Emergence of chaos in a viscous solution of rods

It is shown that the addition of small amounts of microscopic rods in a viscous fluid at low Reynolds number causes a significant increase of the flow resistance. Numerical simulations of the dynamics of the solution reveal that this phenomenon is associated to a transition from laminar to chaotic flow. Polymer stresses give rise to flow instabilities which, in turn, perturb the alignment of the rods. This coupled dynamics results in the activation of a wide range of scales, which enhances the mixing efficiency of viscous flows.

Figure 1

Left: Snapshot of the vorticity field $\omega$ for ${\eta }_{\mathrm{p}}=3$ and $L=1/8$. Black (white) represents negative (positive) vorticity. Right: Snapshot of the component ${\mathcal{R}}_{11}$. White represents 0, black represents 1.

Figure 2

Mean velocity profiles for $L=1/8$ and concentrations ${\eta }_{\mathrm{p}}=1$ (solid), ${\eta }_{\mathrm{p}}=3$ (dotted), and ${\eta }_{\mathrm{p}}=5$ (dashed).

Figure 3

The kinetic energy $\mathcal{E}$ for $L=1/8$ and concentrations ${\eta }_{\mathrm{p}}=1$ (top, solid), ${\eta }_{\mathrm{p}}=3$ (middle, dotted), and ${\eta }_{\mathrm{p}}=5$ (bottom, dashed) divided by the kinetic energy ${\mathcal{E}}_0=F^2{L}^4/2{\nu }^2$ corresponding to ${\eta }_{\mathrm{p}}=0$ and the same value of the force $F$.

Figure 4

The normalized mean injected power $P/P_{\mathrm{lam}}$ as a function of ${\eta }_{\mathrm{p}}$. Inset: The amplitudes of the stresses ${\mathrm{\Pi }}_r$ (brown $\square$), ${\mathrm{\Pi }}_{\nu }$ (blue $•$), and ${\mathrm{\Pi }}_p$ (red $\times$) divided by the amplitude of the total stress ${\mathrm{\Pi }}_{\mathrm{tot}}$ for $L=1/8$ and different values of ${\eta }_{\mathrm{p}}$ (see Table 1).

Figure 5

The kinetic energy spectrum $E\left(k\right)$ for $L=1/4$, ${\eta }_{\mathrm{p}}=3$ (solid red) and $L=1/8$, ${\eta }_{\mathrm{p}}=5$ (dashed blue). The two black dotted segments represent $k^{-4}$ and $k^{-5}$. Inset: The unsigned kinetic-energy dissipation spectrum $2\nu k^{2}E\left(k\right)$ (magenta $+$) and the polymer energy transfer (solid black) for $L=1/4$ and ${\eta }_{\mathrm{p}}=3$.