Junior Applied Mathematics Seminar

Cellular adhesion is a fundamental mechanism underlying diverse collective cell behaviours, from tissue self-organisation in developmental biology to the formation of directional queues that guide cell migration. Modelling such interactions has also proven mathematically rich, motivating the use of continuum partial differential equation models that capture adhesion through nonlocal interaction kernels. These models can, for instance, reproduce classical cell-sorting patterns arising from differential adhesion in mixtures of cell populations. In this talk, we briefly review such models and explain how a local approximation of nonlocal aggregation–diffusion equations can be derived in the limit of short-range interactions. We then discuss recent advances in the field and highlight new results on pattern formation driven by adhesive interactions in migrating and proliferating cell populations, as well as in systems of nonreciprocally interacting cells.

Surfactants are chemicals that preferentially reside at interfaces. Once surfactant molecules have adsorbed to an interface, they reduce the surface tension between the two neighbouring fluids and may induce fluid flow. Surfactants have many household applications, such as in cleaning products and cosmetics, as well as industrial applications, like mineral processing and agriculture. Thus, understanding the dynamics of surfactant solutions is particularly important with regards to improving the efficacy of their applications as well as highlighting how they work. In this seminar, we will explore the spreading of a droplet over a substrate, in which there is constant injection of liquid and soluble surfactant through a slot in the substrate. Firstly, we will see how the inclusion of surfactant alters the spreading of the droplet. We will then investigate the early- and late-time behaviour of our model and compare this with numerical simulations. We shall conclude by briefly examining the effect of changing the geometry of the inflow slot.

WCMB, University of Oxford and University of Bristol

University of Cambridge