Main Article Content



This manuscript addresses the linear stability analysis of a thermoconvective problem in an annular domain. The flow is heated from below, with a linear decreasing horizontal temperature profile from the inner to the outer wall. The top surface of the domain is open to the atmosphere and the two lateral walls are adiabatic. The effects of several parameters in the flow are evaluated. Three different values for the ratio of the momentum diffusivity and thermal diffusivity are considered: relatively low Prandtl number (Pr = 1), intermediate    Prandtl number (Pr = 5) and high Prandtl number (ideally Pr → ∞ , namely Pr = 50). The thermal boundary condition on the top surface is changed by imposing different values of the Biot number, Bi. The influence of the aspect ratio (Γ) is assessed for through by studying several aspect ratios, Γ. The study has been performed for two values of the Bond number (namely Bo = 5 and 50), estimating the perturbation given by thermocapillarity effects on buoyancy effects. Different kind of competing solutions appear on localized zones of the Γ -Bi plane. The boundaries of these zones are made up of co-dimension two points. Co-dimension two points are found to be function of Bond number, Marangoni number and boundary condition but to be independent on the Prandtl number.

Article Details

How to Cite
HOYAS, Sergio et al. ON THE ONSET OF INSTABILITIES IN A BÉNARD-MARANGONI PROBLEM IN AN ANNULAR DOMAIN WITH TEMPERATURE GRADIENT. Thermal Science, [S.l.], mar. 2017. ISSN 2334-7163. Available at: <http://thermal-science.tech/journal/index.php/thsci/article/view/2369>. Date accessed: 20 feb. 2018. doi: https://doi.org/10.2298/TSCI160628277H.
Received 2017-03-10
Accepted 2017-03-14
Published 2017-03-14


