Instability driven by shear thinning and elasticity in the flow of concentrated polymer solutions through microtubes

We present a systematic experimental investigation of the onset of instability in the flow of concentrated, shear-thinning, polyethylene oxide (PEO) solutions through rigid microtubes (diameters $\sim 500\mu \mathrm{m}$ and $2840\mu \mathrm{m}$). Micro-PIV measurements are performed for the flow of PEO solutions ($\sim 2000–6000\mathrm{ppm}$ in water) in the $500\mu \mathrm{m}$ tube to obtain the magnitude of normalized velocity fluctuations, which show a jump at Reynolds number ${\text{Re}}_s$ as low as $\sim 10$, indicative of an instability of the laminar flow, where ${\text{Re}}_s=\left(DV\rho \right)/{\eta }_s$ is the Reynolds number based on the viscosity ${\eta }_s$ evaluated at the maximum shear rate prevailing in the flow, $D$ is the tube diameter, $V$ is the cross-sectional average velocity, and $\rho$ is the fluid density. However, the ratio of peak to center-line velocity does not show any significant shift from the laminar value. Experiments are also carried out to obtain friction factor data for the flow of $2000–4000\mathrm{ppm}$ PEO solutions through a $2.84\mathrm{mm}$ glass tube. Here the transition is inferred from the deviation of the experimental data from the friction factor associated with the laminar flow obtained by fitting the shear-thinning rheology of the polymer solution using the Carreau model. The data for transition Reynolds number ${\text{Re}}_{s,t}$ inferred from micro-PIV and friction factor measurements, for polymer solutions of varying concentrations and for two different tube diameters, collapse reasonably well to yield a scaling law ${\text{Re}}_{s,t}\sim {\left[E_s(1-\beta )\right]}^{-3/4}$, where $E_s=4{\lambda }_s{\eta }_s/\rho D^2$ is the elasticity number based on the fluid relaxation time ${\lambda }_s$ estimated at the maximum shear rate of the flow, and $\beta$ is the ratio of solvent to (shear-rate dependent) total viscosity of the polymer solution. The scaling exponent of $-3/4$ inferred from the present experiments for concentrated polymer solutions is very different from the $-1/2$ exponent reported in previous studies for relatively dilute solutions concerning the onset of elastoinertial turbulence. This suggests that the instability observed in this study for concentrated polymer solutions is qualitatively different from the instability leading to elastoinertial turbulence in relatively dilute solutions. Our study further shows that even rectilinear laminar flows of concentrated polymer solutions become unstable at a relatively low ${\text{Re}}_s\sim 10$, provided the fluid is both sufficiently shear thinning and elastic.

Figure 1

(a) Shear and loss moduli of PDMS used for tube fabrication. The shear modulus is found to be more than $5\times {10}^5$ Pa, and hence the tube wall can be considered as rigid. (b) Shear and loss moduli for 6000 ppm polyethylene oxide solution. The data are fitted to the single-mode Maxwell model to obtain the longest relaxation time of the solution. (c) Fitting of 5000 ppm PEO solution oscillatory data into a multimode Maxwell model. A spectrum of four relaxation times is obtained with ${\lambda }_1=3.33\mathrm{s}$, ${\lambda }_2=1.11\mathrm{s}$, ${\lambda }_3=0.33\mathrm{s}$, and ${\lambda }_4=0.04\mathrm{s}$. Average of the spectrum of relaxation times is $1.2\mathrm{s}$.

Figure 2

(a) Normalized viscosity of 2000 ppm, 3000 and 4000 ppm, and (b) normalized viscosity of 5000 ppm, 6000 ppm, and 6500 ppm PEO solutions as a function of shear rate as measured by using the concentric cylinder geometry. $\eta$ is the shear-rate dependent viscosity of the polymer solution, and ${\eta }_0$ is the zero shear viscosity. The values of ${\eta }_0$ for 2000 ppm, 3000 ppm, 4000 ppm, 5000 ppm, 6000 ppm, and 6500 ppm solutions are 0.022, 0.077, 0.25, 0.45, 0.90, and 1.95 Pa s, respectively.

Figure 3

Comparison of stress fluctuations at low and high shear rates depicting large stress fluctuations in the unstable flow (at high shear rates) of the concentric cylinder flow geometry as compared to the stable regime (at low shear rates). Here $\sigma$ is the instantaneous stress and ${\sigma }_m$ is the mean of stress over time.

Figure 4

(a) Comparison of viscosity measured using the concentric cylinder and cone-and-plate geometries for 4000 ppm PEO solution and (b) shear-rate dependent relaxation time calculated using the first normal stress coefficient by using the cone-and-plate geometry. Here the cone angle is $1^{\circ }$ and radius is $20\text{mm}$.

Figure 5

(a) Dependence of zero shear viscosity on concentration of PEO used. (b) Dependence of zero shear relaxation time on concentration of PEO used.

Figure 6

Onset of instability for PEO solutions of different concentrations: (a) Reynolds number ${\text{Re}}_0$ is calculated using the zero shear viscosity of the individual PEO solutions. (b) Reynolds number ${\text{Re}}_s$ is calculated using the viscosity corresponding to the maximum shear rate in the flow.

Figure 7

Comparison of laminar velocity profiles in the laminar and unstable regimes. (a) ${\text{Re}}_s\sim 1.1$ corresponds to the laminar regime for the flow of 6000 ppm PEO solution, and (b) ${\text{Re}}_s\sim 90$ corresponds to the unstable regime. $V_{\mathrm{loc}}$ is the local velocity, and $V_{\mathrm{avg}}$ is the cross-sectional average velocity in the tube.

Figure 8

Comparison of laminar velocity profiles for 3000 ppm PEO solution at ${\text{Re}}_0=0.4$, taken from micro-PIV measurements and from the solution of Carreau model for shear-thinning fluids. $V_{\mathrm{loc}}$ is the local velocity and $V_{\mathrm{avg}}$ is the cross-sectional average velocity in the tube.

Figure 9

Friction factor charts for the flow of concentrated PEO solutions through a $2.84\mathrm{mm}$ glass tube. (a) Laminar friction factor line obtained by using Carreau model, by taking shear-thinning exponent $m=0.35$ and elasticity $E=400$, compared with the Newtonian laminar line. (b) Friction factor data for 2000 ppm, (c) 3000 ppm PEO, and (d) 4000 ppm PEO solutions. The solid lines in panels (b), (c), and (d) represent the laminar friction data obtained by using the Carreau model for shear-thinning fluids and dotted vertical line is an indicator of the the onset of transition. (e) Friction factor dependence on ${\text{Re}}_s$ i.e., when Reynolds number is based on the maximum shear rate prevalent in the experiments.

Figure 10

Snapshots of the dye stream visualization experiments in the laminar and posttransition regimes. Dye stream appears as a straight line at ${\text{Re}}_s\sim 8$ in panel (a), while panel (b) shows the breakage of dye stream at ${\text{Re}}_s\sim 12$.

Figure 11

(a) Variation of transition Reynolds number, ${\text{Re}}_{0,t}$, with elasticity number when the Reynolds number is calculated using the zero shear viscosity and (b) transition Reynolds number, ${\text{Re}}_{s,t}$, with elasticity number when Reynolds number and elasticity number, $E_s$, are calculated at the maximum shear rates present in the flow. Also shown are results from Poole [29] and Chandra et al. [13].