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Hindered settling velocity and microstructure in suspensions of solid spheres with moderate Reynolds numbers
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Image of FIG. 1.
FIG. 1.

The drag on a simple cubic array of spheres at moderately small Reynolds numbers. The drag is normalized by the Stokes drag and is compared to Sangani and Acrivos’s prediction (dashed line): .39 The squares represent , the upward triangles represent , the downward triangles represent .

Image of FIG. 2.
FIG. 2.

Hindered settling velocities in small Re suspensions. (a) , ; (b) , ; (c) and (d) are the enlarged views of (a) and (b) in the dilute limit. The open symbols were obtained with and the filled symbols were obtained with . The solid lines correspond to Richardson and Zaki’s power-law Eq. (2); the dash-dot lines correspond to Garside and Al-Dibouni’s correlation Eq. (3); the dashed lines are the best power-law fits using and from Table IV. The dotted lines correspond to Batchelor’s asymptote of for dilute low Re suspensions.47

Image of FIG. 3.
FIG. 3.

Hindered settling velocities in suspensions with higher Reynolds numbers. (a) and ; (b) and ; (c) and ; (d) an enlarged view of the dilute regime. The meanings of the symbols and lines follow the definitions in Fig. 2.

Image of FIG. 4.
FIG. 4.

Hindered settling velocities as functions of on a logarithmic scale. (a) [ and 40.0 ]; (b) [ ), 319 , and 815 ]. The lines are the best linear fits yielding the values of and presented in Table IV. All data were obtained from periodic unit cells with .

Image of FIG. 5.
FIG. 5.

and in power laws obtained from simulations. (a) the power-law exponent as a function of Re; (b) the prefactor as a function of Re. The solid and dashed lines in (a) correspond to based on Eq. (2) and based on Eq. (3).

Image of FIG. 6.
FIG. 6.

(Color online) The pair probability density distributions and structure factors in suspensions with . (a), (c), and (e) show the pair probability in dilute , intermediate , and concentrated suspensions; (b), (d), and (f) show the corresponding structure factors, with error bars representing 90% confidence intervals. The triangles represent and the squares represent . The solid lines in the plots of structure factors are for hard-sphere suspensions.54 The results for dilute suspensions [(a) and (b)] were obtained in systems of size . The rest of the simulations were conducted with .

Image of FIG. 7.
FIG. 7.

(Color online) The pair probability density distributions and structure factors in suspensions with . The volume fractions, from top to bottom, are , 0.05, and 0.20. For the definition of symbols and lines, as well as the information on the system size, see the caption of Fig. 6.

Image of FIG. 8.
FIG. 8.

The radial distributions of settling spheres in suspensions with (a) and ; (b) and . The triangles represent , the diamonds represent , and the squares represent .

Image of FIG. 9.
FIG. 9.

The order parameters in settling suspensions. (a) and ; (b) and . The symbols have the same meanings as in Fig. 8.

Image of FIG. 10.
FIG. 10.

A qualitative view of the velocity field around a sphere settling with finite Re and the interaction between a pair of spheres. The fluid velocity due to sphere 1 in the absence of a disturbance due to the other sphere is sketched. It consists of a wake behind the sphere and a radial source flow in other directions. A sphere located at 2 would be in the wake of 1 and thus would be attracted toward 1. At the same time, a lift force would act to push sphere 2 horizontally outward. If the lift force is insufficient to avoid a collision, spheres 1 and 2 would experience a torque orienting them in the horizontal direction. Horizontally oriented spheres (such as at location 3) repel due to the source flow.

Image of FIG. 11.
FIG. 11.

A sequence of images taken from a simulation showing the interaction between a pair of solid spheres, which are painted black, in a suspension of other particles painted gray. The images are displayed in a reference frame moving with the leading sphere. The simulation parameters are , , , and .


Generic image for table
Table I.

The effective hydrodynamic sphere diameters used in this work and the corresponding input diameters and viscosities. The values are in terms of lattice units.

Generic image for table
Table II.

The Archimedes numbers, terminal velocities, and Reynolds numbers based on terminal velocity studied in our simulations. The four middle columns give the terminal velocity and Reynolds number obtained from our simulations for a single sphere in a cubic periodic domain. The Reynolds numbers in the last column are based on an empirical drag law Eq. (11). For and , . For the higher Archimedes numbers, . In all simulations, the sphere diameter is . Each entry in this table is the average of 6–7 runs with the gravity oriented differently relative to the cell axes. The standard deviations in and are less than 2% of the means.

Generic image for table
Table III.

Settling velocities in Stokesian suspensions. and . The simulations were carried out in domains with and . The viscosity of the fluid is .

Generic image for table
Table IV.

Power-law exponent and prefactor that provide the best fits to the settling velocities in concentrated suspensions . The quality of the linear fits is indicated by values. The numbers after signs are the 95% confidence intervals. For comparison, we included the power-law exponents calculated from Eq. (2) and calculated from Eq. (3)


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Hindered settling velocity and microstructure in suspensions of solid spheres with moderate Reynolds numbers