Control of Rayleigh-Bénard Convection in a Fluid Layer with Internal Heat Generation

The control of the onset of convection in a horizontal fluid layer with internal heat generation is studied. The horizontal boundaries of the system are cooled isothermally. The stability of the fluid layer is investigated on the basis of the linear stability theory and the resulting eigenvalues problem is solved numerically. Upon using a feedback proportional control, the heating power of the system is modulated in order to counteract any deviations of the temperature of the fluid from its conductive value. As a result, it is possible to postpone (or advance) significantly the onset of motion. The optimal positions of the thermal sensors can be predicted on the basis of the linear stability theory. The linear stability analysis also reveals the possible existence of Hopf’s bifurcations at the onset of motion. This type of bifurcation can be delayed using differential controllers. Two-dimensional numerical simulations of the full governing equations are carried out and found to agree well with the prediction of the linear stability theory.