• DOI:10.1098/rspa.1954.0186
• Corpus ID: 98030586

Experiments on a gravity-free dispersion of large solid spheres in a Newtonian fluid under shear

@article{Bagnold1954ExperimentsOA,
  title={Experiments on a gravity-free dispersion of large solid spheres in a Newtonian fluid under shear},
  author={Ralph Alger Bagnold},
  journal={Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences},
  year={1954},
  volume={225},
  pages={49 - 63}
}

• R. Bagnold
• Published 6 August 1954
• Geology
• Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences

Dispersions of solid spherical grains of diameter D = 0.13cm were sheared in Newtonian fluids of varying viscosity (water and a glycerine-water-alcohol mixture) in the annular space between two concentric drums. The density σ of the grains was balanced against the density ρ of the fluid, giving a condition of no differential forces due to radial acceleration. The volume concentration C of the grains was varied between 62 and 13 %. A substantial radial dispersive pressure was found to be exerted between the grains. This was measured as an increase of static pressure in the inner stationary drum which had a deformable periphery. The torque on the inner drum was also measured. The dispersive pressure P was found to be proportional to a shear stress λ attributable to the presence of the grains. The linear grain concentration λ is defined as the ratio grain diameter/mean free dispersion distance and is related to C by λ=1(C0/C)12−1 where C0 is the maximum possible static volume concentration. Both the stressesT and P, as dimensionless groups TσD2/λη2, and PσD2/λη 2, were found to bear single-valued empirical relations to a dimensionless shear strain group λ½σD2(dU/dy)lη for all the values of λ< 12(C= 57% approx.) where dU/dy is the rate of shearing of the grains over one another, and η the fluid viscosity. This relation gives Tασ(λD)2(dU/dy)2 and T∝λ12ηdU/dy according as dU/dy is large or small, i.e. according to whether grain inertia or fluid viscosity dominate. An alternative semi-empirical relation F = (1+λ)(1+½λ)ηdU/dy was found for the viscous case, when T is the whole shear stress. The ratio T/P was constant at 0·3 approx, in the inertia region, and at 0.75 approx, in the viscous region. The results are applied to a few hitherto unexplained natural phenomena.