Session RN: Granular Flows IV

Resistive Force Theory for Quasi-static Intruders

Resistive Force Theory was originally developed to calculate forces on slender filaments at low Reynolds number, and recently adapted to drag on cylindrical objects slowly moving through granular media in the horizontal direction. For low Reynolds numbers the contribution of the ends to the drag can be non-negligible, as can be the curvature of the filament. Here we present experimental results for the granular case, and observe surprising similarities between quasi-static granular media and viscous fluids.   

Drag induced lift in granular media

Laboratory experiments and numerical simulation reveal that a submerged intruder dragged horizontally at constant velocity within a granular medium can experience a vertical force, whose sign and magnitude depend on the shape of the intruder and the depth. Simulations show that the lift as well as drag are generated mainly by interaction with the leading surface. Comparing the stress on flat plates with different inclination angles with the surface stresses on the intruders indicates that shape dependent drag and lift can be understood as the sum of the contributions from differential (flat plate) elements. A model similar to Coulomb's wedge method is developed to describe the forces experienced by the flat plates.   

Lift and drag forces in washboard road

When a wheel of plow is dragged at a constant velocity on a granular bed, a ripple pattern known as washboard road forms if the velocity is above a critical value. Although much work has been recently devoted to this topic the underlying mechanisms remain unclear. We have studied the phenomenon using both an experimental setup consisting of a circular track on which a wheel or plow is dragged and 2D DEM simulations. Here we focus on the lift and drag forces exerted by the sand onto the wheel or plow. We found that these forces do not seem to depend on the velocity. We also found a linear relation between the lift and drag forces. These results are typical of static friction which is somewhat surprising considering the complexity of the granular flow advected by the wheel of plow. These results are a first step to the development of a stability analysis of washboard roads.   

Force generation during rotational intrusion into granular media

When legged locomotors move on granular media their limbs intrude into the substrate along paths more complicated than simple vertical or horizontal trajectories. To investigate force generation for paths more representative of typical limb-ground interaction, we rotated simple objects (plate, sphere, rod, c-shaped leg) into granular media and measured the resulting resistive force, $F,$ as a function of the angle, $\theta,$ from maximum penetration depth. For all objects, greatest $F$ occurs not at maximum depth ($\theta = 0$) as expected from the linear dependence of force on depth for vertical penetration, but substantially earlier ($\theta \approx -15^\circ \sim-30^\circ$). The location and magnitude of maximum $F$ depend on intruder geometry. For plate and rod, $F$ is primarily opposite displacement, while for sphere and c-shaped leg $F$ has a substantial inward radial component and is significantly larger than for plate and rod geometries with similar extent. Our data suggest that in granular media, larger yield stresses at fixed depth and with the same projected intruder area can be obtained by adjusting intruder geometry to maximize normal stress. This in turn provides hypotheses for locomotion biology and guidance for design of legged robots and other mobile devices.   

Wedge model of force and flow oscillations in plowed granular media

We develop a model that captures the changing response of granular media with volume fraction, $\phi,$ to a partially submerged vertical plate dragged horizontally at low velocity. In experiment, a bifurcation in force and flow occurs at the onset of grain dilatancy, $\phi_c.$ Below $\phi_c$ rapid irregular fluctuations in the drag force, $F_D$, are observed. Above $\phi_c$ fluctuations in $F_D$ are periodic and increase with $\phi.$ Velocity field measurements indicate $F_D$ fluctuations are correlated with the creation and destruction of shear bands during drag. Shear bands originate at the base of the plate and extend to the surface forming a nearly triangular wedge of material moving with the plate. Our model assumes that $F_D$ originates in the force required to overcome sliding friction and push the wedge of material up the slope defined by the inclination of the shear band. Combined with the fact that shear bands are weaker (stronger) than the bulk material for $\phi > \phi_c$ ($\phi<\phi_c$) our model quantitatively predicts the observed dependence of $F_D$ fluctuations and flow on time and $\phi$ for $\phi>\phi_c$ and gives significant insight into the non-periodic fluctuations observed for $\phi<\phi_c.$   

Time-Dependent Continuum and Molecular Dynamics Simulations of Density Inversion in Shaken Granular Layers

We investigate density inversion in vertically oscillated granular layers using continuum and molecular dynamics simulations. Layers of grains atop a plate that is shaken sinusoidally in the direction of gravity will leave the plate at some time in the cycle if the maximum acceleration of the plate $a_{max}$ exceeds the acceleration of gravity $g$. For some values of shaking frequency $f$ and accelerational amplitude $a_{max}$, a small region near the plate displays time-dependence in response to the sinusoidal shaking, while the bulk of the layer reaches a steady-state. In certain cases, the system exhibits a ``density inversion'' in which a low density granular gas supports a higher density layer of grains. We use three-dimensional simulations of time-dependent continuum equations as well as molecular dynamics simulations to study both the time-dependent and the steady-state regions of the flow.