Onset of Soret-induced convection in a horizontal layer of ternary fluid with fixed vertical heat flux at the boundaries.

The paper deals with the investigation of the onset and weakly nonlinear regimes of the Soret-driven convection of ternary liquid mixture in a horizontal layer with rigid impermeable boundaries subjected to the prescribed constant vertical heat flux. It is found that there are monotonous and oscillatory longwave instability modes. The boundary of the monotonous longwave instability in the parameter plane Rayleigh number Ra - net separation ratio [Formula: see text] at fixed separation ratio of one of solutes consists of two branches of hyperbolic type. One of the branches is located at [Formula: see text], the other one at [Formula: see text]. The oscillatory longwave instability exists at [Formula: see text] only for the heating from below and at [Formula: see text] there exist two oscillatory longwave instability modes: one at [Formula: see text] and the other at [Formula: see text]. Corrections to the Rayleigh number obtained in the higher order of the expansion show that the longwave perturbations can be most dangerous at any values of [Formula: see text]. The numerical solution of the linear stability problem for small perturbations with finite wave numbers confirms this conclusion. The weakly nonlinear analysis shows that all steady solutions are unstable to the modes of larger wavelength and stable to the modes of smaller wavelength, i.e. the solution with maximal possible wavelength is realized.

Stability of the fluid interface in a Hele-Shaw cell subjected to horizontal vibrations.

The stability of the horizontal interface of two immiscible viscous fluids in a Hele-Shaw cell subject to gravity and horizontal vibrations is studied. The problem is reduced to the generalized Hill equation, which is solved analytically by the multiple scale method and numerically. The long-wave instability, the resonance (parametric resonance) excitation of waves at finite frequencies of vibrations (for the first three resonances), and the limit of high-frequency vibrations are studied analytically under the assumption of small amplitudes of vibrations and small viscosity. For finite amplitudes of vibrations, finite wave numbers, and finite viscosity, the study is performed numerically. The influence of the specific natural control parameters and physical parameters of the system on its instability threshold is discussed. The results provide extension to other results [J. Bouchgl, S. Aniss, and M. Souhar, Phys. Rev. E 88, 023027 (2013)10.1103/PhysRevE.88.023027], where the authors considered a similar problem but took into account viscosity in the basic state and did not consider it in the equations for perturbations.