Flow Enhancement due to Elastic Turbulence in Channel Flows of Shear Thinning Fluids

We explore the flow of highly shear thinning polymer solutions in straight geometry. The strong variations of the normal forces close to the wall give rise to an elastic instability. We evidence a periodic motion close the onset of the instability, which then evolves towards a turbulentlike flow at higher flow rates. Strikingly, we point out that this instability induces genuine drag reduction due to the homogenization of the viscosity profile by the turbulent flow.

Figure 1

Flow curve obtained by standard bulk rheometry (large open circles), and best power-low fit (straight solid line) $\sigma =3.73{\dot{\gamma }}^{0.21}$, for $\dot{\gamma }<50{\mathrm{s}}^{-1}$. The small colored symbols are obtained by differentiating the velocity profiles in both 152 and $170\mu \mathrm{m}$ width channels. The wall shear stress ${\sigma }_w$ of each experiment is indicated in the color bar. In the right axes is plotted the Weissenberg number, either from direct normal force measurement, $\mathrm{Wi}=N_1/2\sigma$ (open squares), or from characteristic time estimations $\mathrm{Wi}=\tau \dot{\gamma }$ (colored circles) using step flow experiments. The solid line stands for $\mathrm{Wi}=3.63{\dot{\gamma }}^{0.43}$.

Figure 2

Velocity profiles obtained in a channel of width $w=152\mu \mathrm{m}$. From top left to bottom right, the values of the applied pressure drop are 10, 17.8, 26.7, 36.7, 47.5, 57.7, 80, and 90 mbar, which correspond after end-effect correction to wall shear stress ${\sigma }_w$ of 1.33, 2.36, 3.54, 4.86, 6.28, 7.60, 10.4, and 11.5 Pa. The velocity is normalized by the average maximum velocity measured in the channel center, i.e., $5.57\times 10^{-6}$, $1.14\times 10^{-5}$, $8.22\times 10^{-5}$, $4.17\times 10^{-4}$, $9.19\times 10^{-4}$, $1.62\times 10^{-3}$, $4.41\times 10^{-3}$, $5.43\times 10^{-3}\mathrm{m}/\mathrm{s}$. The mean velocity profile is displayed as circles and is superimposed on a gray-scale image which represents the velocity density probability estimated from correlation counts in a correlation box of width $5.6\mu \mathrm{m}$. The dashed line is the velocity prediction from the liquid bulk rheological power law (see text) where the slip velocity is the single fitting parameter. The solid line represents the best fit to the data using the nonlocal fluidity model (see text).

Figure 3

Left: normalized standard deviation of the maximum velocity in the flow direction (channel width: colored circles $170\mu \mathrm{m}$, open circles $152\mu \mathrm{m}$). Right: standard deviation of the transverse component $v_y$ of the velocity field.

Figure 4

Top: superposition of 26 successive images taken at 10 Hz in the $152\mu \mathrm{m}$ width channel. The channel was mounted on a translation stage that is moving at a velocity close to the mean liquid velocity, in the $x$ direction. Left: ${\sigma }_w=7.8\mathrm{Pa}$, stage velocity $0.83\mathrm{mm}/\mathrm{s}$. Right: ${\sigma }_w=12.3\mathrm{Pa}$, stage velocity $3.75\mathrm{mm}/\mathrm{s}$. Bottom: superposition of all the individual tracer velocity in the $y$ (transverse) direction. Additional pictures and movies are available in the Supplemental Material [16].

Figure 5

Left: characteristic length $\xi$ involved in the nonlocal fluidity model (see text) as a function of the wall shear stress (width of the channel: $152\mu \mathrm{m}$ open circles, $170\mu \mathrm{m}$ colored circles). These values are corresponding to the best fit to the mean velocity profiles displayed in Fig. 2. For comparison is displayed (squares) the characteristic length $\tau \sqrt{v_y^2}$ where $\sqrt{v_y^2}$ is the amplitude of the transverse velocity fluctuations and $\tau$ the relaxation time of the solution for $\sigma ={\sigma }_w/2$. Right: Experimental wall shear rate ${\dot{\gamma }}_w$ normalized by the expected value in the absence of instability ${\dot{\gamma }}_{w_0}={({\sigma }_w/A)}^{1/n}$.