Stability of a liquid layer in a rotating Hele-Shaw reactor under competition of buoyancy effects generated by inertial forces

Technologies that involve studying heat and mass transfer processes in a rotating Hele-Shaw cell can be effectively used in particular in designing microfluidic devices and small-scale flow-type chemical reactors. A quasi-two-dimensional model allows recording density fields by optical methods, and rotation makes it possible to control them through spatially distributed inertial forces. As is known, in the limit of an infinitely thin layer, the Coriolis force disappears in a standard mathematical model. However, the experimental observations of a fluid flow in a rotating Hele-Shaw cell indicate that the Coriolis effect fully manifests itself. The correct derivation of the equation of motion in the Hele-Shaw and Boussinesq approximation leads to the appearance of a term responsible for the buoyancy of the medium element caused by the Coriolis force. In this paper, to study the new effect, we consider the problem of convective stability of a fluid with internal generation of a transport component, which can be either the concentration of a dissolved substance or the temperature of the medium. The study of the system includes finding its basic state and linear analysis of its stability, analysis of weakly nonlinear solutions near the first bifurcation point, direct numerical modeling of nonlinear convection regimes, and evaluation of the general properties of a disturbance spectrum and the branching of solutions near the equilibrium bifurcation. The character of the branching of solutions is determined by applying the method of multiple time scales. The weakly nonlinear analysis indicates that, when the Rayleigh number reaches a critical value, the steady-state equilibrium of the fluid is replaced by oscillatory convection. The finite difference method is used to obtain a complete picture of the nature of nonlinear dynamics. It is shown that the Coriolis force has a stabilizing effect on the basic state of the system. In the strongly nonlinear regimes of convection, the Coriolis buoyancy leads to a complication of the scenario of transition to chaotic convection. It is demonstrated that the transition is accompanied by a series of bifurcations of limit cycles and tori, the final destruction of 2-D or 3-D tori, the appearance of strange attractors of the toroidal type. A stability map is constructed in the Rayleigh number-Ekman number parameter space.