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.
Herrera et al. (Fri,) studied this question.