Heat and mass transfer by convection in multicomponent Navier-Stokes mixtures: absence of subcritical instabilities and global nonlinear stability via the Auxiliary System Method (Q484644)

The author has investigated the problem of mass had heat transfer by convection in horizontal layers filled by Navier-Stokes fluid mixtures with arbitrary number of different chemicals (salts) dissolved in. This problem is investigated in the general case of Navier-Stokes multicomponent fluid mixture heated from below and salted, partly from below and partly from above by \(S_{1}\), \(S_{2}\), \(\ldots\), \(S_{m}\) salts. It is shown that: \(1^{\circ}\) for each Fourier component of the perturbations to the thermal conduction solution there exists an own nonlinear system of equations as auxiliary system; \(2^{\circ}\) subcritical instabilities do not exist and the global nonlinear asymptotic \(L^{2}\)-stability is guaranteed by the condition of linear stability; \(3^{\circ}\) by the Routh-Hurwitz conditions applied to the spectral equation governing the eigenvalues of the linear associated problem rigorous stability conditions are obtained for any number of salts; \(4^{\circ}\) the Auxiliary System Method introduced by the author in [Phys. Fluids 24, No. 10, Paper No. 104101, 17 p. (2012; Zbl 1315.76018)] for the Darcy-Boussinesq fluid mixtures is extended for the Navier-Stokes fluid mixtures; \(5^{\circ}\) the looking hidden symmetries and skew symmetries allows to guarantee the global nonlinear stability by an obtained simple algebraic condition in a closed form.