Shear-thickening suspensions down inclines: from Kapitza to Oobleck waves

Abstract

We investigate experimentally and theoretically the stability of a shear-thickening suspension flowing down an inclined plane. In a previous paper (Darbois Texier et al., Commun. Phys., vol. 3, 2020), we have shown that for particle volume fractions $\phi$ above the discontinuous shear-thickening fraction $\phi _{DST}$ , long surface waves grow spontaneously at a flow Reynolds number much below 1. This motivated a simplified analysis based on a purely inertialess mechanism, called the ‘Oobleck waves’ mechanism, which couples the negatively sloped rheology of the suspension with the free-surface deflection and captures well the experimental instability threshold and the wave speed, for $\phi >\phi _{DST}$ . However, neglecting inertia does not allow us to describe the inertial Kapitza regime observed for $\phi <\phi _{DST}$ , nor does it allow us to discriminate between Oobleck waves and other inertial instabilities expected above $\phi _{DST}$ . This paper fills this gap by extending our previous analysis, based on a depth-averaged approach and the Wyart–Cates constitutive shear-thickening rheology, to account for inertia. The extended analysis recovers quantitatively the experimental instability threshold in the Kapitza regime, below $\phi _{DST}$ , and in the Oobleck waves regime, above $\phi _{DST}$ . By providing additional measurements of the wave growth rate and investigating theoretically the effect of a strain delay in the rheology, it also confirms that the instability observed above $\phi _{DST}$ stems from the non-inertial Oobleck wave mechanism, which is specific to free-surface flows and dominates modes of inertial origin. These results emphasize the variety of instability mechanisms for shear-thickening suspensions and might be relevant to free-surface flows of other complex fluids displaying velocity-weakening rheology.