Boundary-Bound Reactions: Pattern Formation with and without Hydrodynamics

Abstract

We study chemical pattern formation in a fluid between two flat plates and the effect of such patterns on the formation of convective cells. This patterning is made possible by assuming the plates are chemically reactive or release reagents into the fluid, both of which we model as chemical fluxes. We view this as a specific example of boundary-bound reactions. In the absence of coupling with flow, we show that the two-reagent system with nonlinear reactions admits instabilities equivalent to diffusion-driven Turing instabilities. In the other extreme, when chemical fluxes at the two bounding plates are constant, diffusion-driven instabilities do not occur but hydrodynamic phenomena analogous to Rayleigh-Benard convection are possible. Assuming we can influence the chemical fluxes along the domain and select suitable reaction systems, this presents a means for the control of chemical and hydrodynamic instabilities and pattern formation. We study a generic class of models and find conditions for a bifurcation to pattern formation. Afterwards, we present two examples derived from the Schnakenberg reaction. Unlike the classical Rayleigh-Benard instability, which requires a sufficiently large unstable density gradient, a chemo-hydrodynamic instability based on Turing-style pattern formation emerges from a state that is uniform in density. We also find parameters that result in the formation of convective cells whether gravity acts upwards or downwards relative to the reactive plate. The wavenumber of the cells and the direction of the flow at regions of high/low concentration depend on the orientation, hence, different patterns can be elicited by simply inverting the device. More generally, our results suggest methods for controlling pattern formation and convection by tuning reaction parameters. As a consequence, we can drive and alter fluid flow in a chamber without mechanical pumps.