Marangoni-B\'enard convection in circular and elliptical cylinders

The spatial organization of single-fluid Marangoni-B\'enard convection in vertical cylinders with circular and elliptical horizontal cross-section is described. The convection is driven by an imposed heat flux from above and the Marangoni stresses that arise at the free but undeformed surface due to temperature-dependent surface tension. The solutions and their stability characteristics are obtained using branch-following techniques together with direct numerical simulations. The effect of the ellipticity and the symmetries of the grid are discussed.