Instability of microbial droplets growing on viscous substrates

TL;DR

It is found that growth forces stabilize the axisymmetric solution while buoyancy forces destabilize it, and this system of integro-differential equations defined solely on the microbial domain is reformulated.

Abstract

We develop and analyze 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 analyze its morphological stability. We find that growth forces stabilize the axisymmetric solution while buoyancy forces destabilize it. We connect these findings to experimental observations.

Figures (9)

• Figure 1: Schematic of the mathematical model. A growing microbial droplet $\varOmega(t)$ (green) sits on the surface $S$ of a semi-infinite fluid volume $V$ (blue). As the droplet grows, it depletes dense nutrients (red circles) in the fluid which drives buoyant flows.
• Figure 2: Axisymmetric solution for the nutrient concentration field. Nutrients are depleted near the droplet and the concentration increases in concentric ellipsoidal-like contours which become increasingly circular.
• Figure 3: (a) Axisymmetric pressure on the unit disk for growth-dominated flow (green) and buoyancy-dominated flow (purple), each scaled by their value at the origin ($p_{g,0} \approx 1.27$ and $p_{b,0} \approx -0.031$) for visualization. The former is strictly positive while the latter is strictly negative. (b) Radial velocity for growth-dominated flow (green) and buoyancy-dominated flow (purple) exterior to the droplet. The former is strictly positive and monotonically decreasing while the latter exhibits a local maximum near $r \approx 1.1$.
• Figure 4: Cross-sectional streamlines and velocity magnitude in the (a) growth-dominated regime ($Ra = 0$) and (b) buoyancy-dominated regime ($Ra\rightarrow\infty$). In the former, fluid is pulled from below and pushed outward, reminiscent of stagnation point flow. In the latter, a vortex ring forms below the droplet, with the core positioned near the droplet edge.
• Figure 5: Stability coefficient $\sigma_g^m$ for growth-dominated flow ($Ra = 0$). For $m = 0$, which corresponds to a dilation, the coefficient is $\sigma_g^0 = 1/2$, consistent with the $O(1)$ solution $R(t) = R_0e^{t/2}$. For $m = 1$ we have $\sigma_g^1 = 0$, which is reflective of translational invariance. All other coefficients are negative, indicating that growth is stabilizing, and tend towards negative infinity as $O(m^{1/2})$.