Thermally driven film climbing a vertical cylinder

The dynamics of a Marangoni driven film climbing the outside of a vertical cylinder is examined in numerical simulations of a thin film model. The model has three parameters: the scaled cylinder radius $\hat{R},$ upstream film height $h_{\infty}$ and downstream precursor film thickness $b,$ and reduces to the model for Marangoni driven film climbing a vertical plate when $\hat{R} \to \infty.$ The advancing front displays dynamics similar to that along a vertical plate where, depending on $h_{\infty},$ the film forms a 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 below $\hat{R} \approx 1.15$ with $b=0.1.$ The substrate curvature controls the Lax shock height, bounds on $h_{\infty}$ that define the three solutions and the maximum growth rate of perturbations when $\hat{R}=O(1),$ whereas the shape of solutions and the stability of the Lax shock converge to the behavior on a vertical plate when $\hat{R} \ge O(10).$ The azimuthal curvatures of the base state and perturbation, arising from the annular geometry of the film, promote instability of the advancing contact line.