Fig. 4: Instability mechanism.
From: Surface-wave instability without inertia in shear-thickening suspensions
a Depth-averaged forces acting along the flow x direction on a slice of suspension of width dx (shaded in blue): basal force τbdx, projected weight of the slice \(\rho ghdx\sin \theta\) and hydrostatic pressure \(\rho g\cos \theta {h}^{2}(x)/2\), where τb is the basal shear stress, g is gravity, h(x) is the flow thickness, θ is the plane inclination angle. b Positive feedback for a shear-thickening suspension with a S-shaped rheological curve: a local increase of the flow thickness h implies that ∂h/∂x is positive (resp. negative) upstream (resp. downstream) of the perturbation. Force balance (see Eq. (1)) then implies that the basal shear stress τb upstream (resp. downstream) must decrease (resp. increase). When the suspension rheogram is negatively sloped (\({\rm{d}}\dot{\gamma }/{\rm{d}}\tau <0\)), this yields a local increase (resp. decrease) of the flow rate \(\dot{\gamma }\). The combination of these two feedback cycles (gray arrows) induces a net inward mass flux towards the bump (red arrows) that amplifies the initial perturbation (red vertical arrow).
