Drag coefficient for a sedimenting and rotating sphere in a viscoelastic fluid

In this work, we study the effect of sphere rotation on the drag coefficient experienced by a sphere sedimenting through a nearly constant viscosity and elastic Boger fluid, using experiments, numerical simulations, and an asymptotic theory. For the purely sedimenting (nonrotating) sphere, we find that the drag coefficient is larger in the Boger fluid than in the Newtonian case, in accordance with previous studies. In the Boger fluid, rotation of the sphere (around the axis aligned with translation) causes a reduction in the drag coefficient compared to the purely sedimenting case; this trend is observed in the experiments, numerical simulations, and theory. The numerical results indicate that the decrease in the drag due to sphere rotation results from modifications in the pressure contribution to the drag.

Figure 1

Results of steady shear rheometry for the Boger fluid (empty circles) and the Newtonian fluid (filled circles). The red line shows a fit to power-law viscosity model with $m=0.9106$ Pa s and $n=0.963$ for $\dot{\gamma }<$ 20 ${\mathrm{s}}^{-1}$. Also, the first normal stress difference ($N_1$) is shown as a function of shear rate (empty squares). The Oldroyd-B (black solid line) and the FENE-P (black dashed-dotted line) models were used to fit ${\eta }_0$ and $N_1$ of the viscoelastic fluid. The full range of shear rates present in the falling rotating sphere experiments is shown with vertical dashed lines.

Figure 2

(a) Experimental setup; (b) container and sphere.

Figure 3

Computational domain used for simulation.

Figure 4

(a) Terminal velocity $u$ (mm/s) as a function of the frequency of rotation (Hz). Symbols refer to experimental measurements of the spheres with different diameter: 7.99 mm (squares), 8.72 mm (circles), 8.81 mm (triangles), and 9.57 mm (diamonds), whereas solid lines represent Stokes settling speed. (b) Comparison of drag coefficient data for the Newtonian fluid with Stokes law (solid line).

Figure 5

(a) Terminal velocity $u$ (mm/s) as a function of the frequency of rotation (Hz). Symbols refer to experimental measurements of the spheres with different diameter: 7.99 mm (squares), 8.72 mm (circles), 8.81 mm (triangles), and 9.57 mm (diamonds). (b) Comparison of drag coefficient data for the viscoelastic fluid with Stokes law (solid line); crossed out markers represent measurements without rotation.

Figure 6

(a) Comparison of drag coefficient data for the Newtonian (empty markers) and the Boger fluid (filled markers) with Stokes law (solid line). Symbols refer to experimental measurements of the spheres with different diameter: 7.99 mm (squares), 8.72 mm (circles), 8.81 mm (triangles), and 9.57 mm (diamonds). Crossed out markers represent measurements without rotation. (b) Dimensionless drag coefficient ${\mathrm{X}}_{\mathrm{e}}$ as a function of the Weissenberg number $\mathrm{Wi}$ for the Newtonian (empty markers) and Boger fluid (filled markers). The symbols are the same as in (a) with the addition of the experimental measurements obtained by Ref. [6] (open stars).

Figure 7

Change in the normalized drag correction factor, $X_e$, as a function of the rotational Weissenberg number, $\mathrm{Wi}=\lambda \omega$. Shown is the comparison of the theory (lines) to the numerical solutions (symbols) for $\beta ={\eta }_s/{\eta }_0=0.3$ and $\beta =0.7$ in an Oldroyd-B fluid.

Figure 8

Change in the normalized drag correction factor, $X_e$, as a function of the rotational Weissenberg number, $\mathrm{Wi}=\lambda \omega$. Results compare a single experimental trial ($d=7.99$ mm), as open symbols, to numerical simulations, as closed symbols (a line is drawn through the numerical simulation data as a guide to the eye).

Figure 9

Contributions to the drag correction factor, $X_e$, as a function of the rotational Weissenberg number, $\mathrm{Wi}=\lambda \omega$, from simulations.

Figure 10

(a) Pressure contours on the surface of the spherical particle and in the surrounding fluid (shown in the $y-z$ plane). Pressure is minus the hydrostatic pressure and made dimensionless with ${\eta }_0{U}_{\mathrm{Wi}=0}/a$. (b) Contours of the trace of the polymer conformation tensor normalized by the maximum polymer extensibility, $c_{ii}/L^2$, on the surface of the spherical particle and in the surrounding fluid (shown in the $y-z$ plane). (c) Velocity contours of $u_z$, here made dimensionless with $U_{\mathrm{Wi}=0}$, on the surface of the spherical particle and in the surrounding fluid (shown in the $y-z$ plane). Streamlines of $(u_x,u_y,u_z)$ are seeded at $z=\pm 2$ and $x=0$ across a range of $y$ (seed lines are shown in red).

Figure 11

Velocity contours, here made dimensionless with $U_{\mathrm{Wi}=0}$, on the surface of the spherical particle and in the surrounding fluid (shown in the $y-z$ plane). Streamlines of $(0,u_y,u_z)$ are seeded at $z=\pm 1.25$ and $x=0$ across a range of $y$ (seed lines are shown in red).