Linear stability of thermocapillary flow in partially confined half-zones

The thermocapillary flow in cylindrical half-zones is investigated numerically for low-Prandtl-number fluids. We consider a configuration in which part of the cylindrical surface is covered by a very thin rigid film and another part is uncovered, supporting thermocapillary stresses. The steady axisymmetric flow is calculated by finite differences. Besides the aspect ratio, the structure of the toroidal vortex flow depends strongly on the location and relative size of the cylindrical free-surface strip. The stability with respect to arbitrary three-dimensional perturbations of the steady two-dimensional flow is investigated by a linear stability analysis. The critical Reynolds numbers and the physical mechanisms are analyzed depending on the aspect ratio of the liquid bridge, the axial position of the uncovered part of the surface, and the free-surface fraction $ξ$⁠, i.e., the ratio of the height of the free surface to the full height of the liquid zone. When only a small area of the liquid zone is covered by the solid film, i.e., $ξ≈1$⁠, the well-known stationary instability for the conventional unconfined half-zone is recovered, which is driven by a combination of the elliptic and the centrifugal mechanism. Decreasing $ξ$⁠, new instabilities can appear. It is found that all critical modes are likewise driven by the elliptic mechanism, the centrifugal mechanism, or a combination of both. When the unconfined free surface is located adjacent to the hot wall, the linear stability boundary shows an intricate behavior. Four qualitatively different instability modes can arise depending on the free-surface fraction $ξ$ and the aspect ratio. One of the time-dependent modes that is strongly driven by the elliptic mechanism resembles an oscillatory Kelvin mode on a strained vortex. The other oscillatory modes show a different behavior as they are mainly driven by centrifugal mechanisms. Quite generally, the importance of the elliptic mechanism diminishes as $ξ$ is further decreased until, for a very small free-surface fraction $ξ$⁠, an instability appears that is purely centrifugal and stationary. When the free surface is located next to the cold wall, the types of instabilities are the same as for the unconfined half-zone, and the azimuthal wave numbers increase with decreasing $ξ$⁠.