On the Convective Stability and Pattern Formation of Volumetrically Heated Flows with Asymmetric Boundaries

Abstract

Non-linear solutions and their stability are presented for homogeneously heated fluids bounded by rigid conducting and insulating plates. In particular, we sought roll-type solutions emerging from the neutral stability curve for fluids with Prandtl numbers of 0.025, 0.25, 0.705, and 7. We determined the stability boundaries for the roll states in order to identify possible bifurcation points for the secondary flow in the form of regions that are equivalent to the Busse balloon. We also compared the stability exchange between ‘‘up’’ and ‘‘down’’ hexagons for a Prandtl number of $$0.25$$ obtained from weakly non-linear analysis in relation to the fully non-linear analysis, consistent with earlier studies. Our numerical analysis showed that there are potential bistable regions for both hexagons and rolls, a result that requires further investigations with a fully non-linear analysis.