Research

I am an applied mathematician with a background in fluid mechanics. Most of the work I do is in the theoretical modelling of biophysical phenomena, but I am fairly agnostic towards the exact type of problem. Anything can be interesting! The challenge (and the fun) lie in developing models that are as simple as possible, and as complicated as necessary to explain the behaviour of a real-world system. In particular, I care about relating the parameters of my models back to reality and giving experimentalists a guide to test their predictions.

Thematically, my work is divided in three areas: The mechanics of embryonic development, the dynamics of biological and artificial microswimmers, and geophysical topics such as plate tectonics. These areas are united by ideas from low-Reynolds number (i.e. inertialess) hydrodynamics, but concepts from nonlinear physics, stochastic analysis, chemistry, reaction-diffusion systems and complex fluids also show up.

I prefer using a pencil over a computer, but over time I have developed passable skills in scientific programming, including finite- and boundary element methods, and coded in Matlab, Mathematica, Python and Fortran.

Research Projects (explained in simple terms). Click to expand! 🙂

Mechanochemical self-organisation in avian embryos
Demonstrating that tissue mechanics regulate embryonic development
Embryonic development is an incredibly complex yet fascinating process. How is it possible to reliably generated a living, breathing organism from a single cell? This project centres on birds and a particular stage in development called gastrulation. During gastrulation the cells of the embryo rearrange and begin to differentiate into different types. The traditional point of view is that this stage in development is controlled through biochemical signalling and activation of genes. In contrast to this view, we show that there is in fact a mechanical mechanism that dynamically regulates this process in a manner reminiscent of a self-organised Turing system embedded within a viscous fluid. This theory explains elegantly how embryos respond to mechanical and chemical perturbations and experimental data even confirms that the tissue mechanics have a direct effect on gene expression.

Related publications:
- Self-organized tissue mechanics underlie embryonic regulation
Plate tectonics
Analysing how the curvature of Earth affects plate subduction
On the extremely slow time scales of plate tectonics, the flow of the rocky interior of the earth can be modelled as an extremely viscous fluid. In this picture, tectonic plates are essentially slabs floating on top of the mantle, which generally has a different viscosity and density but behaves otherwise the same. Using the numerical boundary element method with certain modifications it is then possible to accurately simulate the subduction dynamics of such plates in more or less idealised circumstances. Our contribution is to include the curvature of the Earth in such modelling. Ultimately it turns out that this leads to an additional resistance to bending, which makes it more difficult for small plates to subduct than for large plates.

Related publications:
- Fluid mechanics of free subduction on a sphere. Part 1. The axisymmetric case
Swimming on the microscale
Understanding how microorganisms propel and interact
Imagine trying to swim in honey. You probably wouldn’t get very far. This is because honey is sticky (or “viscous”) which prevents coasting by eliminating your inertia. Now imagine you are a microbe. It turns out that even water appears as viscous to the microbe as honey appears to you! So how do they swim?

One way is by using non-reversible motions, such as a corkscrew. This is what many bacteria do. Another is by sending a travelling wave down your body, like spermatozoa. In general, the flows that microswimmers create to propel can be categorised into “pushers” and “pullers”, which allows for a fairly general analysis of their interactions with boundaries and each other. I have done some work on numerically simulating the motion of a general microswimmer through a lattice of obstacles, which stands in for a porous medium. It turns out that depending on the type the behaviour is quite different. Pushers, such as bacteria and spermatozoa tend to get stuck or move in random directions, whereas pullers, such as some algae, are mostly unperturbed and go straight. Some other work I have done concerns the propulsion of bacteria in particular, and the way their filaments bundle together during forward motion.

Related publications:
- Active particles in periodic lattices
- Direct versus indirect hydrodynamic interactions during bundle formation of bacterial flagella
Bacterial flagella
Fluid-structure interaction between filaments and fluids
Many bacteria use flagella in order to self-propel. In general, flagella consist of three parts: a rotary ion-driven motor at the base, a short flexible hook, and a slender helical filament. These filaments are passive, they consist of a single protein called flagellin that stacks up to create a self-similar helix in one of eleven different ways (this is called polymorphism). Since the structure is biologically passive except for the action of the motor, the propulsion mechanism becomes an almost purely physical problem in elastohydrodynamics, which has been investigated very widely. I made a small contribution by abstracting the interaction of two flagella to an absolute minimal model of hydrodynamic singularities. This could explain the importance of their direct hydrodynamic interactions compared to global flows around the bacterium in the process of bundle formation. It turns out it’s mostly the latter. This is important, because those kind of flows are absent in many simplified models (e.g. of flagella parallel to a wall) and hence overlooked in theoretical analysis despite being dominant by several orders of magnitude!

Related publications:
- Direct versus indirect hydrodynamic interactions during bundle formation of bacterial flagella
Dynamics of colloids
Controlling the behaviour of artificial active particles
Artificial microswimmers are a popular research topic, because they have potential in noninvasive medicine (such as targeted drug delivery) and industrial applications. These particles usually have a size of a couple of micrometers and the problem is how to control them. Because they are so small, they are subject to the laws of viscous flow, which among other things forces all flows to be reversible. This means, broadly speaking, that a certain action such as cargo delivery can be undone by reversing all the inputs of the system. I made two contributions to the field so far. In one case, an irreversible steric interaction with a wall to irreversibly pick up a cargo particle by a magnetically driven microswimmer and trap it inside a vortex. In the second, I addressed the practicability of common colloid designs which are not degradable and calculated how a degradable colloid would behave. The answer is, broadly speaking, that it can be very difficult to control, because a decrease in size means an increased importance of thermal effects.

Related publications:
- Stochastic dynamics of dissolving active particles
- Irreversible hydrodynamic trapping by surface rollers
Singularities in Stokes flow
Deriving Green's functions in non-trivial geometries
In general, the dynamics of an incompressible fluid are described by the Navier-Stokes equations, a set of non-linear partial differential equations. In general they are very hard to solve, even numerically, and progress can only be made by making simplifications informed by the specifics of a problem — much of the beauty in fluid dynamics lies in this process. My niche is the limit of inertia-less flow (low Reynolds number), which typically applies when the length and velocity scales are very small (e.g. in microfluidics) or the viscosities are very high (e.g. in the mantle of the Earth). In this limit, called Stokes flow, the equations become linear and a whole arsenal of mathematical and physical tools becomes available.

One such tool is the idea of a Green’s function, which corresponds to a point force. Analytical expressions for these in various geometries are extremely useful both for theoretical modelling (because they lead to simple expressions for the interactions between objects with sufficient separation), and numerical computations (because they reduce the problem of solving a differential equation to calculating an integral).

My contribution to this area has been the derivation and calculation of such Green’s functions in various geometries and with different boundary conditions for the flow.

Related publications:
- Stokes flow due to point torques and sources in a spherical geometry
- Fluid mechanics of free subduction on a sphere, 1: The axisymmetric case