Paul B. Umbanhowar

An effective gravitational temperature for sedimentation.

The slow sedimentation of suspensions of solid particles in a fluid results in complex phenomena that are poorly understood. For a low volume fraction (φ) of particles, long-range hydrodynamic interactions result in surprising spatial correlations in the velocity fluctuations; these are reminiscent of turbulence, even though the Reynolds number is very low. At higher values of φ, the behaviour of sedimentation remains unclear; the upward back-flow of fluid becomes increasingly important, while collisions and crowding further complicate inter-particle interactions. Concepts from equilibrium statistical mechanics could in principle be used to describe the fluctuations and thereby provide a unified picture of sedimentation, but one essential ingredient—an effective temperature that provides a mechanism for thermalization—is missing. Here we show that the gravitational energy of fluctuations in particle number can act as an effective temperature. Moreover, we demonstrate that the high-φ behaviour is in fact identical to that at low φ, provided that the suspension viscosity and sedimentation velocity are scaled appropriately, and that the effects of particle packing are included.

Scaling and Dynamics of Sphere and Disk Impact into Granular Media

Direct measurements of the acceleration of spheres and disks impacting granular media reveal simple power law scalings along with complex dynamics which bear the signatures of both fluid and solid behavior. The penetration depth scales linearly with impact velocity while the collision duration is constant for sufficiently large impact velocity. Both quantities exhibit power law dependence on sphere diameter and density, and gravitational acceleration. The acceleration during impact is characterized by two jumps: a rapid, velocity-dependent increase upon initial contact and a similarly sharp depth-dependent decrease as the impacting object comes to rest. Examination of the measured forces on the sphere in the vicinity of these features leads to an experimentally based granular force model for collision. We discuss our findings in the context of recently proposed phenomenological models that capture qualitative dynamical features of impact but fail both quantitatively and in their inability to capture significant acceleration fluctuations that occur during penetration and which depend on the impacted material.

Sensitive dependence of the motion of a legged robot on granular media

Legged locomotion on flowing ground (e.g., granular media) is unlike locomotion on hard ground because feet experience both solid- and fluid-like forces during surface penetration. Recent bioinspired legged robots display speed relative to body size on hard ground comparable with high-performing organisms like cockroaches but suffer significant performance loss on flowing materials like sand. In laboratory experiments, we study the performance (speed) of a small (2.3 kg) 6-legged robot, SandBot, as it runs on a bed of granular media (1-mm poppy seeds). For an alternating tripod gait on the granular bed, standard gait control parameters achieve speeds at best 2 orders of magnitude smaller than the 2 body lengths/s (≈60 cm/s) for motion on hard ground. However, empirical adjustment of these control parameters away from the hard ground settings restores good performance, yielding top speeds of 30 cm/s. Robot speed depends sensitively on the packing fraction φ and the limb frequency ω, and a dramatic transition from rotary walking to slow swimming occurs when φ becomes small enough and/or ω large enough. We propose a kinematic model of the rotary walking mode based on generic features of penetration and slip of a curved limb in granular media. The model captures the dependence of robot speed on limb frequency and the transition between walking and swimming modes but highlights the need for a deeper understanding of the physics of granular media.

Mechanical models of sandfish locomotion reveal principles of high performance subsurface sand-swimming

We integrate biological experiment, empirical theory, numerical simulation and a physical model to reveal principles of undulatory locomotion in granular media. High-speed X-ray imaging of the sandfish lizard, Scincus scincus, in 3 mm glass particles shows that it swims within the medium without using its limbs by propagating a single-period travelling sinusoidal wave down its body, resulting in a wave efficiency, η, the ratio of its average forward speed to the wave speed, of approximately 0.5. A resistive force theory (RFT) that balances granular thrust and drag forces along the body predicts η close to the observed value. We test this prediction against two other more detailed modelling approaches: a numerical model of the sandfish coupled to a discrete particle simulation of the granular medium, and an undulatory robot that swims within granular media. Using these models and analytical solutions of the RFT, we vary the ratio of undulation amplitude to wavelength (A/λ) and demonstrate an optimal condition for sand-swimming, which for a given A results from the competition between η and λ. The RFT, in agreement with the simulated and physical models, predicts that for a single-period sinusoidal wave, maximal speed occurs for A/λ ≈ 0.2, the same kinematics used by the sandfish.

