Internship proposal MASTER 2:

'Coarse-graining of swimmer dynamics in a fluid filled with obstacles'

Supervisors: D. Peurichard, INRIA Paris-team MAMBA, LJLL- UPMC, diane.a.peurichard@inria.fr

P.Degond, Imperial College London, p.degond@imperial.ac.uk,

S.Merino, University of Sussex, s.merino-aceituno@sussex.ac.uk

Location: LJLL, Université Pierre et Marie Curie, Jussieu, Paris, France

Context:

Swarming of animals, for example schooling of fish, flocking of birds, or herding of mammals, is one of the most commonly observed phenomenon in the real world. It corresponds to the formation of largescale coherent structures that emerge from the local interactions between individuals. It is also observed at the level of cells (bacteria, sperm), where the viscosity of the surrounding fluid can not be ignored anymore. Therefore at the microscopic level, these systems become more challenging due to the complex mechanical interplay between the swimmers and the fluid. Particularly, highly non-linear interactions occur between neighboring swimmers through the perturbations that their motions create in the surrounding fluid. The goal of this internship is to perform numerical and theoretical analysis of a novel model that has been introduced in [1] for suspensions of active particles in a viscous fluid.

Mathematical model:

The model features an undulatory swimmer moving in a fluid composed of obstacles. The obstacles are modelled as randomly distributed spheres attached via Hookean springs to the substrate, while a swimmer is modelled as a pair of spheres constrained to be at a given (time dependent) distance. The obstacles positions are perturbed by interactions with the swimmer or the flow. In accordance with physical observations, we suppose that the environment microstructure is of the same length scale as the swimmer. We first derive an evolution equation for the fluid homogenized with the obstacles using classical homogenization theory. The derivation is done by assuming that the length scale of the heterogeneities is small compared to the space variable and that the dynamics of the springs are weaker than the rest of the interactions. Then, using balance of forces, we add a swimmer within this fluid. This approach has for main advantage to preserve the assumptions on the swimmer under the scaling limit. Note that in the largescale limit and under these scaling assumptions, the fluid and obstacles become a continuum medium which -we conjecture- will act as a viscoelastic fluid. As in reality the swimmer experiences the surrounding medium as a set of obstacles, the first goal of this internship will be to perform numerical simulations of the microscopic model and of the derived continuum equations to numerically validate that these governing equations match the limiting behavior of the particle model. Then, according to the student background and motivation, the following routes can be considered:

• • Theoretical analysis of the derived system of PDE and stability analysis to link the swimming behavior with obstacle density and spring stiffness

  • Numerical simulations of the full system and comparison with existing models for viscoelastic fluids

  • Improvement the model’s physicality acting on the model parameters, by considering for instance non-linear springs instead of Hooke’s law for the obstacles, explore different types of obstacles etc..

[1] P. Degond, E. Keaveny, S. Merino-Aceituno, D. Peurichard, Coarse-graining of swimmer dynamics in a fluid filled with obstacles, in preparation

[2] P. Degond, S. Merino-Aceituno, F. Vergnet, H. Yu, Coupled Self-Organized Hydrodynamics and Stokes models for suspensions of active particles, https://arxiv.org/pdf/1706.05666.pdf