Instability of microbial droplets growing on viscous substrates

We develop and analyse a model for a flat microbial droplet growing on the surface of a three-dimensional viscous fluid. The model describes growth-induced stresses at the fluid surface, density variations in the bulk due to nutrient consumption and the resulting fluid flows that arise. We reformulate this free-boundary problem as a system of integro-differential equations defined solely on the microbial domain. From this formulation, we identify an axisymmetric solution corresponding to a radially expanding disk and analyse its morphological stability. We find that growth forces stabilise the axisymmetric solution, while buoyancy forces destabilise it. We connect these findings to experimental observations.