If the shear and normal stresses in the mixture (suspension: mixture of solid and fluid) vary quadratically with the shear rate, the flow is said to satisfy Bagnold’s grain-inertia flow. If this relation is linear, then the motion is said to satisfy Bagnold’s macroviscous flow. These flow laws were derived by Bagnold in 1954 with his pioneering experiments in an annular coaxial cylinder rheometer where he evaluated the effects of grain interaction in the suspension. These types of relationships have been confirmed by many subsequent shear-cell experiments for both wet and dry mixtures and computer simulations.   Bagnold's rheology can be used to describe debris and granular flows down inclined slopes. 
Scaling law for the settlement (depositional) behavior of Bagnold’s fluids
A new scaling law for Bagnold’s fluids is established to relate the settlement time of debris deposition. It is found analytically that the macroviscous fluid settles (comes to a standstill) much faster than the grain-inertia fluid, as manifested by dispersive pressure. Given the same time, the macroviscous fluid is settled 6/5 unit length compared to the unit length settlement of the grain-inertia fluid as measured from the nose-tip of the flowfront that has already settled to the back side of the debris. Therefore, the macroviscous fluid settles (completely stops to flow) 20% faster than the grain-inertia fluid. Due to the dispersive pressure in grain-inertia fluid, the settlement process is delayed by 20% for the grain-inertia fluid than for the macroviscous fluid. This is meaningful because particles are more agitated due to higher dispersive pressure in grain-inertia fluids than in macroviscous fluids. Once the material comes close to rest, these dispersive forces (induced by the quadratic shear rate), are still active for grain-inertia fluid but macroviscous fluid settles relatively faster because it is less dispersive. This provides a tool to approximate and estimate the final settlement time (the time at which the entire fluid body is at rest). These are mechanically important relationships concerning the settlement time and the settlement lengths between the grain-inertia and the macroviscous fluids.
- Bagnold, R.A. (1954). "Experiments on a gravity-free dispersion of large solid spheres in a Newtonian fluid under shear". Proceedings of the Royal Society A 225,: 49.
- Campbell, S.C. (1990). "Rapid granular flows". Annu. Rev. Fluid Mech. 22,: 57.
- Silbert, E.; Ertas, D.; Grest, G. S.; Halsey, T. C.; Levine, D.; Plimpton, S. J. (2001). "Granular flow down an inclined plane: Bagnold scaling and rheology". Phys. Rev. E 64,: 051302.
- Takahashi, T. (2007). Debris Flow: Mechanics, Prediction and Countermeasures. Taylor and Francis, Leiden.