Resistive force theory for sand swimming

We discuss a resistive force theory~\footnote{Maladen et. al, Science, \textbf{325}, 314, 2009} that predicts the ratio of forward speed to wave speed (wave efficiency, $\eta$) of the sandfish lizard as it swims in granular media of varying volume fraction $\phi$ using a sinusoidal traveling wave body motion. In experiment $\eta\approx0.5$ independent of $\phi$ and is intermediate between $\eta \approx 0.2$ for low $Re$ Newtonian fluid undulatory swimmers like nematodes and $\eta \approx 0.9$ for undulatory locomotion on a deformable surface. To predict $\eta$ in granular media, we developed a resistive force model which balances thrust and drag force over the animal profile. We approximate the drag forces by measuring the force on a cylinder (a ``segment'' of the sandfish) oriented at different angles relative to the displacement direction. The model correctly predicts that $\eta$ is independent of $\phi$ because the ratio of thrust to drag is independent of $\phi$. The thrust component of the drag force is relatively larger in granular media than in low $Re$ fluids, which explains why $\eta$ in frictional granular media is greater than in viscous fluids.