Oscillatory regimes and transition to chaos in a Darcy–Brinkman model under quasi-periodic gravitational modulation

Abstract

This research paper examines the chaos control in porous media convection by imposing an external excitation on the system. The excitation is under the form of a quasi-periodic gravitational modulation with two incommensurate frequencies ${\sigma }_1$ and ${\sigma }_2$. This will be accomplished by taking into consideration a two-dimensional rectangular porous layer that is saturated with fluid, heated from below, and subjected to a quasi-periodic vertical gravitational modulation. The model consists of a nonlinear heat equation coupled with a system of equations representing motion under the Darcy–Brinkman law. Utilizing a spectral approach, the problem is simplified into a set of four ordinary differential equations. Three equilibria of the system are given, namely the motionless convection steady state and convection steady states. The local and global stability for the motionless convection steady state were performed. Additionally, the local stability of the other equilibria is fulfilled. The fourth-order Runge–Kutta method is used to solve the system numerically. Numerical simulations have shown that the quasi-periodic gravitational modulation plays an essential role on the fluid dynamics behavior. We find chaotic and oscillating convection regimes depending on the ratio of gravitational modulation frequencies. It was demonstrated that by properly adjusting the frequencies ratio $\eta ={\sigma }_2/{\sigma }_1$, transition from oscillating regime to chaos is observed and vice versa. Those transitions were checked by Poincaré section, Lyapunov exponent or phase diagram. It was concluded that controlling the dynamical behavior of the fluid in porous media may be achieved by implementing an appropriate quasi-periodic gravitational modulation.