Thick film flowing down a non-isothermal vertical cylinder

• Published in: Journal of engineering mathematics 2022-12, Vol.137 (1), Article 2
• Main Authors: Kishal, Divij, Tiwari, Naveen
• Format: Article
• Language: English
• Subjects: Applications of Mathematics; Aspect ratio; Bond number; Computational Mathematics and Numerical Analysis; Droplets; Gas flow; Gas pressure; Mathematical and Computational Engineering; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Normal stress; Stability analysis; Substrates; Theoretical and Applied Mechanics; Traveling waves; Vertical cylinders
• ISSN: 0022-0833; 1573-2703
• DOI: 10.1007/s10665-022-10239-2

A thick film flow on a non-isothermal vertical cylinder due to gravity is analyzed for temporal and spatiotemporal stability. The model equation is derived by neglecting inertial terms and keeping the full curvature term in the normal stress balance. The thermoviscous and thermocapillary effects are also considered. The film aspect ratio with respect to cylinder radius leads to an increase followed by a decrease in the temporal growth rate. For a cooled substrate, the film can be stabilized for a moderate aspect ratio. Spatiotemporal stability analysis is performed using the Briggs criterion. The aspect ratio significantly affects the spatiotemporal stability of the film. Absolutely unstable regimes are found for small and much larger values of the Bond number, with a convectively unstable regime for moderate values. The critical condition is found that leads to a relation between different model parameters such as Marangoni number, thermoviscosity number, and aspect ratio for which the film is absolutely unstable films for all values of Bond numbers. An opposing pressure gradient is also included in the model through normal stress balance, which can arise due to a counter-current gas flow. While this pressure gradient does not affect the temporal growth rates but promotes absolute instability. Non-linear computations are done to validate the growth rates from both linear temporal and spatiotemporal stability analyses. It is observed that the gaussian disturbance leads to traveling waves well beyond the saturation. The droplet size and gap between two subsequent droplets increase with the gas pressure gradient, thermoviscosity number, and Marangoni number.