Separation regimes of two spheres falling in shear-thinning viscoelastic fluids

We experimentally study the time evolution of the distance between two identical spheres settling down in polyacrylamide shear-thinning solutions in mixtures of water and glycerin. In order to analyze the behavior of the spheres under different rheological conditions, we varied the concentration of the polymer and glycerin. To examine the influence of wall effects, the experiments were performed in two containers of different size. We found that depending on the fluids' rheological properties, two different regimes are revealed for the same diluted polymer. In one of them (I), the separation between the spheres grows in time if the initial separation $S_i$ is beyond a critical distance $S_c$, while the separation decreases for an initial distance shorter than $S_c$. In the second observed regime (II), the spheres aggregate if $S_i<S_c$ while they remain unchanged in their separation if $S_i>S_c$, i.e., repulsion is not detected. The transition from one to another regime can be produced at constant polymer concentration by changing the solvent viscosity ${\eta }_s$. In our experiments this shift has been done modifying the glycerin-water proportion. Although these two behaviors were reported in previous works for different viscoelastic fluids, a result that we show is that both regimes can be observed for the same polymer. The elusive regime I, in which far-field repulsion is exhibited, is of particular interest since it has been previously reported only by Riddle et al. [J. Non-Newtonian Fluid Mech. 2, 23 (1997)]. Thus, our results confirm that this behavior is also feasible.

Figure 1

Rheological properties ${\eta }^{\prime }$ (circles) and $2{\eta }^{\prime \prime }/\omega$ (squares) of the fluid P2G60 corresponding to the small-amplitude oscillatory shear flow. Solid and dashed lines are regressions to the multimode linear viscoelasticity and Carreau models, respectively.

Figure 2

Dynamical viscosity ${\eta }^{\prime }$ for the fluids P1G90 and P1G60 as a function of the frequency. Solid and dashed lines are regressions to the multimode linear viscosity and Carreau models, respectively.

Figure 3

Dependence of $\tau$ with $\dot{\gamma }$ for the three fluids considered: P1G60 (asterisks), P2G60 (triangles), and P1G90 (circles), for $\omega >1{\mathrm{s}}^{-1}$.

Figure 4

Scheme of the central vertical plane of the experimental apparatus as seen from the CCD camera. The three draws correspond to different stages along a typical measurement: (a) before the releasing of the first sphere; (b) after the first pulse of voltage has been sent to the driven motor and the first sphere has fallen; and (c) after the second pulse was sent and both spheres are falling.

Figure 5

Temporal evolution of the nondimensional separation, $S/D$, between the spheres for the test fluid P1G60, $D=1\mathrm{mm}$, container A. The initial separations, taken at the initial time $t/t^{*}=0$, are (a) $S_i/D\approx 53.1$, (b) $S_i/D\approx 48.2$, (c) $S_i/D\approx 47.6$, (d) $S_i/D\approx 45.9$, and (e) $S_i/D\approx 43.6$. Deborah and Reynolds numbers in these experiments are all near $\text{De}=0.08$ and $\text{Re}\left(\dot{\gamma }\right)=2\times {10}^{-4}$.

Figure 6

Separation between the spheres as a function of time for the test fluid P2G60, $D=1\mathrm{mm}$, container A. The value of the initial separation for each curve is (a) $S_i/D\approx 19.0$, (b) $S_i/D\approx 17.4$, (c) $S_i/D\approx 13.8$, (d) $S_i/D\approx 11.5$, 11.4, 11.2 (group of three curves), (e) $S_i/D\approx 10.5$, and (f) $S_i/D\approx 9.8$. Deborah and Reynolds numbers in these experiments: $\text{De}=0.9, \text{Re}\left(\dot{\gamma }\right)=1\times {10}^{-7}$.

Figure 7

Separation between the spheres as a function of time for the test fluid P2G60, $D=1\mathrm{mm}$, container B. The value of the initial separation for each curve is (a) $S_i/D\approx 15.2$, (b) $S_i/D\approx 15.1$, (c) $S_i/D\approx 12.8$, 12.9, 13.0, 13.5, 13.6, 14.0 (group of six curves), and (e) $S_i/D\approx 8.8$. Deborah and Reynolds numbers in these experiments: $\text{De}=0.9, \text{Re}\left(\dot{\gamma }\right)=1\times {10}^{-7}$.

Figure 8

Separation between the spheres as a function of time for the test fluid P1G60, $D=1.6\mathrm{mm}$, container A. The value of the initial separation for each curve is (a) $S_i/D\approx 66.3$, (b) $S_i/D\approx 61.5$, (c) $S_i/D\approx 52.9$, (d) $S_i/D\approx 45.5$, (e) $S_i/D\approx 41.0$, (f) $S_i/D\approx 24.4$, and (g) $S_i/D\approx 19.7$. Deborah and Reynolds numbers in these experiments: $\text{De}=2.0, \text{Re}\left(\dot{\gamma }\right)=1\times {10}^{-3}$.

Figure 9

Separation between the spheres as a function of time for the test fluid P1G90, $D=1\mathrm{mm}$, container B. The value of the initial separation for each curve is (a) $S_i/D\approx 35.1$, (b) $S_i/D\approx 23.6$, (c) $S_i/D\approx 20.3$, 19.7 (group of two curves), (d) $S_i/D\approx 15.4$, (e) $S_i/D\approx 10.5$, and (f) $S_i/D\approx 7.6$. Deborah and Reynolds numbers in these experiments: $\text{De}=2.0$ and $\text{Re}\left(\dot{\gamma }\right)=1\times {10}^{-3}$.

Figure 10

Vertical velocities of the spheres for the curve $b$ ($S_i\approx S_c$) in Fig. 5. Thick and thin lines correspond to spheres 1 and 2, respectively.

Figure 11

Vertical velocities of the spheres for the curve $a$ ($S_i>S_c$) in Fig. 5. Same symbols as in Fig. 10.

Figure 12

Vertical velocities of the spheres for the curve $e$ ($S_i<S_c$) in Fig. 5. Same symbols as in Fig. 10.