# Effective drag of a rod in fluid-saturated granular beds

## TL;DR

This study investigates how a rod's drag in a fluid-saturated granular bed varies with speed, depth, and fluid viscosity, revealing a transition from quadratic to linear drag behavior and proposing empirical models involving inertial and viscous numbers.

## Contribution

It introduces a comprehensive analysis of drag in granular beds considering inertial and viscous effects, providing empirical relations for effective friction and viscosity over a wide range of conditions.

## Key findings

- Drag decreases with fluid buoyancy at low speeds.
- Effective friction increases with speed, following a power law in viscous number.
- Effective viscosity scales with fluid viscosity and effective friction.

## Abstract

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 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 $\mu_e$ encountered over six orders of magnitude of speeds. While a constant $\mu_e$ 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 $\mu_e$ in terms of the inertial number $I$ and viscous number $J$ to understand the relative importance of inertia and viscous forces, respectively. For sufficiently large fluid viscosities, we find that the effect of varying the speed, depth, and viscosity can be described by the empirical function $\mu_e = \mu_o + k J^n$, where $\mu_o$ is the effective friction measured in the quasi-static limit, and $k$ and $n$ are material constants. The drag is then analyzed in terms of the effective viscosity $\eta_e$ and found to decrease systematically as a function of $J$. We further show that $\eta_e$ as a function of $J$ is directly proportional to the fluid viscosity and the $\mu_e$ encountered by the rod.