Effect of Heat Flux and Exponential Viscosity Variation on the Stability of Thin Liquid Films over Cylindrical Substrates

In paper manufacturing technology, a Yankee cylinder—a hollow, uniformly heated cylinder, is commonly used to dry thin films of paper pulp. To develop a realistic model of this drying process, we consider the gravity-driven flow of an incompressible, Newtonian thin liquid film along the outer surface of a hollow, vertically-oriented cylindrical wall. This wall is assumed to be infinitely long and to have a small thickness. A heater embedded within the inner side of the cylinder maintains a uniform temperature across the cylindrical surface. The temperature difference between the heated wall and the surrounding liquid generates a constant heat flux, directed radially outward. Since the cylinder is hollow, some heat is lost from the inner surface of the cylindrical wall to the surrounding air filling the core. This heat loss, which was neglected in previous studies, influences the thermal gradient across the cylinder. Given that viscosity is temperature-dependent, we assume it varies exponentially with the local temperature gradient. Applying the long-wave expansion method, we derive a Benney-type nonlinear partial differential equation describing the evolution of the instantaneous film thickness. A normal-mode linear stability analysis is then conducted to determine the onset of instability, characterized by a critical Reynolds number that incorporates the effects of both viscosity variation and heat flux, including the influence of heat loss at the wall-liquid and wall-air interfaces. For the nonlinear regime, we use the Fourier Spectral Method to perform simulations based on the fully nonlinear free-surface equation. Results from the nonlinear analysis show strong agreement with findings from the corresponding linear stability analysis.