Codimension-three bifurcations in a Bénard-Marangoni problem

[EN] This Brief Report studies the linear stability of a thermoconvective problem in an annular domain for relatively low (∼1) Prandtl (viscosity effects) and Biot (heat transfer) numbers. The four possible patterns for the instabilities, namely, hydrothermal waves of first and second class, longitudinal rolls, and corotating rolls, are present in a small region of the Biot-Prandtl plane. This region can be split in four zones, depending on the sort of instability found. The boundary of these four zones is composed of codimension-two points. Authors have also found two codimension-three points, where some of the former curves intersect. Results shown in this Brief Report clarify some reported experiments, predict new instabilities, and, by giving a deeper insight into how physical parameters affect bifurcations, open a gateway to control those instabilities. Hoyas, S.; Gil, A.; Fajardo, P.; Pérez Quiles, MJ. (2013). Codimension-three bifurcations in a Bénard-Marangoni problem. Physical Review E. 88(015001). doi:10.1103/PhysRevE.88.015001 Smith, M. K., & Davis, S. H. (1983). Instabilities of dynamic thermocapillary liquid layers. Part 1. Convective instabilities. Journal of Fluid Mechanics, 132, 119-144. doi:10.1017/s0022112083001512 Herrero, H., & Mancho, A. M. (1998). Influence of aspect ratio in convection due to nonuniform heating. Physical Review E, 57(6), 7336-7339. doi:10.1103/physreve.57.7336 RILEY, R. J., & NEITZEL, G. P. (1998). Instability of thermocapillary–buoyancy convection in shallow layers. Part 1. Characterization of steady and oscillatory instabilities. Journal of Fluid Mechanics, 359, 143-164. doi:10.1017/s0022112097008343 Hoyas, S., Herrero, H., & Mancho, A. M. (2002). Thermal convection in a cylindrical annulus heated laterally. Journal of Physics A: Mathematical and General, 35(18), 4067-4083. doi:10.1088/0305-4470/35/18/306 Hoyas, S., Herrero, H., & Mancho, A. M. (2002). Bifurcation diversity of dynamic thermocapillary liquid layers. Physical Review E, 66(5). doi:10.1103/physreve.66.057301 Garnier, N., & Chiffaudel, A. (2001). Two dimensional hydrothermal waves in an extended cylindrical vessel. The European Physical Journal B, 19(1), 87-95. doi:10.1007/s100510170352 Hoyas, S., Mancho, A. M., Herrero, H., Garnier, N., & Chiffaudel, A. (2005). Bénard–Marangoni convection in a differentially heated cylindrical cavity. Physics of Fluids, 17(5), 054104. doi:10.1063/1.1876892 SCHWABE, D., ZEBIB, A., & SIM, B.-C. (2003). Oscillatory thermo