Stability of a flow down an incline with respect to three-dimensional disturbances: the question of Squire conjecture for Newtonian or generalized Newtonian fluids

The Squire conjecture, which states that the two-dimensional instabilities are the more dangerous, is questioned here for a flow down an incline, making use of numerical stability analysis and Squire relationships when available. For a Newtonian fluid, it is shown that oblique wave instabilities are never the dominant instabilities. Both the Squire relationships and the particular variations of the two-dimensional wave marginal curve with regard to the inclination angle are needed to obtain this result validating the Squire conjecture. For a generalized Newtonian fluid, a similar result has only been obtained for a reduced stability problem where some term connected to the perturbation of viscosity is neglected. For the general stability problem, however, no Squire relationship can be derived and the numerical stability results show that the thresholds for oblique waves can be smaller than the thresholds for two-dimensional waves, particularly for large obliquity angles and strong non-Newtonian behaviours. The Squire conjecture is then clearly not valid in this case.