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

Pritchard, D. (2009) The linear stability of double-diffusive miscible rectilinear displacements in a Hele-Shaw cell. European Journal of Mechanics - B/Fluids, 28 (4). pp. 564-577. ISSN 0997-7546 (https://doi.org/10.1016/j.euromechflu.2009.01.004)

[thumbnail of p08heleshaw.pdf]
Preview
PDF. Filename: p08heleshaw.pdf
Preprint

Download (266kB)| Preview

Abstract

We investigate the viscous instability of a miscible displacement process in a recti-linear geometry, when the viscosity contrast is controlled by two quantities whichdiuse at dierent rates. The analysis is applicable to displacement in a porousmedium with two dissolved species, or to displacement in a Hele-Shaw cell with twodissolved species or with one dissolved species and a thermal contrast. We carry outasymptotic analyses of the linear stability behaviour in two regimes: that of smallwavenumbers at intermediate times, and that of large times.An interesting feature of the large-time results is the existence of regimes in whichthe favoured wavenumber scales with t−1/4, as opposed to the t−3/8 scaling foundin other regimes including that of single-species ngering. We also show that theregion of parameter space in which the displacement is unstable grows with time,and that although overdamped growing perturbations are possible, these are neverthe fastest-growing perturbations so are unlikely to be observed. We also interpretour results physically in terms of the stabilising and destabilising mechanisms actingon an incipient nger.

ORCID iDs

Pritchard, D. ORCID logoORCID: https://orcid.org/0000-0002-9235-7052;