Linear and nonlinear Marangoni stability of a thin film falling down the inside or the outside of a vertical cylinder with slip

Background: Research on thin liquid films has been of interest since many years ago due to their impact on applications of coating and cooling of walls. It is of interest to find passive ways to modify the flow stability. An important way to change the stability is by means of the chemical treatment of the substrate or else by means of very small topography (roughness) of the substrate. Both have the ability to produce a mean slip at the wall–liquid interface. Methods: The small wavenumber and large radius of the cylinder approximation is used to derive a nonlinear evolution equation that describes the free surface deformations of the liquid film falling down a vertical cylinder. This equation is linearized and its linear stability is investigated using normal modes including azimuthal perturbations. The nonlinear free surface deformations are investigated numerically by means of a nonlinear normal modes expansion method. Significant Findings: The linear and nonlinear Marangoni instability of a liquid film falling down, inside or outside, a vertical cylinder in the presence of slip at the wall–liquid interface is investigated. It is found that the throttling effect (necking) in the presence of slip is able to change the stability results found in previous papers on films falling down flat inclined walls. A nonlinear evolution of the free surface deformation is derived under the small wavenumber and large radius of the cylinder. The curves of linear growth rate, maximum growth rate and critical Marangoni number are calculated including the azimuthal modes of instability. It is found that from the point of view of the linear growth rate the flow destabilizes with slip in a wavenumber range k < k + . However, slip stabilizes for larger wavenumbers k > k + up to the critical (cutoff) wavenumber. From the point of view of the maximum growth rate, it is found that for flow outside the cylinder slip cannot stabilize the flow, in agreement with results derived for a vertical wall in previous papers. However, due to the throttling effect (necking) slip is able to stabilize the flow inside the cylinder up to a certain Reynolds number, in contrast to previous results for a vertical wall. Explicit formulas were derived for the intersections where slip may change its stability properties. All this findings are also corroborated with nonlinear numerical results of the evolution equation.