Dynamics of a thermally driven film climbing the outside of a vertical cylinder.

The dynamics of a film climbing the outside of a vertical cylinder under the competing effects of a thermally driven surface tension gradient and gravity is examined through numerical simulations of a thin-film model for the film height. The model, including boundary conditions, depends on three parameters, the scaled cylinder radius R[over ̂], the upstream film height h_{∞}, and the downstream precursor film thickness b, and reduces to the model for Marangoni driven film climbing a vertical plate in the limit R[over ̂]→∞. The axisymmetric advancing front displays dynamics similar to that found along a vertical plate where, depending on h_{∞}, the film forms a single Lax shock, an undercompressive double shock, or a rarefaction-undercompressive shock. A linear stability analysis of the Lax shock reveals the number of fingers that form along the contact line increases linearly with cylinder circumference while no fingers form for sufficiently small cylinders (below R[over ̂]≈1.15 when b=0.1). The substrate curvature controls the height of the Lax shock, bounds on h_{∞} that define the three distinct solutions, and the maximum growth rate of contact line perturbations to the Lax shock when R[over ̂]=O(1), whereas the three solutions and the stability of the Lax shock converge to the behavior one observes on a vertical plate when R[over ̂]≥O(10). An energy analysis reveals that the azimuthal curvatures of the base state and perturbation, which arise from the annular geometry of the film, promote instability of the advancing contact line.