Effects of the variation of viscosity on the stability of thin liquid film flows along a uniformly heated substrate under heat flux boundary condition

Abstract

The effects of variable viscosity on the stability of gravity-driven, Newtonian, thin liquid film flowing down a uniformly heated substrate under heat flux (HF) boundary condition is investigated. HF boundary condition allow us to consider the heat loss from the system at the solid–air interface as well as the heat flux by the rigid wall to the surrounding liquid, both of which effects the temperature gradient on the solid–liquid interface. This model is more realistic in comparison with specified temperature (ST) boundary condition/Dirichlet condition. The underlying assumption of ST boundary condition is that the heat flux at the solid–liquid interface is equal to the heat loss at the solid–air interface. It results in vanishing temperature gradient on the top surface of the rigid wall. Consequently, both the heat flux at solid–liquid and solid–air interfaces have no influence on the thin liquid film flow over the rigid substrate. Considering exponential variation of viscosity, together with the linear variation of surface tension, an evolution equation is constructed, using long-wave expansion technique. This evolution equation captures the effect of the variation of viscosity, thermocapillarity, and heat flux at the solid–air interface, through the parameters, Kμ (coefficient of dynamical viscosity), MBs (products of film Marangoni and free surface Biot number), and Bw (wall film Biot number), respectively. Using normal mode approach, the linear stability analysis reveals the destabilizing behavior of Kμ, MBs, and stabilizing effect of Bw. Using multiple-scale analysis, the weakly nonlinear study demarcates the supercritical (subcritical) stable (unstable) zones and their dependence on Kμ and Bw. Finally, the numerical simulation of the evolution equation, by the spectral method over a periodic domain, confirms the results obtained by the linear and weakly nonlinear study.