Does surface tension stabilize liquid films inside a rotating horizontal cylinder? Part 2: Multidimensional disturbances

We examine the stability of a thin film of viscous fluid on the inside surface of a cylinder with horizontal axis, rotating about this axis. Depending on the parameters involved, the dynamics of the film can be described by several asymptotic models, one of which was examined by Benilov [J. Fluid Mech. 501:105-124 (2004)]. It turned out that the linearized stability problem for this model admits infinitely many neutrally stable eigenmodes, which form a complete set. Despite that, the film is unstable with respect to exploding disturbances, which grow infinitely in a finite time. The present paper examines the effect of surface tension on the stability of the film. Two cases are considered: short-scale disturbances (such that the axial wavelength lambda is much smaller than the radius R of the cylinder) and long-scale disturbances (for which lambda greater than or similar to R). In the former case, surface tension is a stabilizing influence, because it regularizes the exploding solutions and makes all eigenmodes asymptotically (not just neutrally) stable. The latter case was previously examined by Acrivos and Jin [J. Eng. Math. 50:99-120 (2004)], who showed that surface tension destabilizes some of the eigenmodes. We argue, however, that the corresponding growth rate is much smaller than that of the so-called inertial instability.