Peter K. Haff

Grain flow as a fluid-mechanical phenomenon

The behaviour of granular material in motion is studied from a continuum point of view. Insofar as possible, individual grains are treated as the ‘molecules’ of a granular ‘fluid’. Besides the obvious contrast in shape, size and mass, a key difference between true molecules and grains is that collisions of the latter are inevitably inelastic. This, together with the fact that the fluctuation velocity may be comparable to the flow velocity, necessitates explicit incorporation of the energy equation, in addition to the continuity and momentum equations, into the theoretical description. Simple ‘microscopic’ kinetic models are invoked for deriving expressions for the ‘coefficients’ of viscosity, thermal diffusivity and energy absorption due to collisions. The ‘coefficients’ are not constants, but are functions of the local state of the medium, and therefore depend on the local ‘temperature’ and density. In general the resulting equations are nonlinear and coupled. However, in the limit s « d, where s is the mean separation between neighbouring grain surfaces and d is a grain diameter, the above equations become linear and can be solved analytically. An important dependent variable, in this formulation, in addition to the flow velocity u, is the mean random fluctuation (‘thermal’) velocity v of an individual grain. With a sufficient flux of energy supplied to the system through the boundaries of the container, v can remain non-zero even in the absence of flow. The existence of a non-uniform v is the means by which energy can be ‘conducted’ from one part of the system to another. Because grain collisions are inelastic, there is a natural (damping) lengthscale, governed by the value of d, which strongly influences the functional dependence of v on position. Several illustrative examples of static (u = 0) systems are solved. As an example of grain flow, various Couette-type problems are solved analytically. The pressure, shear stress, and ‘thermal’ velocity function v are all determined by the relative plate velocity U (and the boundary conditions). If v is set equal to zero at both plates, the pressure and stress are both proportional to U^2, i.e. the fluid is non-Newtonian. However, if sufficient energy is supplied externally through the walls (v ≠ 0 there), then the forces become proportional to the first power of U. Some examples of Couette flow are given which emphasize the large effect on the grain system properties of even a tiny amount of inelasticity in grain–grain collisions. From these calculations it is suggested that, for the case of Couette flow, the flow of sand is supersonic over most of the region between the confining plates.

Simulation of Eolian Saltation

Saltation is important in the transport of sand-sized granular material by wind and in the ejection of dust from the bed both on Earth and on Mars. The evolution of the saltating population and all its characteristic profiles is calculated from inception by pure aerodynamic entrainment through to steady state. Results of numerical simulations of single-grain impacts into granular beds are condensed into analytic expressions for the number and speeds of grains rebounding or rejected (splashed) from the bed. A model is combined with (i) this numerical representation, (ii) an expression for the aerodynamic entrainment rate, and (iii) the modification of the wind velocity profile by saltating grains. Calculated steady state mass fluxes are within the range of mass fluxes measured in wind tunnel experiments; mass flux is nonlinearly dependent on the shear velocity. Aerodynamically entrained grains in the system are primarily seeding agents; at steady state, aerodynamic entrainment is rare. The time for the entire system to reach steady state is roughly 1 second, or several long-trajectory hop times.

Wind modification and bed response during saltation of sand in air

A model of eolian sediment transport has been constructed, a special case of which is that corresponding to sand-sized mineral grains subjected to moderate winds: saltation. The model consists of four compartments corresponding to (1) aerodynamic entrainment, (2) grain trajectories, (3) grain-bed impacts, and (4) momentum extraction from the wind. Each sub-model encapsulates the physics of the process, and is constrained where necessary by experimental data. When combined, the full model allows simulation of eolian saltation from inception by aerodynamic entrainment to steady state.

Ion‐beam‐induced atomic mixing

Calculations based on the diffusion model are presented of atomic mixing by ion bombardment. This mixing is assumed to have its basis, as does sputtering, in the collision cascades generated by the primary beam. Sharp interfaces within a target are seen to be smoothed by ion bombardment. Mixing may place fundamental limits on the resolution of ion microprobes.

