Exogenous-endogenous surfactant iteraction yields heterogenous spreading in complex branching networks

Experiments¹ have shown that surfactant introduced at the entrance of a maze filled with a shallow layer of milk can induce a flow in the liquid which spontaneously finds the solution path, effectively solving the maze with minimal flow into dead-end sections. Here, we test the hypothesis that this maze-solving behaviour is due to the dynamics of a second species of surfactant endogenous to the liquid. Lubrication theory can be used to derive equations capturing the Marangoni flow induced by surfactants, which reduce to a nonlinear diffusion equation² describing the spread of the exogenous surfactant in a large-gravity and low-Reynolds-number limit. Treating both species of surfactant as a single concentration field, we solve this equation simultaneously with a kinematic equation to track the exogenous front locations.

To solve the governing equation on a network representing the topology of the maze, we construct a mass-conserving mimetic-finite-difference (MFD) scheme using tools from discrete calculus and graph theory. We find that the simulation qualitatively captures the maze-solving behaviour observed in the experiment, with the result dependent on only two fitting parameters: the mass ratio of exogenous to endogenous surfactant, and the initial ratio of the surfactant concentrations. A modal decomposition of the MFD Laplacian operator shows that only three dominant modes are needed to qualitatively capture the key behaviour of maze solving, and the experimentally observed behaviour of receding in dead end sections.

¹ Temprano-Coleto et al., Phys Rev Fluids. 3, 100507 (2018)

² Jensen and Halpern, J. Fluid Mech. 372, 273 (1998)