Vibration effects on weak turbulent natural convection in a porous layer heated from below

The objective of the current theoretical analysis is to observe the effect the vibrations have on heat convection over a wide range of parameter values. Analysing the different regimes and their corresponding dynamics leads to the evaluation of the Nusselt number that provides an answer to how the heat transfer is affected by the vibrations. Solutions for the amplitudes of stream function and temperature, as a function of time presented in this paper are evaluated via Fortran programming, by using the subroutine DVERK by Hull et al. and applying a fifth- and a sixth-order Runge–Kutta–Verner numerical method of solution. The results are presented in graphs showing the amplitudes as a function of time as well as on phase diagrams with the aim of identifying the different parameter regimes and possibly identifying the regimes that enhance the heat transfer. In order to establish the stability type of the results and the type of attracting trajectory, the results were used to evaluate their corresponding Lyapunov exponents.