[1] Bénard, H., Les tourbillons cellulaires dans une nappe liquid, Rev. Gen. Sci. Pure Appl., 11 (1900), pp. 1261-1271.
[2] Zhang, W., Zhou, J., Optimal control of a viscous shallow water equation, Advances in Mathematical Physics, vol 2013 (2013), 715959.
[3] Es Sakhy, R., El Omari, K., Le Guer, Y., Blancher, S, Rayleigh–Bénard–Marangoni convection in an open cylindrical container heated by a non-uniform flux, International Journal of Thermal Sciences, 86 (2014), pp. 198-209.
[4] Mercier, J. F., Normand, C, Buoyant‐thermocapillary instabilities of differentially heated liquid layers, Physics of Fluids (1994-present), 8 (1996), 6, pp. 1433-1445.
[5] Mancho, A. M., Herrero, H., Burguete, J, Primary instabilities in convective cells due to nonuniform heating, Physical Review E, 56 (1997), 3, pp. 2916-2923.
[6] Herrero, H., Mancho, A. M, Influence of aspect ratio in convection due to nonuniform heating. Physical Review E, 57 (1998), 6, pp. 7336-7339.
[7] Ezersky, A. B., Garcimartin, A., Burguete, J., Mancini, H. L., Pérez-García, C, Hydrothermal waves in Marangoni convection in a cylindrical container, Physical Review E, 47 (1993),2, pp. 1126-1131.
[8] Bammou, L., Blancher, S., Le Guer, Y., El Omari, K., Benhamou, B., Linear stability analysis of Poiseuille–Bénard–Marangoni flow in a horizontal infinite liquid film, International Communications in Heat and Mass Transfer, 54 (2014), pp. 126-131.
[9] Pardo, R., Herrero, H., Hoyas, S., Theoretical study of a Bénard–Marangoni problem. Journal of Mathematical Analysis and Applications, 376 (2011), 1, pp. 231-246.
[10] Hoyas, S., Herrero, H., Mancho, A. M, Thermal convection in a cylindrical annulus heated laterally. Journal of Physics A: Mathematical and General, 35 (2002), 18, pp. 4067-4083.
[11] Hoyas, S., Herrero, H., Mancho, A. M., Bifurcation diversity of dynamic thermocapillary liquid layers, Physical Review E, 66 (2002), 5, 057301.
[12] Peng, L., Li, Y. R., Shi, W. Y., Imaishi, N, Three-dimensional thermocapillary–buoyancy flow of silicone oil in a differentially heated annular pool, International Journal of Heat and Mass Transfer, 50 (2007), 5, pp. 872-880.
[13] Shi, W., Liu, X., Li, G., Li, Y. R., Peng, L., Ermakov, M. K., Imaishi, N., Thermocapillary convection instability in shallow annular pools by linear stability analysis, Journal of Superconductivity and Novel Magnetism, 23 (2010), 6, pp. 1185-1188.
[14] Wertgeim, I. I., Kumachkov, M. A., Mikishev, A. B., Periodically excited Marangoni convection in a locally heated liquid layer, The European Physical Journal Special Topics, 219 (2013), 1, pp. 155-165.
[15] Hoyas, S., Mancho, A. M., Herrero, H., Garnier, N., Chiffaudel, A., Bénard–Marangoni convection in a differentially heated cylindrical cavity, Physics of Fluids (1994-present), 17 (2005), 5, 054104.
[16] Garnier, N., Chiffaudel, A., Two dimensional hydrothermal waves in an extended cylindrical vessel, The European Physical Journal B-Condensed Matter and Complex Systems, 19 (2001), 1, pp. 87-95.
[17] Torregrosa, A. J., Hoyas, S., Pérez-Quiles, M. J., Mompó-Laborda, J. M., Bifurcation diversity in an annular pool heated from below: Prandtl and biot numbers effects, Communications in Computational Physics, 13 (2013), 2, pp. 428-441.
[18] Hoyas, S., Gil, A., Fajardo, P., Pérez-Quiles, M. J.. Codimension-three bifurcations in a Bénard- Marangoni problem, Physical Review E, 88 (2013), 1, 015001.
[19] Eckert, E. R. G., Goldstein, R. J., Ibele, W. E., Patankar, S. V., Simon, T. W., Kuehn, T. H., Strykowski, P.J., Tamma, K.K., Bar-Cohen, A., Heberlein, J.V.R., Davidson, J.H., Bischof, J., Kulacki, F.A., Kortshagen, U., Garrick, S., Heat transfer—a review of 1997 literature, International Journal of Heat and Mass Transfer, 43 (2000), 14, pp. 2431-2528.
[20] O’Shaughnessy, S. M., Robinson, A. J., Heat transfer near an isolated hemispherical gas bubble: The combined influence of thermocapillarity and buoyancy, International Journal of Heat and Mass Transfer, 62 (2013), pp. 422-434.
[21] Hoyas, S., Fajardo, P., Gil, A., Perez-Quiles, M. J., Analysis of bifurcations in a Bénard–Marangoni problem: Gravitational effects, International Journal of Heat and Mass Transfer, 73 (2014), pp. 33-41.
[22] Hoyas, S., Fajardo, P., Pérez-Quiles, M. J., Influence of geometrical parameters on the linear stability of a Bénard-Marangoni problema, Physical Review E, 93 (2016), 4, 043105.
[23] Smith, M. K., Davis, S. H., Instabilities of dynamic thermocapillary liquid layers. Part 1. Convective instabilities, Journal of Fluid Mechanics, 132 (1983), pp. 119-144.
[24] Favre, E., Blumenfeld, L., Daviaud, F., Instabilities of a liquid layer locally heated on its free surface, Physics of Fluids (1994-present), 9 (1997), 5, pp. 1473-1475.
[25] Chandrasekhar, S., Hydrodynamic and Hydromagnetic Stability. International Series of Monographs on Physics, Oxford-Clarendon Press, 1961.
[26] Jimenez, J., Hoyas, S., Turbulent fluctuations above the buffer layer of wall-bounded flows, Journal of Fluid Mechanics, 611 (2008), pp. 215-236.
[27] Orszag, S. A. Comparison of pseudospectral and spectral approximation. Studies in Applied Mathematics, 51 (1972), 3, pp. 253-259.
[28] Canuto, C., Hussaini, M. Y., Quarteroni, A., Zang, T. A., Spectral methods in Fluid Dynamics, Springer-Verlag, 1988.
[29] Bernardi, C., Maday, Y, Approximations spectrales des problemes aux limites elliptiques. Springer- Verlag, 1992.
[30] Navarro, M. C., Herrero, H., Hoyas, S., Chebyshev collocation for optimal control in a thermoconvective flow, Comm. Comput. Phys, 5 (2009), 2-4, pp. 649-666.