Phase Field Models

Melting ice in sea water is fiendishly tricky problem to simulate. On top of all the fluid dynamics going on, you also have to wrestle with a domain that is evolving over time, according to the temperature (and salt) gradients at the boundary! Trying to simulate these complicated geometries in a spectral code like Dedalus might seem foolhardy. But phase-field models provide a way.

Phase-field models are smooth models of phase transitions, which regularise the infinitesimal boundary conditions of Stefan problems over a finite distance $\varepsilon$. This smoothness allows you to simulate them with much more general codes. But it comes at the cost of accuracy. Actual melting and freezing phenomena really are smoothed over a finite thickness, but physical length scales are orders of magnitude smaller than could be feasible on a computer. The idealised boundary conditions of Stefan problems become an accurate model at macroscopic scales. Phase-field models seem limited to an error of order $\varepsilon$.

But there is a way forward for phase-field models! By carefully tuning the equations, using sophisticated multiple-scales matched-asymptotics, it is possible to accelerate the convergence of phase-field models to $\mathcal{O}(\varepsilon^2)$. The issue is that this analysis becomes rapidly more complicated as you add additional effects, like dissolution, or fluid dynamics.

By combining accelerated phase-field models with the improved volume-penalty method, and automating the asymptotic analysis with the elegent differential geometry of the signed-distance function in Mathematica, we can build higher-order accurate phase-field models of melting and dissolution in fluid dynamics. In the header video, I use these equations in Dedalus to simulate the melting of a block of ice in moving warm salty water (as for icebergs in the ocean). All sorts of interesting effects occur, like the generation of recirculating vortices that accelerate melt rates and lead to complicated new geometries.