Chemical fronts in Hele-Shaw cells: Linear stability analysis based on the three-dimensional Stokes equations

We present linear stability results based on the three-dimensional Stokes equations for chemically reacting, propagating fronts giving rise to an unstable density stratification in a Hele-Shaw cell. The results are compared with the experiments in M. Böckmann and S. C. Müller [Phys. Rev. Lett. 85, 2506 (2000)], as well as with a corresponding linear stability analysis based on the Darcy equations that was performed in A. De Wit [Phys. Rev. Lett. 87, 054502 (2001)]. The reason for the good agreement between these earlier Darcy data and the experimentally observed growth rates is found in the relatively low experimental value of the Rayleigh number, Ra=79, for which the flow is approximately of Poiseuille type. Already for Ra values as low as 300, we observe a discrepancy between the stability results based on the Darcy and Stokes equations, respectively, with the Darcy results overpredicting both the most amplified wavenumber, as well as the corresponding growth rate, by about a factor of two. This indicates that three-dimensional effects quickly gain importance as Ra increases, so that the stability analysis needs to be based on the full, three-dimensional Stokes equations. The stability results based on the Stokes equations furthermore demonstrate the stabilizing influences of both an increasing interfacial thickness, as well as increasing frontal propagation velocities, confirming the earlier Darcy-based findings by De Wit. An argument in terms of vorticity is forwarded to explain the latter effect. A more rapidly advancing front deposits vorticity over a wider layer of fluid particles, so that the concentrated regions of vorticity needed for rapid instability growth cannot form. Somewhat surprisingly, however, slowly propagating fronts are seen to be more unstable than nonreacting fronts of equivalent thickness, as the chemical reaction leads to the formation of more compact perturbations in the interfacial region.