Drag law for an intruder in granular sediments

We investigate the drag experienced by a spherical intruder moving through a medium consisting of granular hydrogels immersed in water as a function of its depth, size, and speed. The medium is observed to display a yield stress with a finite force required to move the intruder in the quasistatic regime at low speeds before rapidly increasing at high speeds. In order to understand the relevant time scales that determine drag, we estimate the inertial number I given by the ratio of the time scales required to rearrange grains due to the overburden pressure and imposed shear and the viscous number J given by the ratio of the time scales required to sediment grains in the interstitial fluid and imposed shear. We find that the effective friction mu(e) encountered by the intruder can be parametrized by I = root rho(g) / P-p nu(i) where.g is the density of the granular hydrogels, nu(i) is the intruder speed, and P-p is the overburden pressure due to the weight of the medium, over a wide range of I where the Stokes number St = I-2 / J >> 1. We then show that mu(e) can be described by the function mu(e)( I) = mu(0) + alpha I-beta, where mu(0), alpha, and beta are constants that depend on the medium. This formula can be used to predict the drag experienced by an intruder of a different size at a different depth in the same medium as a function of its speed.