Study of primary and secondary instabilities arising due to a chemical reaction in a two-component Rayleigh–Bénard system

Abstract

Manifestation of stationary and oscillatory convection and secondary instabilities due to a chemical reaction in a two-component convective fluid system is reported in the paper by considering idealistic as well as physically realistic boundaries. Using a normal mode solution, analytical expression of the critical Rayleigh number for a stationary and oscillatory disturbances, and the natural frequency are reported. The range of parameters is identified where oscillatory motion happens. Further, the parameters’ range for existence of oscillatory regime is found to be larger for rigid boundaries compared to that of free boundaries. Furthermore, for both the boundaries, parameters’ range for this regime increases when the chemical reaction rate increases, leading to the conclusion that the oscillatory motion emerges as the most preferred mode in the two-component system due to the presence of a chemical reaction and the size of this domain is directly proportional to the chemical reaction rate. The marginal stability plots depict that the oscillatory and stationary regimes respectively correspond to Hopf and direct pitchfork bifurcations. The critical Rayleigh number and the wave number where codimension two bifurcation exists are documented in the paper for fixed values of parameters. It is shown that the codimension two bifurcation that arose in the problem is not a Takens–Bogdanov bifurcation. In a stationary regime, the domain for secondary instabilities of Eckhaus and zigzag is obtained using the spatio-temporal Newell–Whitehead–Segel equation. These instabilities grow with increasing chemical reaction rate. In the oscillatory regime, the complex Ginzburg–Landau equation is used to predict the appearance of the Benjamin–Feir instability.