Stability of a flow down an incline with respect to two-dimensional and three-dimensional disturbances for Newtonian and non-Newtonian fluids
Abstract
Squire’s theorem, which states that the two-dimensional instabilities are more dangerous than the three-dimensional
instabilities, is revisited here for a flow down an incline, making use of numerical stability analysis
and Squire relationships when available. For flows down inclined planes, one of these Squire relationships
involves the slopes of the inclines. This means that the Reynolds number associated with a two-dimensional
wave can be shown to be smaller than that for an oblique wave, but this oblique wave being obtained for a larger
slope. Physically speaking, this prevents the possibility to directly compare the thresholds at a given slope. The
goal of the paper is then to reach a conclusion about the predominance or not of two-dimensional instabilities at
a given slope, which is of practical interest for industrial or environmental applications. For a Newtonian fluid,
it is shown that, for a given slope, oblique wave instabilities are never the dominant instabilities. Both the Squire
relationships and the particular variations of the two-dimensional wave critical curve with regard to the inclination
angle are involved in the proof of this result. For a generalized Newtonian fluid, a similar result can only be
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 relationships 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
at a given slope, particularly for large obliquity angles and strong shear-thinning behaviors. The conclusion is
then completely different in that case: the dominant instability for a generalized Newtonian fluid flowing down
an inclined plane with a given slope can be three dimensional.
Origin | Files produced by the author(s) |
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