Unsteady sedimentation of a sphere in wormlike micellar fluids

The unsteady sedimentation of a sphere in wormlike micellar fluids is studied experimentally through shear and extensional rheometry, sphere trajectory tracking, and particle image velocimetry. Unsteady sphere sedimentation characterized by fluctuations in the sphere settling velocity was observed for a range of sphere size and density in two non-shear-banding wormlike micellar solutions, a cetylpyridinium chloride (CpCl)-sodium salicylate (NaSal) solution and a cetyltrimethylammonium p-toluenesulfonate (CTAT)-NaCl solution. The onset of the transition from steady to unsteady sphere motion is characterized by an extensional Deborah number, $\mathrm{D}{\mathrm{e}}_{\mathrm{ext}}$, defined locally in the negative wake of the falling sphere. This instability criterion is in agreement with previous findings by Mohammadigoushki and Muller [J. Rheol. 60, 587 (2016)] in the wormlike micelle system of cetyltrimethylammonium bromide (CTAB) and NaSal, and appears to be universally valid independent of micelle chemistry or solution rheology (e.g., shear banding or not). Moreover, the frequency at which the sphere velocity fluctuates is found to be linearly correlated with an average shear Deborah number $\mathrm{D}{\mathrm{e}}_s$, which is a measure of the overall flow strength. This suggests that a constant critical strain is accumulated before the flow instability takes place in each velocity oscillation. The velocity fluctuations are found to become increasingly disordered with increasing elastic Mach number, $\mathrm{M}{\mathrm{a}}_e$, indicating that the interactions between the flow instability and elastic wave propagation result in more chaotic velocity fluctuations.

Figure 1

(a) Small amplitude oscillatory shear rheology for the two wormlike micelle solutions. Filled symbols (and solid lines) correspond to G′, open symbols (and dotted lines) correspond to G″. The lines are the fit to a single mode Maxwell model. (b) Steady shear rheology for the two solutions.

Figure 2

(a) Schematic diagram of the dripping onto substrate (DoS) experiment. (b) Sample images of the filament thinning from the DoS experiment for the CpCl-NaSal solution (top) and CTAT-NaCl solution (bottom). (c) The radius of the filament neck, nondimensionalized with the nozzle radius, as a function of time during the thinning process for the two solutions. The solid lines indicate the best fits to Eq. (2).

Figure 3

(a) The velocity of a sphere as a function of time as it settles in the CTAT-NaCl solution. Data for three spheres of different densities are shown. (b) The corresponding power spectra for the data in (a).

Figure 4

(a) The velocity of a sphere as a function of time as it settles in the CpCl-NaSal solution. Data for four spheres of different densities and/or size are shown. (b) The corresponding power spectra for the data in (a) for the three highest $\mathrm{D}{\mathrm{e}}_s$.

Figure 5

Velocity vector maps from PIV for flow around a 3/32 inch stainless steel sphere settling in CTAT-NaCl. The sedimentation is unsteady under these conditions; the average terminal velocity yields $\mathrm{D}{\mathrm{e}}_s=4.2$. Velocity vectors are scaled such that an arrow of length $l/a=1/3$ corresponds to 1 mm/s, where $a$ is the radius of the sphere.

Figure 6

For flow around a 3/32 inch stainless steel sphere settling in CTAT-NaCl: (a) The axial velocity $v_z$ in the fluid as a function of axial position. The axial position has been non-dimensionalized so that the center of mass of the sphere is always located at $\left(z\text{−}{z}_{\mathrm{sphere}}\right)/a=0$. (b) The local extensional strain rate ${\dot{\varepsilon }}_{zz}$, defined in Eq. (3), as a function of axial position. (c) The maximum strain rate, ${\dot{\varepsilon }}_{M,z}$ as a function of time t. The points indicated I through V correspond to the times shown in Fig. 5.

Figure 7

Unsteady sedimentation of a 1/8 inch ceramic sphere in CpCl-NaSal; $\mathrm{D}{\mathrm{e}}_s=16.4$. (a) The center-of-mass motion of the sphere from particle tracking velocimetry as a function of time. (b) The instantaneous maximum strain rate ${\epsilon }_{\dot{M},z}$ as a function of time obtained (as in Fig. 6) from PIV measurements. (c) Velocity vector maps from PIV showing the velocity field at the three time points I–III indicated in (b).

Figure 8

Sphere sedimentation experiments in CpCl-NaSal, CTAT-NaCl, and CTAB-NaSal (MM2016: Ref. [15]) plotted as (a) $\mathrm{D}{\mathrm{e}}_s$ versus Re and (b) $\mathrm{D}{\mathrm{e}}_{\mathrm{ext}}$ versus Re. In both plots open symbols denote steady sedimentation, while filled symbols denote unsteady sedimentation.

Figure 9

The characteristic frequency $f$ of the sphere sedimentation velocity vs. Deborah numbers based on shear (a) and extension (b). Filled symbols, periodic; half-filled symbols, periodic but with high-order frequencies; open symbols, irregular velocity fluctuations with no apparent characteristic frequency.

Figure 10

The characteristic frequency $f$ nondimensionalized with the shear relaxation time, as a function of the viscoelastic Mach number ${\mathrm{Ma}}_e$