Elastic Instabilities of Polymer Solutions in Cross-Channel Flow

When polymer molecules pass near the hyperbolic point of a microchannel cross flow, they are strongly stretched. As the strain rate is varied at low Reynolds number ($<{10}^{-2}$), tracer and particle-tracking experiments show that molecular stretching produces two flow instabilities: one in which the velocity field becomes strongly asymmetric, and a second in which it fluctuates nonperiodically in time. The flow is strongly perturbed even far from the region of instability, and this phenomenon can be used to produce mixing.

Figure 1

Dye advection patterns for a cross-channel flow with two inputs and two outputs at low $\mathrm{Re}(<{10}^{-2})$ for (a) Newtonian fluid, and (b) PAA flexible polymer solution (strain rate $\dot{\epsilon }=0.36{\mathrm{s}}^{-1}$, Deborah number $\mathrm{De}=4.5$), where the interface between dyed and undyed fluid is deformed by an instability. (c),(d) Particle streak lines and velocity field magnitudes corresponding to (a),(b), showing the symmetry-breaking instability.

Figure 2

(a)–(c) Dye advection patterns for the PAA solution in the time-dependent regime ($\dot{\epsilon }=2.13{\mathrm{s}}^{-1}$, $\mathrm{De}=26.4$) at 1.75 s intervals. (d) Velocity magnitude, averaged over the central 23% of the intersection region, for PAA at various $\dot{\epsilon }$ and (in one case) for the Newtonian fluid. The flexible polymer solution becomes time dependent for sufficiently large $\dot{\epsilon }$. (e) Corresponding power spectra of the velocity measurements for $\dot{\epsilon }=0.36{\mathrm{s}}^{-1}$ and $\dot{\epsilon }=2.13{\mathrm{s}}^{-1}$.

Figure 3

Velocity fields at (a) $\dot{\epsilon }=0.1{\mathrm{s}}^{-1}$ ($\mathrm{De}=1.2$, steady), (b) $\dot{\epsilon }=0.36{\mathrm{s}}^{-1}$ ($\mathrm{De}=4.5$, steady and asymmetric), and (c) $\dot{\epsilon }=2.13{\mathrm{s}}^{-1}$ ($\mathrm{De}=26.4$, time dependent and asymmetric). The region shown is $600\times 600\mu \mathrm{m}$. (d) Normalized rms deviation $\Delta {\tilde{V}}_{\mathrm{rms}}$ of PAA velocity fields from the Newtonian case at the same shear rate (see text), along with a fit to the expected square root law for a normal bifurcation.

Figure 4

(a) Longitudinal velocity (normalized by the maximum value) as a function of the cross-channel coordinate at 4 channel widths downstream of the hyperbolic point, showing the strong distortion caused by the symmetry-breaking instability. (b) Modified apparatus with an extra $T$ at each outlet to promote mixing. (c)–(e) Dye snapshots showing the stretching and folding of fluid elements in one of the end regions that results from the time-dependent instability, at $\dot{\epsilon }=2.13{\mathrm{s}}^{-1}$ ($\mathrm{De}=26.4$).