Boundary conditions for high-shear grain flows

Boundary conditions are developed for rapid granular flows in which the rheology is dominated by grain–grain collisions. These conditions are v_0=constdv_0/dy and u_0 = constdu_0/dy, where v and u are the thermal (fluctuation) and flow velocities respectively, and the subscript indicates that these quantities and their derivatives are to be evaluated at the wall These boundary conditions are derived from the nature of individual grain–wall collisions, so that the proportionality constants involve the appropriate coefficient of restitution ew for the thermal velocity equation, and the fraction of diffuse (i.e. non-specular) collisions in the case of the flow-velocity equation. Direct application of these boundary conditions to the problem of Couette-flow shows that as long as the channel width h is very large compared with a grain diameter d it is permissible to set v=0 at the wall and to adopt the no-slip condition. Exceptions occur where d/h is not very small, when the wall is not rough, and when the grain–wall collisions are very elastic. Similar insight into other flows can be obtained qualitatively by a dimensional analysis treatment of the boundary conditions. Finally, the more difficult problem of self-bounding fluids is discussed qualitatively.

The grain-bed impact process in aeolian saltation

SummaryWe report the results of impact experiments in which high velocity steel spheres (BBs) were directed against a loose bed of similar particles. The purpose of these experiments is to shed some light on the collision processes which occur when saltating sand grains driven by the wind strike the bed. The scattered particles fall into two categories: a single high energy rebound which scatters quasi-specularly, and a number of low energy recoils. The high energy rebound is identified with the “successive saltation” particle of Rumpel, and the low energy recoils are interpreted as creeping, or reptating particles. These observations provide information on the “splash function” of Ungar and Haff, which describes the response of a bed to grain impact and which plays a central role in the theory of saltation.

Computer simulation of the mechanical sorting of grains

Abstract A two-dimensional system of inelastic frictional disks all of equal diameter save one was studied by computer simulation. A single large disk was placed on the bottom of a container and covered by 30 smaller disks. When the container was agitated to induce a shear motion in the disk assembly, the large particle showed a tendency to rise toward the surface. This sorting process was mediated by shear-driven rotational motion, the large grain rolling up on top of neighboring small grains. The grain-grain friction coefficient μ is a critical parameter in this kind of sorting process, since if μ is too small, the large grain cannot get sufficient purchase to roll without slipping.

Possible new sputtering mechanism in track registering materials

The ’’ion explosion’’ model of track production in dielectric materials in investigated as a possible source of sputtered particles at high bombarding energies.

Ring and plasma: The enigmae of Enceladus

Abstract The E ring associated with the Kronian moon Enceladus has a lifetime of only a few thousand years against sputteringly by slow corotating O ions. The existence of the ring implies the necessity for a continuous supply of matter. Possible particle source mechanisms on Enceladus include meteoroidal impact ejection and geysering. Estimates of ejection rates of particulate debris following small meteoroid impact are on the order of 3 × 10−18 g cm−2 sec−1, more than an order of magnitude too small to sustain the ring. A geyser source would need to generate a droplet supply at a rate of approximately 10−16 g cm−2 sec− in order to account for a stable ring. Enceladus and the ring particles also directly supply both plasma and vapor to space via sputtering. The absence of a 60 eV plasma at the Voyager 2 Enceladus L-shell crossing, such as might have been expected from sputtering, cannot be explained by absorption and moderation of plasma ions by ring particles, because the ring is too diffuse. Evidently, the effective sputtering yield in the vicinity of Enceladus is on the order of, or smaller than, 0.4, about an order of magnitude less than the calculated value. Small scale surface roughness may account for some of this discrepancy.

A model for surface layer composition changes in sputtered alloys and compounds

Under the assumption that extracted beam energy is quickly shared among secondary cascade members of all masses, we present a model which accounts quantitatively for recently observed equilibrium surface enrichments in heavy atoms following ion bombardment of alloys and compounds. Assuming strong radiation‐driven diffusion, effects of which are directly observed, and given the time required to reach equilibrium, we can calculate the thickness Δx of the enriched layer. Alternatively, knowing Δx, a calculation of the equilibration time constant is possible.