Effective drag of a rod in fluid-saturated granular beds

We measure the drag encountered by a vertically oriented rod moving across a sedimented granular bed immersed in a fluid under steady-state conditions. At low rod speeds, the presence of the fluid leads to a lower drag because of buoyancy, whereas a significantly higher drag is observed with increasing speeds. The drag as a function of the depth is observed to decrease from being quadratic at low speeds to appearing more linear at higher speeds. By scaling the drag with the average weight of the grains acting on the rod, we obtain the effective friction μ_{e} encountered over six orders of magnitude of speeds. While a constant μ_{e} is found when the grain size, rod depth, and fluid viscosity are varied at low speeds, a systematic increase is observed as the speed is increased. We analyze μ_{e} in terms of the inertial number I and viscous number J to understand the relative importance of inertia and viscous forces, respectively. For sufficiently high fluid viscosities, we find that the effect of varying the speed, depth, and viscosity can be described by the empirical function μ_{e}=μ_{o}+kJ^{n}, where μ_{o} is the effective friction measured in the quasistatic limit, and k and n are material constants. The drag is then analyzed in terms of the effective viscosity η_{e} and found to decrease systematically as a function of J. We further show that η_{e} as a function of J is directly proportional to the fluid viscosity and the μ_{e} encountered by the rod.