Collective Behaviour in 2D Microswimmer Suspensions

Abstract

This thesis investigates the fluid dynamics and phase behaviour of active microswimmers in unbounded two-dimensional suspensions. The first part derives analytical expressions for fluid correlations, fluid velocity variance, and energy spectra of non-interacting swimmers. While expressions derived from the Stokes equations, hence ignoring inertial effects, capture qualitative trends, they show quantitative discrepancies. This work demonstrates that these deviations mostly originate from the neglect of inertia, which in 2 dimensions is required to regularise the solutions at long distances. By instead employing the Oseen equations, which incorporate the leading-order effect of fluid inertia, these discrepancies are largely resolved. This indicates that, even at low Reynolds numbers, inertia is an essential regularisation factor in 2D systems, as it is required to govern the far-field flow behaviour.
The second part examines interacting swimmers via lattice Boltzmann (LB) simulations, which undergo a transition from disordered swimming to a quasi-stationary steady state characterised by system-spanning fluid correlations and mesoscopic, nematically ordered swimmer patches. A theoretical expression for the critical density is derived, specifically accounting for finite-size effects. Simulations reveal that the transition occurs only above the critical density and for a sufficiently large persistence length $l_p$. Further investigation suggests a complex interplay between density, system size, and persistence length, and show that the observed transition is broadly reminiscent of a continuous, second order phase transition. While a similar transition occurs in 3D LB and 2D continuum models, the 2D LB transition investigated here is uniquely distinguished by a different transition behaviour for high and low lp values.

Popular Abstract

From the bacteria in our gut to the algae in the ocean and the sperm cells that create life, we are surrounded by tiny organisms that move on their own, known as active matter. Understanding how they stir the fluids around them, and how that moving fluid, in turn, moves the organisms, is crucial for fields ranging from medicine to the design of tiny, self-moving robots.
On the microscopic scale, where these organisms live, water doesn't feel like a splashy liquid; it feels more like thick honey. This changes the rules of motion completely. Because these swimmers are so small, their own momentum is almost instantly cancelled out by the 'stickiness' (viscosity) of the fluid. This means as soon as a bacterium stops making swimming motions, it stops moving. In this microscopic world, inertia can be neglected. This is a sharp contrast to the macroscopic world of humans and fish, where one can stop swimming and continue to glide through the water for some time. This absence of gliding creates a unique physics where the formulas we use for fish or human swimmers simply do not apply.
This thesis combines theoretical work, where we develop formulas to describe the behaviour of those organisms, with simulations. In the simulations, we model individual swimmers, their speed, how they stir the fluid, and how they are moved by the flow of the fluid caused by other swimmers, to verify our analytical formulas. Rather than looking at behaviour in three dimensions (3D), we investigate the behaviour in two dimensions (2D), such as in thin films or on surfaces. We found that this change in dimensions significantly alters how these systems behave.
In the first part, we discovered that while describing microswimmers with standard formulas that neglect inertia works to an extent, we obtain much better results, that are closer to what we observe in simulations, when we include inertia. This reveals that inertia plays a much more important role in 2D than previously assumed.
In the second part, we examined through simulations, what happens when swimmers gather in large groups. We found that once they reach a certain density, they stop moving individually and instead begin to move collectively, like we already know from a school of fish or a flock of birds. The collective motion of swimmers in 2D leads to the fluid moving in system-spanning patterns, while the swimmers themselves form smaller, aligned patches. The collective behaviour depends on the number of swimmers per area, and how far a swimmer can move before changing direction (its persistence length). Again, we noticed a subtle difference between the dimensions. In both 2D and 3D, the transition becomes sharper as the persistent length increases. However, different from 3D, in 2D the transition looks different for small and large persistence lengths.
Ultimately, this work shows that the dimensions of a system have an influence on its behaviour. Even when we see similar phenomena in 2D and 3D, the underlying details and rules that govern them are fundamentally different.

• author: Hoffmann, Judith
• supervisor: Joakim Stenhammar
• organization: Department of Chemistry; Computational Chemistry
• course: KEMR20 20261
• year: 2026
• type: H2 - Master's Degree (Two Years)
• subject: Chemistry
• keywords: active matter, microswimmer, simulations, physical chemistry
• language: English
• id: 9238097
• date added to LUP: 2026-06-17 09:59:05
• date last changed: 2026-06-17 09:59:05