Frontal polymerization in thin layers: Hydrodynamic effects and asymptotic dynamics.

Buoyancy-driven convection currents arise from temperature gradients in thermal frontal polymerization (FP) when the spatially localized polymerization reaction travels perpendicularly to the gravity field. We propose a theoretical study of the system dynamics under adiabatic conditions. The polymer and the reactant mixture are considered to be in the same liquid phase, but the viscosity can increase with the degree of polymerization. We find that the reaction zone propagates as a hot spot-like pattern with a broken symmetry in both the vertical and horizontal directions. Furthermore, the system can reach an asymptotic dynamics characterized by a front with a steady shape that propagates at constant speed with a steady vortex surrounding it. As the strength of the vortex is increased, either by decreasing the reactants' viscosity or by increasing the layer's thickness, we observe a transition between (i) a passive regime predicted by pure reaction-diffusion and hydrodynamic models and (ii) an active chemo-hydrodynamic regime where such models separately break down. In the active regime (ii), the front speed decreases as convection intensifies. By means of a scaling analysis, we explain how hydrodynamic currents might lower the velocity of a polymerization wave. As the viscosity of the polymer is enlarged, the flow is shifted ahead of the reaction zone and becomes more symmetrical with respect to the middle of the system, as recently observed in solid-liquid FP experiments [Y. Gao et al., Phys. Rev. Lett. 130, 028101 (2023) and Y. Gao et al., Int. J. Heat Mass Transf. 240, 126622 (2025)].