Taylor Couette instability in disk suspensions

We study the stability of dilute suspensions of spheroids in Taylor Couette flow. We focus on axisymmetric perturbations and on the limiting cases of thin disks and long rods. It is found that in the non-Brownian limit, the rods have a negligible effect on the stability, while the disks are destabilizing. The instability is driven by a tilting of the disks, which draws energy from the base flow into azimuthal velocity fluctuations. The resulting instability mode has a wavelength which is smaller than the unstable Newtonian mode. These findings may serve to understand experiments using clay suspensions in the literature.

Figure 1

Absolute values of the dilute spheroid suspension material constants ${\alpha }_i$, divided by the spheroid volume fraction $c$. These quantities are defined in Eq. (5), and they are functions of the aspect ratio $r_a$. The relations are taken from Ref. [12] and are listed in the Appendix. The black (gray) lines indicate positive (negative) ${\alpha }_i$.

Figure 2

Coordinate system $r,\varphi ,z$, base flow field $U_{\varphi }\left(r\right)$, and particle orientation unit vector $\mathit{n}$.

Figure 3

Spheroid contribution to the effective viscosity ${\mathrm{\Sigma }}_{r\varphi }/G$, scaled by the value at zero shear $\nu {\alpha }_0$ [Eq. (21)], as a function of the Peclet number $\mathrm{Pe}$ [Eq. (6)] for disks with $r_a={10}^{-3}$, computed by solving Eqs. (9)–(12) and (20). The shear rate $G$ is defined in Eq. (13).

Figure 4

(a) Taylor number $\mathrm{Ta}$ [Eq. (1)] in Newtonian flow, for which the maximum of the growth rates [eigenvalue of matrix in Eq. (28)] equals zero, as a function of the spanwise wave number $k$. (b) Growth rates $\lambda$ [eigenvalues of matrix in Eq. (32)] as a function of the Taylor number $\mathrm{Ta}$ [Eq. (1)] for various concentrations ${\alpha }_1$ of non-Brownian disks, using $k\mathrm{\Delta }R/\pi =1$.

Figure 5

The critical Taylor number $\mathrm{Ta}$ [Eq. (40)] as a function of the concentration parameter ${\alpha }_0$ [Eq. (21)], for disks with an aspect ratio of $r_a={10}^{-3}$ (solid line) and for rods with an aspect ratio of $r_a={10}^3$ (dashed line), and using a Peclet number [Eq. (6)] of $\mathrm{Pe}={10}^9$, a spanwise wave number of $k=\pi /\mathrm{\Delta }R$, and a radius ratio of $R_2/R_1=1.05$.

Figure 6

The critical Taylor number $\mathrm{Ta}$ [Eq. (40)] as a function of the Peclet number $\mathrm{Pe}$ [Eq. (6)] for disks with an aspect ratio of $r_a={10}^{-3}$ (solid line) and for rods with an aspect ratio of $r_a={10}^3$ (dashed line), and a concentration parameter [Eq. (21)] of ${\alpha }_0=0.1$, using a spanwise wave number of $k=\pi /\mathrm{\Delta }R$ and a radius ratio of $R_2/R_1=1.05$.

Figure 7

The critical Taylor number $\mathrm{Ta}$ [Eq. (40)] as a function of the spanwise wave number, $k=\pi /\mathrm{\Delta }R$, for disks with an aspect ratio of $r_a={10}^{-3}$ (solid line) and for rods with an aspect ratio of $r_a={10}^3$ (dashed line), a Peclet number [Eq. (6)] of $\mathrm{Pe}={10}^9$, a concentration parameter [Eq. (21)] of ${\alpha }_0=3\times {10}^{-3}$, and a radius ratio of $R_2/R_1=1.05$. The vertical, dotted lines indicate the minima of the curves.

Figure 8

Most unstable eigenmode in a suspension of rods, using an aspect ratio of $r_a={10}^3$, a Peclet number [Eq. (6)] of $\mathrm{Pe}={10}^9$, a concentration parameter [Eq. (21)] of ${\alpha }_0=0.1$, a spanwise wave number of $k=\pi /\mathrm{\Delta }R$, a radius ratio of $R_2/R_1=1.05$, and a Taylor number [Eq. (40)] at the critical value of $\mathrm{Ta}\approx 42$.

Figure 9

Most unstable eigenmode in a suspension of disks, using an aspect ratio of $r_a={10}^{-3}$, a Peclet number [Eq. (6)] of $\mathrm{Pe}={10}^9$, a concentration parameter [Eq. (21)] of ${\alpha }_0=0.1$, a spanwise wave number of $k=\pi /\mathrm{\Delta }R$, a radius ratio of $R_2/R_1=1.05$, and a Taylor number [Eq. (40)] at the critical value of $\mathrm{Ta}\approx 2.5$.

Figure 10

The critical Taylor number $\mathrm{Ta}$ [Eq. (40)] as a function of the Peclet number $\mathrm{Pe}$ [Eq. (6)], for rods with an aspect ratio of $r_a={10}^3$, using a concentration parameter [Eq. (21)] of ${\alpha }_0=2.35$, a spanwise wave number of $k=\pi /\mathrm{\Delta }R$, and a radius ratio of $R_2/R_1=1.05$. Comparison between present result (line) and results from Ref. [10] (markers). The dashed line indicates the Newtonian value of $\mathrm{Ta}\approx 42.4$.

Figure 11

The critical Taylor number $\mathrm{Ta}$ [Eq. (40)] as a function of the fluid index $n$ [Eq. (43)], for disks with an aspect ratio of $r_a=3.1\times {10}^{-3}$, using a Peclet number [Eq. (6)] of $\mathrm{Pe}=1.4$, a spanwise wave number of $k=\pi /\mathrm{\Delta }R$, and a radius ratio of $R_2/R_1=1.05$. Comparison between present result (solid line) and results from Ref. [11] (markers).