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Stability of liquid film falling down a vertical non-uniformly heated wall. (English) Zbl 1149.76024
Summary: We study the stability of a viscous film flowing down a vertical non-uniformly heated wall under gravity. The wall temperature is assumed linearly distributed along the wall, and the free surface is taken to be adiabatic. A long wave perturbation method is used to derive the nonlinear evolution equation for the falling film. Using the method of multiple scale, the nonlinear stability analysis is performed for travelling wave solution of the evolution equation. The complex Ginzburg-Landau equation is determined to discuss the bifurcation analysis of the evolution equation. The results indicate that the supercritical unstable region increases and the subcritical stable region decreases with the increase in Peclet number. It has been also shown that the spatial uniform solution corresponding to the sideband disturbance may be stable in the unstable region.
MSC:
76E17Interfacial stability and instability (fluid dynamics)
76E30Nonlinear effects (fluid mechanics)
76A20Thin fluid films (fluid mechanics)