Upper and lower estimates of multicomponent convection instability threshold via auxiliary Bénard problems

Abstract

The onset of convection in a $m$-component Navier–Stokes fluid mixture, filling a horizontal layer $\texttt L$ heated from below and salted partly from above and partly from below, $\forall m \in \mathbb N$, is investigated. The difficulties of handling this nonlinear problem, grow drastically with $m$ and only under special circumstances the instability threshold can be given in algebraic simple closed form. Then the problem of finding, in simple algebraic closed forms, suitable lower and upper estimates of the instability threshold arises. In the present paper, for any type of boundary (rigid-rigid, one rigid-one free, free-free), it is shown that: 1) a linearization principle in the energy norm holds; 2) upper and lower estimates of the instability threshold can be obtained via auxiliary classical Bénard problems. The estimates obtained appear to be of interest not only for theoreticians but also for experimentalists investigating natural phenomena and/or industrial processes related to the onset of convection.