Rheological Measurements in Liquid-Solid Flows

The behavior of liquid-solid flows varies greatly depending on fluid viscosity, particle and liquid inertia, and collisions between particles. While particle collisions in inviscid fluids can be understood statistically, liquid-solid flows are complicated by the fluid viscosity and forces acting on the particles (e.g. lift, drag, added mass). These flows were first studied by Bagnold, whose investigation found two different flow regimes: a macro-viscous regime where the shear and pressure forces are proportional to the shear rate, and a grain-inertia regime defined by a dependance on the square of the shear rate [1, 2]. The scaling relations he developed have been used to model and understand natural phenomena since

Investigation of f/2 and f/4 Waves in Granular Beds Subject to Vertical, Sinusoidal Oscillations

When a deep bed of granular material is subject to vertical, sinusoidal oscillations, a number of phenomena appear including two regimes of standing surface waves that form at one-half and one-quarter of the oscillation forcing frequency. These waves are referred to as f/2 and f/4 waves where f is the oscillation frequency. This paper presents the results from experiments and computer simulations designed to study the wavelength and wave amplitude dependence of the surface waves on the vibration parameters, collision coefficient of restriction, and the particle bed depth

Effects of vertical vibration on hopper flows of granular material

This paper examines the flow of granular material through a wedge-shaped hopper subject to vertical, sinusoidal oscillations. Experiments and discrete element computer simulations were conducted to investigate particle trajectories within and mass discharge rates from the hopper. With the hopper exit closed, side wall convection cells are observed in both the experiments and simulations. The convection cells are oriented such that particles move up along the inclined walls of the hopper and down along the centerline. Results from the computer simulation indicate that the convection cells are a result of the dilation of the granular bed during free fall and interaction with hopper walls. Measurements of the mean mass discharge rate for various vibration parameters were also made in both the experiments and simulations. The ratio of the mass discharge rate for a vibrating hopper to the mass discharge rate for a non-vibrating hopper scales with the oscillation velocity amplitude and exhibits a maximum value just greater than one for oscillation velocity amplitudes less than 0.5. The ratio is less than one for larger velocity amplitudes. A simple model taking into account the change in the effective gravity acting on the granular material over an oscillation cycle is examined. A significant deficiency in the model is that is assumes no material discharges from the hopper during part of each oscillation cycle for acceleration amplitudes greater than gravitational acceleration. Data from the simulations indicate that although the discharge rate from the hopper varies throughout an oscillation cycle, it never equals zero. The simulation was also used to examine particle horizontal position and velocity profiles at the hopper exit. Lastly, preliminary observations of the effects of localized vibration on a granular material in a closed hopper are presented

Rheological measurements of large particles in high shear rate flows

This paper presents experimental measurements of the rheological behavior of liquid-solid mixtures at moderate Stokes and Reynolds numbers. The experiments were performed in a coaxial rheometer that was designed to minimize the effects of secondary flows. By changing the shear rate, particle size, and liquid viscosity, the Reynolds numbers based on shear rate and particle diameter ranged from 20 to 800 (Stokes numbers from 3 to 90), which is higher than examined in earlier rheometric studies. Prior studies have suggested that as the shear rate is increased, particle-particle collisions also increase resulting in a shear stress that depends non-linearly on the shear rate. However, over the range of conditions that were examined in this study, the shear stress showed a linear dependence on the shear rate. Hence, the effective relative viscosity is independent of the Reynolds and Stokes numbers and a non-linear function of the solid fraction. The present work also includes a series of rough-wall experiments that show the relative effective viscosity is also independent of the shear rate and larger than in the smooth wall experiments. In addition, measurements were made of the near-wall particle velocities, which demonstrate the presence of slip at the wall for the smooth-walled experiments. The depletion layer thickness, a region next to the walls where the solid fraction decreases, was calculated based on these measurements. The relative effective viscosities in the current work are larger than found in low-Reynolds number suspension studies but are comparable with a few granular suspension studies from which the relative effective viscosities can be inferred

Effects of inertia and turbulence on rheological measurements of neutrally buoyant suspensions

