Stability of a non-Newtonian falling film due to three-dimensional disturbances

In this paper, we apply the normal modes method to study the linear stability of a liquid film flowing down an inclined plane, taking into account the complex rheology of the media. We consider generalized Newtonian liquids; the conditions of the Squire theorem do not hold for this case. We check if the flow is unstable due to three-dimensional (3D) disturbances that propagate at a certain angle to the flow direction but stable for the two-dimensional (2D) ones. We derived the generalized Orr-Sommerfeld equation, considered a long-wave approximation, and proved that 3D long-wave disturbances are less growing than the 2D ones for any rheological law. We solved the problem for finite wavenumbers numerically and found that for low inclination angles of the plane, instability due to 3D disturbances prevails. In this case, the shear mode of instability dominates, and the surface tension destabilizes the flow. For shear-thickening liquids, the critical Reynolds number decreases down to zero.