Jeffrey fluid saturating a heated porous layer: Dynamical and microcontroller execution probing

Abstract

In this paper, a dynamical investigation and microcontroller execution on a Jeffrey fluid saturating a porous layer with bottom heating and periodic modulation of gravity is carried out. The continuity equation, the energy conservation equation and the Boussinesq-Darcy approximation are used to describe the Jeffrey fluid saturating a porous layer with bottom heating and periodic gravity modulation. Thanks to the truncated Galerkin expansion method, the partial differential equations obtained from the modelling is reduced to four first order ordinary differential equations. This four-dimensional system without gravity modulation has three or one steady states depending on the scaled thermal Rayleigh number and the non-dimensional ratio of relaxation time to the retardation time of the fluid. From the stability analysis of the obtained equilibrium points in the four-dimensional system without gravity modulation, it is revealed that the steady states are stable or unstable depending on the non-dimensional ratio of relaxation time to retardation time of the fluid. Without gravity modulation, the four-dimensional system experiences Hopf bifurcation, steady convection, periodic convections, seven different shapes of chaotic convections, bistable period-1-convection, bistable period-3-convections and coexisting convections. With the sinusoidal periodic gravity modulation, the four-dimensional system displays Hopf bifurcation, steady convections, periodic convections, bursting convections, quasiperiodic convections and three different shapes of chaotic convections. A microcontroller execution of the four-dimensional system is used to establish the convection characteristics spotted in the four-dimensional system without and with sinusoidal periodic gravity modulation during the numerical simulations.