Weakly nonlinear stability of Marangoni convection in a liquid bridge

Marangoni convection arising in a liquid column, bridging between concentric, circular parallel plates with different but uniform temperatures, is examined on its linear and weakly nonlinear stability. The analyses are conducted for small and moderate Prandtl numbers $0.001\le P\le 10$. Our attention is focused on a relatively low liquid bridge with $h/r_0=1$ where $h$ is the height and $r_0$ is the radius. The buoyancy effect is ignored and perfectly insulating condition is imposed on the surface of the liquid bridge. Linear stability analysis revealed that the critical condition was given by different azimuthal wavenumbers, $m=1$, 2, and 3, depending on the Prandtl number. The critical condition is given by steady solutions for $0 < P < 0.0578$ and by oscillatory solutions for $P > 0.0578$. Weakly nonlinear analysis identifies the stable region of the secondary solutions bifurcating from the linear critical curve.