Thermally influenced capillary ridge formation in a two-layer fluid system on an inclined plate

This study explores the formation of capillary ridges in a two-layer flow of immiscible viscous Newtonian fluids down a non-uniformly heated inclined plate. Temperature gradients induced by the heated substrate generate surface-tension variations that, in turn, produce thermocapillary (Marangoni) stresses, driving fluid flow and leading to liquid accumulation at the advancing front and the formation of pronounced capillary ridges. A comprehensive mathematical model is developed based on the conservation of mass, momentum, and energy, together with appropriate boundary and interfacial conditions. Long-wave theory is employed based on the aspect ratio of the fluid domain to derive the transient thin-film equations. The resulting coupled non-linear equations are discretized using the finite volume method and solved with a nonlinear solver. The study systematically investigates how key parameters, including temperature gradients, inclination angle, and fluid properties, influence the formation and morphology of capillary ridges. The results reveal that increased thermal gradients enhance Marangoni stresses and ridge prominence. The findings provide new insights into the interplay between thermal effects and flow behavior, offering valuable guidance for optimizing thin-film processes in applications such as coating technologies and heat-transfer systems.