(105a) Two Sphere Aggregation in a Shear Thinning Fluid

Particulate flows occur throughout nature and are encountered in all industries. It is important to understand how particles migrate in liquids since this has a consequence on the transport and processing of particle suspensions. A simple system of two spheres settling at low particle Reynolds in a Newtonian fluid is well described. The method requires the calculation of forces acting on each sphere, including the contribution from gravity, and the force associated with the second sphere, which can be determined by the Stokes stream function.

Unlike Newtonian liquids, non-Newtonian liquids provide shear dependent fluid properties which create opportunity to transport some of the more challenging particulate suspensions, i.e. dense phase transport of nuclear sludge particulates. The shear responsive viscosity of a non-Newtonian fluid potentially allows for particle resuspension under high shear, and particle stability under low shear. However, it has been observed that the low Reynolds settling of spheres in a shear thinning fluid leads to the formation of particle chains, with particle aggregation shown for a minimum of two spheres. This is in contrast to the low Reynolds settling of two spheres in a Newtonian fluid, where the particles settle at an equivalent velocity and maintain a constant particle-particle separation.

Previous research has attributed the particle aggregation to a “corridor of reduced viscosity” which forms behind the leading sphere, causing the trailing sphere to experience a reduced viscosity allowing faster settling to occur, eventually leading to particle aggregation. Despite the importance of particle settling in non-Newtonian fluids, limited work has been completed. Attempts to predict the sphere velocities have been made by empirical fits. The current study aims to replicate the methodology applied to determine the settling velocities of spheres in Newtonian fluids and extend the approach to non-Newtonian fluids.

The fluid velocity around a 1.5 mm stainless steel ball bearing settling in a 50 mm × 50 mm × 300 mm square glass column (Vitrocom) was visualised using Particle Imaging Velocimetry (PIV, Dantec Dynamics). A DualPower 65-15 laser was used to illuminate 1-20 μm PPMA-Rhodamine B tracer particles (25 mg in 100 μL water per 1 L solution). A FlowSense EO 2M Camera captured the spheres settling and the images were processed using the DynamicStudio software to produce vector maps. These vector maps were further processed to elucidate the velocity profile along the line of particle motion. The settling velocities of the two spheres at different separations were also calculated directly from the location of the sphere during settling. Using a Point Grey Chameleon3 camera at 30 fps and a custom MATLAB® script, the spheres were identified using peak analysis and the sphere settling velocity determined using locations over 9 sequential images.

The Newtonian method was validated using a 95 wt% glycerol + 5 wt% water solution. The experimental fluid velocity decayed more rapidly than that predicted by the Stokes stream function velocity. The fluid velocity given by Stokes was half the sphere velocity (V = 0.5V_S) at a distance of 2.88 sphere radii, whereas the PIV data decayed to the same sphere velocity in 2.36 sphere radii, although this was attributed to the finite column size and non-zero Reynolds number. A DigiDEMâ„¢ simulation was executed with parameters equivalent to the experiment. The resulting velocity profile was in good agreement with the experimental velocity profile. Using the PIV fluid velocity profiles, sphere velocities at different separations were calculated and compared to experimentally determined velocities. There was no significant difference with the calculated velocities within the scatter of the experimental data.

A TA Instruments DHR-2 rheometer was used to characterise the flow properties of a 0.25 wt% xanthan + 60 wt% glycerol + 40 wt% water solution. The prepared solution was found to be highly shear thinning, viscoelastic and exhibited a long recovery time, but low thixotropy. A steady state stress sweep showed a flow curve which could be described by the Carreau model. A stress hold test was used to measure the recovery of the fluid after shearing the fluid at 10 Pa for 2 s which mimicked the settling of a sphere past a single point. Fitting the data using an exponential decay, the time constant of the fluid was found to be 173 s.

Since the Newtonian fluid method does not account for the time dependent viscosity, a viscosity ratio was included in the gravitational force term. Using the fluid velocity around a single sphere in a non-Newtonian fluid, the sphere velocities were calculated using the new method (non-Newtonian fluid) for different particle separations. These calculated velocities were in good agreement with the experimentally determined sphere velocities.

The new method for non-Newtonian fluids uses i) the single sphere velocity, ii) doublet velocity, iii) fluid velocity profile around a single sphere and iv) the time dependent viscosity to successfully predict the settling velocity of two spheres settling along their line of centres in a shear thinning fluid. These four parameters are much easier to obtain than completing a number of settling experiments at different particle separations. Furthermore, it has been shown that the fluid velocity profile scales to the sphere radius and density, so this method may be extended to predict the velocities of different sized spheres with little additional work. This is particularly promising as most particulate suspensions in industry are non-uniform, and there is very little research published which considers the settling of different sized spheres.