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

Published online by Cambridge University Press:  22 March 2023

Baptiste Darbois Texier ,
Henri Lhuissier Open the ORCID record for Henri Lhuissier [Opens in a new window] ,
Bloen Metzger Open the ORCID record for Bloen Metzger [Opens in a new window]  and
Yoël Forterre Open the ORCID record for Yoël Forterre [Opens in a new window]
Baptiste Darbois Texier
Affiliation:
Aix-Marseille University, CNRS, IUSTI, 13453 Marseille, France Université Paris-Saclay, CNRS, FAST, 91405 Orsay, France
Henri Lhuissier
Affiliation:
Aix-Marseille University, CNRS, IUSTI, 13453 Marseille, France
Bloen Metzger
Affiliation:
Aix-Marseille University, CNRS, IUSTI, 13453 Marseille, France
Yoël Forterre*
Affiliation:
Aix-Marseille University, CNRS, IUSTI, 13453 Marseille, France
*
†Email address for correspondence: yoel.forterre@univ-amu.fr.
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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.

JFM classification

Complex Fluids: Suspensions Interfacial Flows (free surface): Thin films
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 959 , 25 March 2023 , A27
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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References

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