The linear stability of double-diffusive miscible rectilinear displacements in a Hele-Shaw cell

We investigate the viscous instability of a miscible displacement process in a rectilinear geometry, when the viscosity contrast is controlled by two quantities which diffuse at different rates. The analysis is applicable to displacement in a porous medium with two dissolved species, or to displacement in a Hele-Shaw cell with two dissolved species or with one dissolved species and a thermal contrast. We carry out asymptotic analyses of the linear stability behaviour in two regimes: that of small wavenumbers at intermediate times, and that of large times.An interesting feature of the large-time results is the existence of regimes in which the favoured wavenumber scales with t−1/4t−1/4, as opposed to the t−3/8t−3/8 scaling found in other regimes including that of single-species fingering. We also show that the region of parameter space in which the displacement is unstable grows with time, and that although overdamped growing perturbations are possible, these are never the fastest-growing perturbations so are unlikely to be observed. We also interpret our results physically in terms of the stabilising and destabilising mechanisms acting on an incipient finger.