Mixing of Non-Newtonian Fluids in Steadily Forced Systems

We investigate mixing in a viscoelastic and shear-thinning fluid—a very common combination in polymers and suspensions. We find that competition between elastic and viscous forces generates self-similar mixing, lobe transport, and other characteristics of chaos. The mechanism by which chaos is produced is evaluated both in experiments and in a simple model. We find that chaotic flow is generated by spontaneous oscillations, the magnitude and frequency of which govern the extent of chaos and mixing.

Figure 1

Flow patterns of a tank ($D_T=24\mathrm{c}\mathrm{m}$ and $H=32\mathrm{c}\mathrm{m}$) stirred by an axysymmetric disc. (a) Newtonian fluid (glycerin) at $\mathrm{R}\mathrm{e}=10$ reveals concentric rings and no signs of chaos. (b) Lobe formation and chaotic motion in a 1%-aqueous solution of CMC ($\lambda =0.55\mathrm{s}$) at $\mathrm{R}\mathrm{e}=13.5$ ($Wi=2.75$). Shear rate dependent viscosity data are fitted to $\eta ={\eta }_0[1+(\kappa \dot{\gamma }{)}^2{]}^{-n/2}$, where ${\eta }_o=3000\mathrm{c}\mathrm{P}$ is the zero-shear-rate viscosity, $n=0.7$ is the power law index, and $\kappa =1.2$ is a time constant. Yellow and blue lines show flow direction. Red line delimits the high-shear region. Green line contours a single lobe.

Figure 2

Periodic flow oscillations of a shear-thinning elastic fluid $\mathrm{R}\mathrm{e}=13.5$, $Wi=2.75$): snapshots of the mixing structure of 1% CMC 10 s apart. (a) initial, (b) $\Delta t=10\mathrm{s}$, (c) $\Delta t=20\mathrm{s}$. Yellow line identifies the upper segregated region. Green arrow (b) points to the region where lobes are created. Dashed red square (c) delimits area of interest for modeling. Periodicity of velocity field is also revealed via PIV (d) and power spectra (e) (arrow points to well-defined peak).

Figure 3

Mixing efficiency worsens with agitation speed. (a) The rate by which a passive dye spreads is inversely related to agitation speed (Re and $Wi$). (b) Lobe-area measurements indicate that the extent of chaos decreases as frequency (agitation speed) is increased. Map results show that lobe area decreases with map frequency ($\tau$). Insert: oscillation frequency increases with agitation speed.

Figure 4

(a) A 2D square lattice (black arrows) is periodically perturbed with a secondary map (red dashed lines) to mimic the experiments. The region of interest is the left lower quadrant of the tank [red dashed line on Fig. 2c]. Map is able to capture main features of the flow; compare (a) to (b). As frequency increases ($\tau$), lobe area decreases and dye coverage diminishes for (b) $\tau =15$, (c) $\tau =20$, and (d) $\tau =25$. Lobe area vs $\tau$ data is presented in Fig. 3b.