Effect of the horizontal aspect ratio on thermocapillary convection stability in annular pool with surface heat dissipation

摘要

A linear stability analysis of the thermoconvective problem of a thin liquid film contained in an annular domain has been conducted. The influence of the horizontal aspect ratio on the solution has been considered by keeping a fixed external wall while the internal radius of the annular domain was modified. The parameter used in the study, Γ_h, has been defined as the ratio of the internal radius to the domain depth. The other control parameter of the study is the Prandtl number ranging from 0.7 to 50, i.e. characteristic of fluids as air to n-butanol. The study has been performed for different Bond (Bo) regimes ranging from 0.0 for surface tension dominated flows to 67 for buoyancy dominated ones. Three different kind of bifurcations are found in the Γ_h - Pr plane for large Bonds, while for low Bonds only two of them appear. In the case of pure buoyancy or surface tension flows, for every Γ_h there exists a Prandtl number such that oscillatory and stationary coexist in a co-dimension two bifurcation point. These transitions show a strong dependency with the Bond number. Indeed, the lower transition disappears for low Bo and the upper one disappears with intermediate Bo values. Furthermore, there is a non-linear dependency of the number of structures of the growing bifurcation with Γ_h These co-dimension two lines show a strong dependency with Bo. Firstly, looking at the frontier between HWI and LR regions, for large Bo numbers, Pr increases with Γ_h, while for low Bo the trend is reversed. Additionally, this transition only appears in the extreme Bo cases, for the central values of the considered, no transition is found. Similarly, the second transition found only appears for Bo larger than 30.