James T. Jenkins

Grad's 13-Moment System for a Dense Gas of Inelastic Spheres

Recent theories for rapid deformations of granular materials have attempted to exploit the similarities between the grains of deforming granular mass and the molecules of a disequilibrated gas. Methods from the kinetic theory may then be used to determine, for example, the form of the balance laws for the means of density, velocity, and energy and to calculate specific forms for the mean fluxes of momentum and energy and, in these dissipative systems, the mean rate at which energy is lost in collisions.

Kinetic theory for plane flows of a dense gas of identical, rough, inelastic, circular disks

Grad’s method of moments is employed to derive balance laws and constitutive relations for plane flows of a dense gas consisting of identical, rough, inelastic, circular disks. Two temperatures are involved; these are proportional to the kinetic energies associated with fluctuations in translational velocity and spin, respectively. When the single particle velocity distribution function is assumed to be close to a two‐temperature Maxwellian, two distinct theories are obtained. One applies when the particles are almost smooth and the collisions between them are nearly elastic; the other applies to nearly elastic particles that, in a collision, almost reverse the relative velocity of their points of contact. I both cases energy is nearly conserved in collisions.

The role of particle collisions in pneumatic transport

We analyse the dilute, steady, fully developed flow of relatively massive particles in a turbulent gas in the context of a vertical pipe. The idea is that the exchange of momentum in collisions between the grains and between the grains and the wall plays a significant role in the balance of forces in the particle phase. Consequently, the particle phase is considered to be a dilute system of colliding grains, in which the velocity fluctuations are produced by collisions rather than by the gas turbulence. The balance equations for rapid granular flow are modified to incorporate the drag force from the gas, and boundary conditions, based on collisional exchanges of momentum and energy at the wall, are employed. The turbulence of the gas is treated using a one-equation closure. A numerical solution of the resulting governing equations provides velocity and turbulent energy profiles in agreement with the measurements of Tsuji et al. (1984).

Kinetic theory for binary mixtures of smooth, nearly elastic spheres

Existing Chapman–Enskog solution procedures for binary mixtures of hard, perfectly elastic spheres are extended to hard, slightly dissipative spheres, and the associated constitutive relations are calculated. Then a steady, homogeneous shear flow is analyzed and the behavior of the mixture viscosity is determined as the diameter ratio, volume ratio, and total volume fraction are varied.

On two-phase sediment transport: sheet flow of massive particles

A model is presented for concentrated sediment transport that is driven by strong, fully developed turbulent shear flows over a mobile bed. Balance equations for the average mass, momentum and energy for the two phases are phrased in terms of concentration–weighted (Favre averaged) velocities. Closures for the correlations between fluctuations in concentration and particle velocities are based on those for collisional grain flow. This is appropriate for particles that are so massive that their fall velocity exceeds the friction velocity of the turbulent fluid flow. Particular attention is given to the slow flow in the region of high concentration above the stationary bed. A failure criterion is introduced to determine the location of the stationary bed. The proposed model is solved numerically with a finite–difference algorithm in both steady and unsteady conditions. The predictions of sediment concentration and velocity are tested against experimental measurements that involve massive particles. The model is further employed to study several global features of sheet flow such as the total sediment transport rate in steady and unsteady conditions.

Collisional sheet flows of sediment driven by a turbulent fluid

We consider a sheet flow in which heavy grains near a packed bed interact with a unidirectional turbulent shear flow of a fluid. We focus on sheet flows in which the particles are supported by their collisional interactions rather than by the velocity fluctuations of the turbulent fluid and introduce what we believe to be the simplest theory for the collisional regime that captures its essential features. We employ a relatively simple model of the turbulent shearing of the fluid and use kinetic theory for the collisional grain flow to predict profiles of the mean fluid velocity, the mean particle velocity, the particle concentration, and the strength of the particle velocity fluctuations within the sheet. These profiles are obtained as solutions to the equations of balance of fluid and particle momentum and particle fluctuation energy over a range of Shields parameters between 0.5 and 2.5. We compare the predicted thickness of the concentrated region and the predicted features of the profile of the mean fluid velocity with those measured by Sumer et al. (1996). In addition, we calculate the volume flux of particles in the sheet as a function of Shields parameter. Finally, we apply the theory to sand grains in air for the conditions of a sandstorm and calculate profiles of particle concentration, velocity, and local volume flux.

Balance Laws and Constitutive Relations for Plane Flows of a Dense, Binary Mixture of Smooth, Nearly Elastic, Circular Disks

We derive balance laws and constitutive relations for plane flows of a dense, binary mixture of smooth, nearly elastic, circular disks. The disks may differ in size and mass and in the coefficients of restitution characterizing the energy lost in collisions between like and unlike pairs. We focus attention on those parts of the fluxes and sources of momentum and energy that are due to collisions. To calculate them, we suppose that the complete pair distribution function for two colliding disks is the product of Maxwellian velocity distributions for each disk and a factor that incorporates the effects of excluded area and collisional shielding. In an Appendix we provide constitutive relations calculated in a similar way for a dense, binary mixture of smooth, nearly elastic, spheres.

Kinetic theory for identical, frictional, nearly elastic spheres

We derive a simple kinetic theory for collisional flows of identical, slightly frictional, nearly elastic spheres that is based on a physically realistic model for a frictional collision between two spheres. When the coefficient of friction is small, the equations of balance for rotational momentum and energy can be solved in approximation. This permits the rotational temperature to be related to the translation temperature and the introduction of an effective coefficient of restitution in the rate of dissipation of translation fluctuation energy. With this incorporation of the additional loss of translational energy to friction and the rotational degrees of freedom, the structure of the resulting theory is the same as for frictionless spheres.

Static equilibrium configurations of a model red blood cell.

SummaryThe membrane of the red blood cell is modeled as a fluid shell which resists bending and changes in area. The differential equations governing the mechanical equilibrium of such a membrane are derived and axisymmetric solutions are obtained numerically.

Boundary Conditions for Rapid Granular Flow: Flat, Frictional Walls

We employ Coulomb friction and both tangential and normal restitution in a model for a collision between a homogeneous sphere and a flat wall. We calculate the impulse and change in kinetic energy in typical collisions and use a particularly simple velocity distribution function to obtain the rates at which momenta and energy are supplied to the flow over a unit area of the wall. From these, we determine boundary conditions that relate the shear stress and energy flux in the flow at the wall to the normal stress, slip velocity, and fluctuation energy and to the parameters that characterize a collision.