On the stability of transient penetrative convection driven by internal heating coupled with a thermal boundary condition

The transient stability of penetrative convection induced by internal heating coupled with a Robin (or third kind) boundary condition is investigated by adopting different transient approaches. A Robin boundary condition introduces a heat exchange coefficient as an additional parameter. In the system, a destabilising boundary flux is determined by the temperature increase of the fluid adjacent to the boundary, which is in turn controlled by the internal heat source, thus the boundary flux increases gradually with time. The study shows that the stability properties of the flow are independent of the Prandtl number and the transient approach being used. Further, the results of the present study demonstrate the effect of varying the heat exchange coefficient on the critical conditions and the formation of primary convection rolls. Interestingly, the horizontal and vertical length scales of the convection rolls are found to be insensitive to the value of the heat exchange coefficient, whilst the onset time of the penetrative convection is strongly dependent on this parameter.