Maxwell–Cattaneo double-diffusive convection: the general case

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Abstract
Double-diffusive convection, in which the density of a fluid is dependent on two fields that diffuse at different rates (such as temperature and salinity), has been widely studied in areas as diverse as the oceans and stellar atmospheres. Under the assumption of classical Fickian diffusion for both heat and salt, the evolution of temperature and salinity is governed by parabolic advection–diffusion equations. In reality, there are small additional terms in these equations that render them hyperbolic (the Maxwell–Cattaneo (M–C) effect). Although these corrections are nominally small, they represent a singular perturbation, and hence can lead to significant effects when the underlying differences of salinity and temperature are large. In an earlier paper (Hughes, Proctor & Eltayeb, J. Fluid Mech. , vol. 927, 2021, p. A13), we investigated the linear stability of a double-diffusive fluid layer subject to the M–C effect in either the temperature or the salinity equation (but not both). Here we consider the general, and much more complicated, case in which the M–C effect influences both temperature and salinity. We find that, as in the earlier paper, oscillatory instability is indeed facilitated (and in fact made possible when the salinity gradient is destabilising, where the classical problem has no oscillatory instability) when the salinity gradients are sufficiently large. The scalings that emerge from the earlier paper, however, are not necessarily representative of those in the general case, thus justifying the present study. In addition, we have found a remarkable singular situation when the ratio of the M–C effects is equal to the ratio between the heat and salinity diffusivities, near which the critical wavenumber is sharply reduced. In addition to determining the stability boundaries we have also investigated the growth rates of unstable modes and shown that these are on a par with those of classical double-diffusive convection.

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