Ashish V. Orpe

Surface flow of granular materials: model and experiments in heap formation

Granular surface flows are important in industrial practice and natural systems, but the understanding of such flows is at present incomplete. We present a combined theoretical and experimental study of quasi-two-dimensional heap formation by pouring particles continuously at a point. Two cases are considered: open systems and closed systems. Experimental results show that the shear rate in the flowing layer is nearly independent of the mass flow rate, and the angle of static friction at the bed-layer interface increases with flow rate. Predictions of the model for the flowing layer thickness and interface angles are in good agreement with experiments.

Continuum model of mixing and size segregation in a rotating cylinder: concentration-flow coupling and streak formation

The effect of segregation and concentration-flow coupling on structure development in binary mixtures of different sized particles (S-systems) in a rotating cylinder is studied. The system is a prototype of tumbling mixers widely used in industry for mixing, coating and reaction. Experiments with S-systems have shown the formation of radial streaks of the small particles when the size ratio is large; however, an explanation of this phenomenon is not available. A continuum model is presented here for the flow in the layer using mass, momentum and species balance equations averaged across the layer. The stress is assumed to be a sum of the Bagnold stress and the Coulomb frictional stress; the temperature and total solids volume fraction are assumed to be uniform across the layer. We consider the case of a large difference in particle sizes so that segregation upon flow is instantaneous and a step concentration profile exists at all points in the flowing layer with the smaller particles forming the lower layer. The velocity profile is assumed to be piecewise linear with continuity of stress at the interface between the small and large particles. The model predicts the time varying velocity, layer thickness and concentration fields in the system. The predictions are compared to experimental flow visualization studies. Conditions for the formation of streaks are investigated.

Segregation of granular materials in rotating cylinders

Mixtures of flowing granular materials containing particles of different sizes or different densities have a tendency to segregate. We focus here on the segregation that occurs in the cross-section of the cylinder using short cylinders (axial length is much smaller than the diameter) ensuring small variation along the cylinder axis. The typical structure formed in these systems is a radial core of the smaller or more dense particles. Coupling of composition with flow can lead to instability causing a pattern of radial streaks. The predictions of hard sphere mixture theories are briefly discussed first. Results of experimental and modelling studies of segregation are reviewed for density-induced radial segregation of equal sized particles, and size-induced radial streak formation. Experimental results for size-induced radial segregation are presented.

Rheology of surface granular flows

Surface granular flow, comprising granular material flowing on the surface of a heap of the same material, occurs in several industrial and natural systems. The rheology of such a flow was investigated by means of measurements of velocity and number-density profiles in a quasi-two-dimensional rotating cylinder, half-filled with a model granular material - monosize spherical stainless-steel particles. The measurements were made at the centre of the cylinder, where the flow is fully developed, using streakline photography and image analysis. The stress profile was computed from the number-density profile using a force balance which takes into account wall friction. Mean-velocity and root-mean-square (r.m.s.)-velocity profiles are reported for different particle sizes and cylinder rotation speeds. The profiles for the mean velocity superimpose when distance is scaled by the particle diameter d and velocity by a characteristic shear rate γ C = [g sin(β m - β s )/d cos β 3 ] 1/2 and the particle diameter, where β m is the maximum dynamic angle of repose and β s is the static angle of repose. The maximum dynamic angle of repose is found to vary with the local flow rate. The scaling is also found to work for the r.m.s. velocity profiles. The mean velocity is found to decay exponentially with depth in the bed, with decay length λ = 1.1d. The r.m.s. velocity shows similar behaviour but with λ = 1.7d. The r.m.s. velocity profile shows two regimes: near the free surface the r.m.s. velocity is nearly constant and below a transition point it decays linearly with depth. The shear rate, obtained by numerical differentiation of the velocity profile, is not constant anywhere in the layer and has a maximum which occurs at the same depth as the transition in the r.m.s. velocity profile. Above the transition point the velocity distributions are Gaussian and below the transition point the velocity distributions gradually approach a Poisson distribution. The shear stress increases roughly linearly with depth. The variation in the apparent viscosity η with r.m.s. velocity u shows a relatively sharp transition at the shear-rate maximum, and in the region below this point the apparent viscosity η ∼ u -1.5 . The measurements indicate that the flow comprises two layers: an upper low-viscosity layer with a nearly constant r.m.s. velocity and a lower layer of increasing viscosity with a decreasing r.m.s. velocity. The thickness of the upper layer depends on the local flow rate and is independent of particle diameter while the reverse is found to hold for the lower-layer thickness. The experimental data is compared with the predictions of three models for granular flow.

