Linear stability of shear-thinning fluid down an inclined plane

Abstract The stability of shear-thinning fluid down an inclined plane is investigated theoretically with application to process engineering (coating process) and natural hazards as well (debris flow impact). The power-law model is used to describe the fluid rheological behavior. A linear stability analysis is brought out for building a generalized Orr-Sommerfeld model with appropriate definition of non-dimensional numbers in order to overcome the inconsistency of the existing shallow-water models. The solution to the secular equation showed that at zeroth order, there is no instability with respect to the long wave perturbations considered. Waves propagate without dispersion, at the same dimensional speed for any wavenumber. Moreover, the interface and the stream function are in phase, while the components of the perturbation velocities are respectively in phase and in quadrature of phase. Finally, the perturbation celerity decreases for increasing power law index, with a value of 2 as minimum for the Newtonian fluid. At first order, the effect of the different forces acting on the flow has been pointed out. It was particularly shown that pressure and surface tension have a stabilizing effect, while inertia and rheofluidification have a destabilizing effect. Moreover, the relative variation of critical Reynolds number increases with increasing reduced wavenumber for all values of slope tested while it decreases with increasing power-law index for all values of reduced wavenumber tested.