Friction-Induced Velocity Fields for Point Parts Sliding on a Rigid Oscillated Plate

We show that small-amplitude periodic motion of a rigid plate causes point parts in frictional contact with the plate to move as if they are in a position-dependent velocity field. Further, we prove that every periodic plate motion maps to a unique velocity field. By allowing a plate to oscillate with six degrees of freedom, we can create a large family of programmable velocity fields. We examine in detail the class of plate motions described by sinusoidal linear and angular accelerations with a single frequency. We hypothesize that this simple class can generate all velocity fields that have constant and linear terms with respect to position, as well as some quadratic fields with respect to position. This set includes fields with isolated sinks and squeeze lines that can be used to perform tasks such as sensorless part orientation. Several of these fields have been verified on our programmable parts-feeding oscillatory device (PPOD). The PPOD is a parallel manipulator similar to a Stewart platform, but with flexures as joints. An iterative learning control algorithm is described that moves the platform with the six-degree-of-freedom periodic motion that creates the desired velocity field.

Force and flow transition in plowed granular media

We use plate drag to study the response of granular media to localized forcing as a function of volume fraction ϕ. A bifurcation in the force and flow occurs at the onset of dilatancy ϕc. Below ϕc rapid fluctuations in the drag force F(D) are observed. Above ϕc fluctuations in F(D) are periodic and increase in magnitude with ϕ. Velocity field measurements indicate that the bifurcation in F(D) results from the formation of stable shear bands above ϕc which are created and destroyed periodically during drag. A friction-based wedge flow model captures the dynamics for ϕ>ϕc.

Granular impact and the critical packing state

Impact dynamics during collisions of spheres with granular media reveal a pronounced and nontrivial dependence on volume fraction ϕ. Postimpact crater morphology identifies the critical packing state ϕcps, where sheared grains neither dilate nor consolidate, and indicates an associated change in spatial response. Current phenomenological models fail to capture the observed impact force for most ϕ; only near ϕcps is force separable into additive terms linear in depth and quadratic in velocity. At fixed depth the quadratic drag coefficient decreases (increases) with depth for ϕϕcps). At fixed low velocity, depth dependence of force shows a Janssen-type exponential response with a length scale that decreases with increasing ϕ and is nearly constant for ϕ>ϕcps.

Periodic, aperiodic, and transient patterns in vibrated granular layers

Experiments on vertically vibrated granular layers in evacuated containers reveal a variety of patterns for acceleration amplitudes above a critical value (≈2.5g). Stripes, squares, hexagons, spirals, triangles, and targets, as well as particle-like localized excitations (“oscillons”) and fronts (“kinks”) between regions with different vibrational phase are observed as the layer depth and the container oscillation frequency and amplitude are varied. A zig-zag instability, unstable hexagons, phase-disordered patterns, and “two-phase” squares are also observed. With a few noteworthy exceptions, the patterns are essentially independent of the lateral boundary conditions.

MR Imaging of Reynolds Dilatancy in the Bulk of Smooth Granular Flows

Dense granular matter has to expand in order to flow, a phenomenon known as dilatancy. Here we perform, by means of Magnetic Resonance Imaging (MRI), direct measurements of the evolution of the local packing density of a slow and smooth granular shear flow generated in a split-bottomed geometry. The degree of dilatancy is found to be surprisingly strong. For flows without appreciable transient, the dilated zone follows the region of large strain rate, while for flows with a strong transient, the dilated zone extends also into the region where transient flow took place. In all cases, the dilated zone slowly spreads as a function of time. These findings suggest that the local packing density is governed by the total amount of local strain experienced since the start of the experiment.

Stratification, segregation, and mixing of granular materials in quasi-two-dimensional bounded heaps.

Segregation and mixing of granular mixtures during heap formation has important consequences in industry and agriculture. This research investigates three different final particle configurations of bidisperse granular mixtures--stratified, segregated and mixed--during filling of quasi-two-dimensional silos. We consider a large number and wide range of control parameters, including particle size ratio, flow rate, system size, and heap rise velocity. The boundary between stratified and unstratified states is primarily controlled by the two-dimensional flow rate, with the critical flow rate for the transition depending weakly on particle size ratio and flowing layer length. In contrast, the transition from segregated to mixed states is controlled by the rise velocity of the heap, a control parameter not previously considered. The critical rise velocity for the transition depends strongly on the particle size ratio.