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Abstract
Despite the great relevance of particle dynamics in non-Newtonian fluids, particles settling in non-Newtonian fluids has received limited scientific attention. The simple case of two vertically aligned spheres settling in a thixotropic xanthan fluid is considered using solutions of xanthan + glycerol + water and 0.75 mm radius stainless steel spheres. Chaining phenomenon is observed where the trailing sphere velocity is increased due to a corridor of reduced viscosity created by the leading sphere. Our experimental methods are first validated against a Newtonian fluid of glycerol + water in which the sphere velocities remained equal. Using the fluid velocity induced by a single sphere, the two sphere velocities are approximated by Faxén's first law. In a non-Newtonian (thixotropic) xanthan fluid, a viscosity ratio is introduced to account for the reduced fluid viscosity encountered by the trailing sphere due to the slow structure recovery of the fluid after shear thinning. The viscosity ratio is an exponential decay function dependent on the time between spheres normalised by the fluid recovery time, a parameter which is experimentally determined. When the sphere-sphere separation is large, the viscosity ratio tends towards unity. The predicted settling velocities show good agreement with experimental data.
Highlights
Particle suspensions are ubiquitous in many industrial processes and the unique flow properties of non-Newtonian fluids may be advantageously exploited to process ‘difficult’ particles
We assume (i) the leading sphere velocity is equal to the single sphere velocity before interactions are accounted for, as is the case for a Newtonian fluid, (ii) the trailing sphere velocity can be multiplied by a before hydrodynamic interactions are calculated, to account for the reduced viscosity the trailing sphere experiences due to thixotropy (Arigo and McKinley [9] found that for spheres of different size and density settling in a viscoelastic fluid exhibiting a negative wake, the normalised VZ/VS against z/r curves collapsed to a single curve [9]), and (iii) the fluid viscosity decrease due to shear thinning is not significant
The calculated and measured velocities for the leading sphere are remarkably similar; within 1 standard deviation of the experimental data when d/r > 7. This suggests that the fluid viscosity decrease, due to shear thinning of the xanthan fluid induced by the settling of a r = 0.75 mm stainless steel sphere, is small even at small sphere-sphere separations
Summary
Particle suspensions are ubiquitous in many industrial processes and the unique flow properties of non-Newtonian fluids may be advantageously exploited to process ‘difficult’ particles. Time-dependent effects can be important in non-Newtonian fluids It has been noted in several studies that the settling velocity of a sphere depends on the history of the fluid. Daugan et al [3] studied the settling of two vertically aligned spheres in xanthan + glycerol + water solutions of varying concentrations and described the settling velocity using the Newtonian equation (V1 = V2 = VS (1 + 3/2 × r/d)), where the Stokes settling velocity VS was replaced by an experimental settling velocity, and the trailing sphere velocity (V2) increase was described by introducing a multiplication factor (V2 = 1.2 × V1 for all fluids). The authors noted that 87% of the experimental data was within 20% of the calculated velocity They suggested that errors in calculating average velocities over a smaller distance resulted in Journal of Non-Newtonian Fluid Mechanics 271 (2019) 104146 higher uncertainties at shorter time intervals [5]. Experimental data of two spheres settling in xanthan + glycerol + water solutions is used for validation of the method
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