Drag force on a spherical intruder in a granular bed at low Froude number

The drag force on an object, or “intruder,” in a granular material arises from interparticle friction, as well as the cyclic creation and buckling of force chains within the material. In contrast to fluids, for which drag forces are well understood, there is no straightforward relationship between speed and mean drag force in granular materials. We investigate spherical intruder particles of varying radii moving at low speeds through granular beds. The system can be parametrized using the dimensionless Froude number $\text{Fr}=2v/\sqrt{gR}$, for intruders of radius $R$ moving at a speed $v$. For frictional systems, we find the drag force obeys a linear relationship with $\text{Fr}$ for low Froude numbers above $\text{Fr}>1$. For $\text{Fr}<1$ we observe a deviation from this linear trend. This transition can be explained by considering the characteristic inertial and gravitational granular time scales of the system. We show that a suitably normalized measure of dissipated power obeys a linear relationship with the imposed intruder velocity, independent of the intruder dimensions. This is found to hold even for particles with no friction, identifying a relationship between the imposed motion of the intruder and the resistance of the granular material to purely geometric rearrangements.

Figure 1

Schematic cutaway of the setup used (not shown to scale). A large spherical intruder particle of radius $R$ is moved through a bed of particles at an imposed speed along the $x$ axis, $0.6$ m above the base of the bed. The domain is $2\times 2\text{m}$ in the $x$ and $z$ directions, respectively, with a free surface approximately $1.3$ m above the base. Periodic boundary conditions are applied in the $x$ and $z$ directions. The lower boundary condition consists of a layer of fixed particles.

Figure 2

Cross section of particle bed showing particles shaded by speed $(\text{m}/\text{s})$ at 4 s into a simulation. The large central intruder particle is 100 mm in radius and has an imposed velocity of $1\text{m}/\text{s}$ along the $x$ axis. All particles have a friction angle of ${15}^{\circ }$. Periodic boundary conditions are applied in the $x$ and $z$ (not shown) directions.

Figure 3

Instantaneous force measured on an intruder particle 100 mm in diameter moving at $1\text{m}/\text{s}$ (gray line) and cumulative average force from 1 s onwards (black line). The inset shows an example of multiple stick-slip cycles occurring in the instantaneous drag force from 2 to $2.5$ s.

Figure 4

Mean nondimensionalized force in $x$ direction on intruder particle against intruder for a system with a friction angle of ${15}^{\circ }$ (upper) and ${30}^{\circ }$ (lower). The solid line is a best-fit spline. Four intruder particle radii are shown, $50,75,100$, and 125 mm.

Figure 5

Mean nondimensionalized force in $x$ direction on intruder particle against Froude number for a system with a friction angle of $0^{\circ }$. The solid line is a fit to a function of the form $\text{F}-\mathrm{const}\propto \sqrt{\text{Fr}}$. Four intruder particle radii are shown, 50, 75, 100, and 125 mm.

Figure 6

Normalized contact number $\kappa$ against Froude number for both frictionless particles and particles with friction angles $7.5^{\circ }$, ${15}^{\circ }$, and ${30}^{\circ }$. The solid line is a fit to the data of the form $\kappa =1+\frac{1}{5}\text{Fr}$.

Figure 7

Power dissipated per contact, $F\sqrt{gR}/n$, against imposed intruder speed $v$. The relationship is dependent on intergrain friction, but linear for all cases considered. The value at $v=0$ gives the minimum power per contact which must be maintained to allow motion of the intruder within the bed $\alpha$. The slope of the lines represents the constant effective frictional force per contact $\beta$. The zero friction case has both a nonzero intercept and slope, showing an effective frictional force in the absence of any intergrain friction.