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

Abstract

Dispersions of solid spherical grains of diameter D = 0$\cdot $13 cm 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 $\sigma $ of the grains was balanced against the density $\rho $ 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 T attributable to the presence of the grains. The linear grain concentration $\lambda $ is defined as the ratio grain diameter/mean free dispersion distance and is related to C by $\lambda =\frac{1}{(C_{0}/C)^{\frac{1}{3}}-1}$, where C$_{0}$ is the maximum possible static volume concentration. Both the stresses T and P, as dimensionless groups T$\sigma $D$^{2}$/$\lambda \eta ^{2}$ and P$\sigma $D$^{2}$/$\lambda \eta ^{2}$, were found to bear single-valued empirical relations to a dimensionless shear strain group $\lambda ^{\frac{1}{2}}\sigma $D$^{2}$(dU/dy)/$\eta $ for all the values of $\lambda $ < 12 (C = 57% approx.) where dU/dy is the rate of shearing of the grains over one another, and $\eta $ the fluid viscosity. This relation gives T $\propto \ \sigma $($\lambda $D)$^{2}$ (dU/dy)$^{2}$ and T $\propto \ \lambda ^{\frac{3}{2}}\eta $dU/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 $\scr{T}$ = (1 + $\lambda $) (1 + $\frac{1}{2}\lambda $) $\eta $dU/dy was found for the viscous case, when $\scr{T}$ is the whole shear stress. The ratio T/P was constant at 0$\cdot $3 approx. in the inertia region, and at 0$\cdot $75 approx. in the viscous region. The results are applied to a few hitherto unexplained natural phenomena.