Stability of traveling waves of a thermoviscous liquid film down the outer surface of a cylinder

A liquid film coating a cylindrical substrate is unstable to interfacial disturbances governed by Rayleigh -plateau instability. These initial disturbances grow in time and, beyond saturation, develop into a train of droplets of equal amplitude that move downstream at the same velocity. The traveling wave profiles, formed by a thermoviscous fluid as it flows down the exterior of a nonisothermal vertical cylinder, are computed as a function of various parameters. Further, this study investigates the stability behavior of these traveling waves. An unstable film leads to the formation of an axially asymmetric pattern. The influence of the thermoviscous nature of the fluid, along with the Biot number, Marangoni number, and Bond number, has been investigated. Linear stability analysis is performed by perturbing the axisymmetric traveling wave solution in the azimuthal direction. It is shown that while a gravity mode exists at small wave numbers, a thermocapillary mode exists at large wave numbers. The streamline patterns are shown to be consistent with the presence of the thermocapillary mode. The nonlinear analysis shows that the chosen pattern of instability does not correspond to the largest growth rate but is governed by the wavelength conforming to the periodic confinement in the azimuthal direction. An interesting pattern evolves in the presence of more than one such conforming wavelength.