Erosion of a granular bed driven by laminar fluid flow

Motivated by examples of erosive incision of channels in sand, we investigate the motion of individual grains in a granular bed driven by a laminar fluid to give us new insights into the relationship between hydrodynamic stress and surface granular flow. A closed cell of rectangular cross-section is partially filled with glass beads and a constant fluid flux Q flows through the cell. The refractive indices of the fluid and the glass beads are matched and the cell is illuminated with a laser sheet, allowing us to image individual beads. The bed erodes to a rest height h r which depends on Q . The Shields threshold criterion assumes that the non-dimensional ratio θ of the viscous stress on the bed to the hydrostatic pressure difference across a grain is sufficient to predict the granular flux. Furthermore, the Shields criterion states that the granular flux is non-zero only for θ > θ c . We find that the Shields criterion describes the observed relationship h r ∝ Q 1/2 when the bed height is offset by approximately half a grain diameter. Introducing this offset in the estimation of θ yields a collapse of the measured Einstein number q * to a power-law function of θ − θ c with exponent 1.75 ± 0.25. The dynamics of the bed height relaxation are described well by the power-law relationship between the granular flux and the bed stress.

Solid-fluid transition in a granular shear flow

The rheology of a granular shear flow is studied in a quasi-2D rotating cylinder. Measurements are carried out near the midpoint along the length of the surface flowing layer where the flow is steady and nonaccelerating. Streakline photography and image analysis are used to obtain particle velocities and positions. Different particle sizes and rotational speeds are considered. We find a sharp transition in the apparent viscosity (eta) variation with rms velocity (u). Below the transition depth we find that the rms velocity decreases with depth and eta proportional to u(-1.5) for all the different cases studied. The material approaches an amorphous solidlike state deep in the layer. The velocity distribution is Maxwellian above the transition point and a Poisson velocity distribution is obtained deep in the layer. The results indicate a sharp transition from a fluid to a fluid + solid state with decreasing rms velocity.

SURFACE GRANULAR FLOWS: TWO RELATED EXAMPLES

Granular surface flows are common in industrial practice and natural systems, however, theoretical description of such flows is at present incomplete. Two prototype systems involving surface flow are compared: heap formation by pouring at a point and rotating cylinders. Continuum models for analysis of these flows are reviewed, and experimental results for quasi-2D systems are presented. Experimental results in both systems are well described by continuum models.

Velocity correlations in dense granular flows observed with internal imaging.

We show that the velocity correlations in uniform dense granular flows inside a silo are similar to the hydrodynamic response of an elastic hard-sphere liquid. The measurements are made using a fluorescent refractive-index-matched interstitial fluid in a regime where the flow is dominated by grains in enduring contact and fluctuations scale with the distance traveled, independent of flow rate. The velocity autocorrelation function of the grains in the bulk shows a negative correlation at short time and slow oscillatory decay to zero similar to simple liquids. Weak spatial velocity correlations are observed over several grain diameters. The mean square displacements show an inflection point indicative of caging dynamics. The observed correlations are qualitatively different at the boundaries.

Friction of a slider on a granular layer: nonmonotonic thickness dependence and effect of boundary conditions.

We investigate the effective friction encountered by a mass sliding on a granular layer as a function of bed thickness and boundary roughness conditions. The observed friction has minima for a small number of layers before it increases and saturates to a value that depends on the roughness of the sliding surface. We use an index-matched interstitial liquid to probe the internal motion of the grains with fluorescence imaging in a regime where the liquid has no significant effect on the measured friction. The shear profiles obtained as a function of depth show a decrease in slip near the sliding surface as the layer thickness is increased. We propose that the friction depends on the degree of grain confinement relative to the sliding surfaces.

Lubrication effects on the flow of wet granular materials

We investigate the dynamics of a partially saturated grain-liquid mixture with a rotating drum apparatus. The drum is partially filled with the mixture and then rotated about its horizontal axis. We focus on the continuous avalanching regime and measure the impact of the volume fraction and viscosity of the liquid on the dynamic surface angle. The inclination angle of the surface is observed to increase sharply to a peak and then decrease as a function of liquid volume fraction. The height of the peak is observed to increase with rotation rate. For higher liquid volume fractions, the inclination angle of the surface can decrease with viscosity before increasing. The viscosity where the minimum occurs decreases with the rotation rate of the drum. Limited measurements of the flow depth were made, and these were observed to show only fractional changes with volume fraction and rotation speeds. We show that the qualitative features of our observations can be understood by analyzing the effect of lubrication forces on the time scale over which particles come in contact.