Nonlinear stability analysis and critical conditions for roll waves in power-law fluid films flowing down inclined planes

Abstract

The nonlinear stability of roll waves in power-law fluid films on inclined planes is investigated using the Kármán momentum integral method and Whitham’s modulation theory. Depth-integrated governing equations and modulation equations are derived, yielding an integro-differential stability criterion dependent on mean and critical depths. Linear analysis shows that shear-thinning fluids have lower critical Reynolds numbers than Newtonian fluids: for n  = 0.4, the linear stability threshold parameter α _s  = 11.25 compared to α _s  = 3 for n  = 1. Nonlinear analysis reveals dual mechanisms of roll wave generation – linear instability and subcritical bifurcation. Asymptotic analysis overcomes the singularity at maximum amplitude, enabling the systematic construction of nonlinear stability diagrams. We identify a critical power-law index n _cr ≈ 0.7 for α  = 2.0, below which shear-thinning unexpectedly promotes stable wave formation. For strongly shear-thinning fluids ( n  = 0.4), a critical inclination parameter α _cr  ≈ 0.2 marks the onset of a regime where milder slopes favor wave stability – a nonlinear phenomenon arising from the competition between inertia and dissipation. Only roll waves with amplitudes exceeding a saturation threshold can propagate stably. These findings provide theoretical guidance for engineering control of non-Newtonian film flows.