Horizontal gradient effects on the flow stability in a cylindrical container in a Bèrnard–Marangoni problem

This study assesses the flow stability of the Bénard–Marangoni (BM) problem, a thermoconvective scenario occurring in an annular domain. The system is heated from below, with a linearly decreasing horizontal temperature profile from the inner to the outer wall. The top surface of the domain is open to the atmosphere, while the lateral walls are adiabatic. The analysis focuses on the effects of the Bond number, which represents capillarity or buoyancy effects, and the horizontal temperature gradient on the flow stability for three different Prandtl numbers, indicative of viscosity effects in fluids ranging from typical gases to $n$-butanol. Three different models for heat transmission to the atmosphere are also considered using the Biot number. The results indicate that, for the two largest Prandtl numbers (50 and 5), multiple competing solutions emerge in localized regions of the Bond-temperature gradient plane. The boundaries between these regions include co-dimension two points, where two solutions coexist, and at least one co-dimension three point for each Prandtl and Biot number combination. These transitions show a strong dependency on both the Bond and Prandtl numbers. Additionally, as anticipated, the solution space is more complex for the smallest Prandtl number ($\mathrm{Pr}=1$), with seven competing solutions identified in the plane.