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Phase plane and bifurcation analysis of thin wavy films under shear



A long-wave equation for film thickness as a function of position is derived for a general case incorporating viscous, surface tension, and interfacial shear effects. The derivation considers both the parabolic and the power-law velocity profiles. The analysis is aimed at revealing the wave velocity that induces infinitely long (homoclinic) periods as well as substrate thickness and wave peak amplitude. Phase plane analysis shows that at Re ≫ 1, due to time-scale separation, the homoclinic velocity is near that at the Hopf bifurcation. That enables analytical derivation of the wave characteristics.

Comparison with experimental results in the range of Re-310–3, 100 with countercurrent gas flow, shows encouraging agreement. At very high Re the wave velocity suggests the onset of turbulence, in agreement with theory. Phase plane analysis predicts also that the wave shape consists of a simple peak with a steep front, with short waves riding on the main wave at low Re.

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