Flow of a model shear-thickening micellar fluid past a falling sphere

We present quantitative measurements of a dilute micellar solution past a falling sphere. The dilute micellar solution that consists cetyltrimethylammonium bromide and 5-methyl salicylate (CTAB/5mS) in deionized water exhibits shear-thickening behavior beyond a critical shear rate of $\dot{{\gamma }_c}\approx 0.4$ (1/s). Previous experiments have demonstrated that this micellar solution forms unentangled rodlike micelles at equilibrium [Davies et al., J. Am. Chem. Soc. 128, 6669 (2006)]. At a vanishingly small Reynolds number $\mathrm{Re}=0.03$, the drag coefficient for the falling sphere is similar to that of a Newtonian fluid. However, surprisingly, for conditions that correspond to $0.09\le \text{Re}\le 9.86$, falling spheres experience a significant drag reduction. Moreover, an unusually extended wake which spans over a long distance $>80a$ downstream of the sphere is detected by particle image velocimetry. These unusual results could be rationalized by invoking the phenomenon of flow-induced structure formation. We hypothesize that strong shear and/or extensional flows around the falling sphere could trigger aggregation of rodlike micelles into giant wormlike structures. Such wormlike micelles may induce significant sphere drag reduction and extended elastic wakes in the rear of sphere. This interpretation is consistent with the steady shear and transient extensional experiments, whereby a strong shear and elongational thickening have been recovered.

Figure 1

Steady-state shear viscosity as a function of shear rate for a dilute micellar solution of CTAB/5mS at room temperature.

Figure 2

Snapshots illustrating the filament-thinning process in the CaBER experiments for a CTAB/5mS solution. The lower panel shows a close-up view of the cylindrical filament that clearly indicates formation of the beads-on-string phenomenon.

Figure 3

(a) Normalized filament diameter as a function of time for a CTAB/5mS solution in CaBER experiments. (b) Transient Trouton ratio as a function of time. The inset in panel (b) shows the transient strain rate as a function of time.

Figure 4

(a) Velocity of the sphere center of mass as a function of time in a dilute micellar solution. (b) Drag coefficient as a function of Reynolds number for falling spheres in a CTAB/5mS solution ($\square$), in simulations of Dhole et al. [17] ($\diamond$), and in Newtonian fluids (continuous curve).

Figure 5

Normalized velocity vector maps for nylon spheres falling in the dilute CTAB/5mS micellar solution at different Reynolds numbers.

Figure 6

Normalized steady-state velocity of the dilute CTAB/5mS micellar fluid on the sphere axis parallel to the direction of sphere motion. The continuous curves show the steady-state velocity of a shear-thickening power-law model with power-law index $n=2$ as calculated by COMSOL.

Figure 7

(a), (b) Strain rate as a function of distance from the sphere center of mass. (c) Temporal evolution of maximum wake strain rate (${\dot{\varepsilon }}_{max}$) for different spheres over a wide range of inertia.