Movies

Contact-Line Dynamics

A simulation of three-phase Cahn-Hilliard dynamics. Correctly defining an ‘‘interface’’ in a smooth PDE is slightly nontrivial. The overall goal is to understand the dynamics of these interfaces in the limit that the smoothness tends to zero. For this we need to combine singular perturbation theory for PDEs with the elegant differential geometry of the signed distance coordinates.

A simulation of three-phase Cahn-Hilliard dynamics in three dimensions. The yellow, magenta and cyan surfaces divide two phases, and meet at a three-phase contact line. The extra spatial dimenionsion leads to all sort of extra interesting phenomena!

Liquid-Liquid Phase Separation

An experiment on temperature-induced liquid-liquid phase-separation in a polymer solution. The equilibrium crescent shape of these compound droplets enable cheap manufacture of nanovials for precise single-cell analysis.

We simulate the phase-separation process in two dimensions, comparing models with (bottom) and without (top) fluid dynamics. We find that accounting for fluid stresses (second row), as well as buoyant forces (third and fourth rows) is critical to accurately reproducing the final shape of phase-separating droplets.

Three-dimensional simulations of phase separating fluids confirms the importance of fluid dynamics to liquid-liquid phase separation.

Signed Distance Coordinates

Much of my research involves analysing singular perturbations of PDEs arising in fluid mechanics. These singular perturbations lead to boundary layers and shocks. We can understand their behaviour through multiple scales matched asymptotics. But to do so in arbitrary geometries we need useful coordinate systems that maximally simplify all the differential geometric terms that need to be considered. We achieve this using an orthogonal coordinate system for neighbourhoods of boundary layers induced by the signed distance function (coloured level sets).

Signed distance coordinates are of course curvilinear, so quite a few extra terms crop up in the Navier-Stokes equations! But fortunately they are orthogonal, giving a maximally simplified orthogonal conforming coordinate system for arbitrary smooth surfaces.

And if there are singularities in our surfaces, we can again derive orthogonal signed distance coordinates for the vicinities of singular curves. Interestingly, these are not equivalent to an extension of the standard Frenet-frame. We actually need to twist our coordinate system if our curve isn’t torsion free.

Iceberg Melting

A big application of my research is developing methods to simulate multiphase problems arising in ice-ocean interactions in geophysical fluid dynamics. This is a time-lapse movie of an experiment of ice melting in warm salty water. The interesting fluid advection patterns generated by the geometry in turn lead to fascinating and nonuniform ice melting.

Changing the geometry of the ice alters the fluid flow, which feeds back on the geometry through altered melting patterns.

Phase-Field Models

To simulate these experiments I developed a higher-order accurate phase-field model coupling fluid flow to melting and dissolution of ice in warm salt water. These simulations reveal the patterns missed in our experiments, showing how vortex generation by strong shear flows can generate localised increases in melt rates — with sizeable predictions for altered iceberg melt rates!

Volume Penalty Method

A visualisation of simple fluid flow past an object implemented using the volume penalty method. I derived a straightforward correction that achieves second order numerical accuracy at no extra cost.

Dead Water

Using the volume penalty method we can simulate all sorts of interesting fluid-solid interactions. The ‘‘dead water effect’’, involving increased boat drag due to internal wave generation in stratified seas, is one fun example.