance of surface tension for the interactions between these droplets and other cellular components, there is currently no reliable technique for their measurement in live cells. To address this, we develop a high-throughput flicker spectroscopy technique. Applying it to a class of cellular droplets known as stress granules, we find their interface fluctuations cannot be described by surface tension alone. It is necessary to consider elastic bending deformation and a non-spherical base shape, suggesting that stress granules are viscoelastic droplets with a structured interface, rather than simple Newtonian liquids. Moreover, given the broad distributions of surface tension and bending rigidity observed, different types of stress granules can only be differentiated via large-scale surveys, which was not possible previously and our technique now enables.

PDE Workshop in Stability Analysis for Nonlinear PDEs will be running Monday 15th - Friday 19th August.

Location: L3, AWB

Our goal was to bring together leading experts in the stability analysis of nonlinear partial differential equations across multi-scale applications. Some of the topics to be addressed include: 

• Stability analysis of shock wave patterns of reflections/diffraction.
• Stability analysis of vortex sheets, contact discontinuities, and other characteristic discontinuities for multidimensional hyperbolic systems of conservation laws.
• Stability analysis of particle to continuum limits including the quantifying asymptotic/mean-field/large-time limits for pairwise interactions and particle limits for general interactions among multi-agent systems
• Stability analysis of asymptotic limits with emphasis on the vanishing viscosity limit of solutions from multidimensional compressible viscous to inviscid flows with large initial data.