For low-Reynolds-number shear flows of neutrally buoyant suspensions, the shear stress is often modelled using an effective viscosity that depends only on the solid fraction. As the Reynolds number (Re) is increased and inertia becomes important, the effective viscosity also depends on the Reynolds number itself. The current experiments measure the torque for flows of neutrally buoyant particles in a coaxial-cylinder rheometer for solid fractions, ϕ, from 10 % to 50 % and Reynolds numbers based on particle diameter from 2 to 1000. For experiments for Reynolds of O(10) and solid fractions less than 30%, the effective viscosity increases with Reynolds number, in good agreement with recent numerical simulations found in the literature. At higher solid fractions over the same range of Re, the results show a decrease in torque with shear rate. For Reynolds numbers greater than 100 and lower solids concentrations, the effective viscosity continues to increase with Reynolds number. However, based on comparisons with pure fluid measurements the increase in the measured effective viscosity results from the transition to turbulence. The particles augment the turbulence by increasing the magnitude of the measured torques and causing the flow to transition at lower Reynolds numbers. For the highest solid fractions, the measurements show a significant increase in the magnitude of the torques, but the effective viscosity is independent of Reynolds number

Shear Stress Measurements of Non-Spherical Particles in High Shear Rate Flows

The behavior of liquid-solid flows varies greatly depending on fluid viscosity; particle and liquid inertia; and collisions and near-collisions between particles. Shear stress measurements were made in a coaxial rheometer with a height to gap ratio (b/r0) of 11.7 and gap to outer radius ratio (h/b) of 0.166 that was specially designed to minimize the effects of secondary flows. Experiments were performed for a range of Reynolds numbers, solid fractions and ratio of particle to fluid densities. With neutrally buoyant particles, the dimensional shear stress exhibits a linear dependence on Reynolds number: the slope is monotonic but a non-linear function of the solid fraction. Though non-neutrally buoyant particles exhibit a similar linear dependence at higher Reynolds numbers, at lower values the shear stress exhibits a non-linear behavior in which the stress increases with decreasing Reynolds number due to particle settling

Reply to comment by B. Andreotti et al. on "Solving the mystery of booming sand dunes"

This reply addresses three main issues raised in the comment of Andreotti et al. [2008]. First, the turning of ray paths in a granular material does not preclude the propagation of body waves and the resonance condition described by Vriend et al. [2007]. The waveguide model still holds in the dune for the observed velocities, even with a velocity increase with depth as implied by Andreotti et al. [2008]. Secondly, the method of initiation of spontaneous avalanching does not influence the booming frequency. The frequency is independent of the source once sustained booming starts; it depends on the subsurface structure of the dune. Thirdly, if all data points from Vriend et al. [2007] are included in the analysis (and not an average or selection), no correlation is observed between the sustained booming frequency and average particle diameter

Solving the mystery of booming sand dunes

Desert booming can be heard after a natural slumping event or during a sand avalanche generated by humans sliding down the slip face of a large dune. The sound is remarkable because it is composed of one dominant audible frequency (70 to 105 Hz) plus several higher harmonics. This study challenges earlier reports that the dunes’ frequency is a function of average grain size by demonstrating through extensive field measurements that the booming frequency results from a natural waveguide associated with the dune. The booming frequency is fixed by the depth of the surficial layer of dry loose sand that is sandwiched between two regions of higher compressional body wave velocity. This letter presents measurements of the booming frequencies, compressional wave velocities, depth of surficial layer, along with an analytical prediction of the frequency based on constructive interference of propagating waves generated by avalanching along the dune surface

A contact model for normal immersed collisions between a particle and a wall

The incompressible Navier–Stokes equations are solved numerically to predict the coupled motion of a falling particle and the surrounding fluid as the particle impacts and rebounds from a planar wall. The method is validated by comparing the numerical simulations of a settling sphere with experimental measurements of the sphere trajectory and the accompanying flow field. The normal collision process is then studied for a range of impact Stokes numbers. A contact model of the liquid–solid interaction and elastic effect is developed that incorporates the elasticity of the solids to permit the rebound trajectory to be simulated accurately. The contact model is applied when the particle is sufficiently close to the wall that it becomes difficult to resolve the thin lubrication layer. The model is calibrated with new measurements of the particle trajectories and reproduces the observed coefficient of restitution over a range of impact Stokes numbers from 1 to 1000