Unstable density stratification of miscible fluids in a vertical Hele-Shaw cell: influence of variable viscosity on the linear stability

Abstract

The influence of viscosity variations on the density-driven instability of two miscible fluids in a vertical Hele-Shaw cell is investigated by means of a linear stability analysis. Dispersion relations are presented for different Rayleigh numbers, viscosity ratios and interfacial thickness parameters of the base concentration profile. The analysis employs the three-dimensional Stokes equations, and the results are compared with those obtained from the variable density and viscosity Hele-Shaw equations. While the growth rate does not depend on which of the two fluids is the more viscous, the maxima of the eigenfunctions are always seen to shift towards the less viscous fluid. For every parameter combination, the dominant instability mode is found to be three-dimensional. With increasing viscosity ratio, the instability is uniformly damped. For a fixed viscosity ratio, both the growth rate and the most unstable wavenumber increase monotonically with the Rayleigh number, until they asymptotically reach a plateau.

Surprising findings are obtained regarding the effects of varying the interface thickness. At higher viscosity ratios the largest growth rates and unstable wavenumbers are observed for intermediate thicknesses. This demonstrates that for variable viscosities thicker interfaces can be more unstable than their thinner counterparts, in contrast to the constant viscosity case. The reason behind this behaviour can be traced to the influence of the gap width on the vertical extent of the perturbation eigenfunctions. For thick interfaces, the eigenfunction can reside almost entirely within the interfacial region. In that way, the perturbation maximum is free to shift towards the less viscous fluid, i.e. into a locally more unstable environment. In contrast, for thin interfaces, the eigenfunction is forced to extend far into the viscous fluid, which leads to an overall stabilization. While the Hele-Shaw analysis also captures this ‘optimal’ growth for intermediate interface thicknesses, the growth rates differ substantially from those obtained from the full Stokes equations. Compared to the Hele-Shaw results, growth rates obtained from the modified Brinkman equation are seen to yield better quantitative agreement with the Stokes results.