Dynamics and Stability of a Thin Liquid Film Falling over a Vertical Cylinder with Linearly Varying Wall Temperature

Linear stability of a gravity-driven thin liquid film flowing along the outer surface of a vertical cylinder with a linearly varying wall temperature is investigated.The mathematical model is developed within the framework of lubrication theory, accounting for the effects of gravity, inertia, surface tension, and thermocapillary forces.Temporal linear stability analysis is performed to examine the influence of the Reynolds number (𝑅𝑒), Bond number (𝐵𝑜), Marangoni number (𝑀𝑎), and Prandtl number (𝑃𝑟) on the evolution of interfacial disturbances for both linearly increasing and decreasing wall-temperature profiles.The results show that the stability characteristics are strongly dependent on the direction of the imposed temperature gradient.For a linearly increasing wall temperature, increasing 𝑀𝑎, 𝑅𝑒, and 𝑃𝑟 enhances disturbance growth, destabilising the film, whereas increasing 𝐵𝑜 suppresses disturbance growth and stabilises the flow.In contrast, a linearly decreasing wall temperature yields the opposite thermal response: increasing the magnitude of negative 𝑀𝑎 and 𝑃𝑟 suppress disturbance growth.The stabilising effect of the Bond number is observed for both temperature-gradient configurations.The theoretical predictions are further validated through nonlinear simulations applying numerical analysis, showing good agreement between the analytical and numerical results.