Inertioelastic Flow Instability at a Stagnation Point

A number of important industrial applications exploit the ability of small quantities of high molecular weight polymer to suppress instabilities that arise in the equivalent flow of Newtonian fluids, a particular example being turbulent drag reduction. However, it can be extremely difficult to probe exactly how the polymer acts to, e.g., modify the streamwise near-wall eddies in a fully turbulent flow. Using a novel cross-slot flow configuration, we exploit a flow instability in order to create and study a single steady-state streamwise vortex. By quantitative experiment, we show how the addition of small quantities (parts per million) of a flexible polymer to a Newtonian solvent dramatically affects both the onset conditions for this instability and the subsequent growth of the axial vorticity. Complementary numerical simulations with a finitely extensible nonlinear elastic dumbbell model show that these modifications are due to the growth of polymeric stress within specific regions of the flow domain. Our data fill a significant gap in the literature between the previously reported purely inertial and purely elastic flow regimes and provide a link between the two by showing how the instability mode is transformed as the fluid elasticity is varied. Our results and novel methods are relevant to understanding the mechanisms underlying industrial uses of weakly elastic fluids and also to understanding inertioelastic instabilities in more confined flows through channels with intersections and stagnation points.

Popular Summary

Simple fluids (like water) often display instabilities in the form of swirling eddies or vortices, which can be suppressed by adding tiny quantities of polymers (long, flexible molecules). In pipes, this helps reduce the flow resistance and can be of great practical benefit. Polymer additives help transport oil across Alaska, increase sewer capacity during heavy rainfall, and can even improve blood circulation. However, studying how polymers and vortices interact is challenging; generally, vortices fluctuate significantly, while the effects of polymers at low concentrations are subtle. Better understanding of polymer-vortex interactions can optimize the use of polymer additives in industrial and biomedical applications, from large pipes to lab-on-a-chip devices. We have developed methods to make measurements on a single, steady, stationary vortex using flow visualization techniques performed on a laboratory microscope.

By adding increasing amounts of a flexible polymer (starting from just one part per million) to water-based solvents, we have been able to directly measure the effect of the polymer on the formation and development of the vortex that appears in the outlet of a cross-shaped channel as the flow rate is increased. While the addition of the polymer causes vortex formation at a lower flow rate than in pure water, it also significantly reduces the strength of the resulting vortex. Numerical simulations of the flow allow us to understand both of these effects in terms of localized regions in the flow where the polymer molecules respond elastically.

Our methods can be readily extended to study the dynamics of vortex formation and interactions and also to study similar 3D and time-dependent flow phenomena.

Figure 1

The cross-slot geometry, where inflow is along the $y$ direction (indicated by the red arrows) and outflow is along the $x$ direction (indicated by the blue arrows). (a) Schematic diagram of a cross-slot device showing the characteristic channel dimensions and the coordinate system, with origin at the geometric center of the crossing channels. A spiral vortex forms along the $x$ axis for $\mathrm{Re}>{\mathrm{Re}}_c$. (b) A scheme illustrating a design for a microfluidic cross-slot device allowing direct observation of the $z\text{−}y$ plane at $x=0$. The device is vertically mounted on an inverted microscope with a long working distance lens, enabling a direct view of the stagnation point region at the center of the geometry where the spiral vortex instability forms. (c) A prototype microfluidic cross-slot device for illustrative purposes (with channel dimensions $w=d=1\mathrm{mm}$), fabricated in fused silica using selective laser-induced etching: scale bar 5 mm. (d) Photograph illustrating the experimental setup with the cross slot mounted on an inverted microscope.

Figure 2

Shear viscosity measurements with a stress-controlled rotational rheometer (Anton Paar MCR 502) for PEO solutions of various concentrations in two different solvents: water and 8 wt % aqueous PEG. Curve fitting for the shear-thinning fluids is done with the Carreau-Yasuda GNF model [Eq. (1)]. The diagonal dashed line represents $10\times$ the minimum sensitivity of the rheometer.

Figure 3

Confocal microscope images of dye-advection patterns taken in the $x\text{−}y$ center plane ($z=0$ plane) of the cross-slot device for aqueous PEO solutions of various El. Fluid dyed with rhodamine B enters from the left-hand inlet channel and undyed fluid enters through the right-hand inlet channel. (a) Stable flow with a symmetric interface (Newtonian or viscoelastic, $\mathrm{Re}<10$). (b) Inertial instability of Newtonian fluid ($\mathrm{El}=0$, $\mathrm{Re}=80$). (c) Inertioelastic instability with 0.001 wt % PEO solution ($\mathrm{El}=0.00018$, $\mathrm{Re}=79$). (d) Inertioelastic instability with 0.01 wt % PEO solution ($\mathrm{El}=0.011$, $\mathrm{Re}=33$). (e) Elastic instability with 0.1 wt % PEO solution ($\mathrm{El}=0.68$, $\mathrm{Re}=49$). (f) Elastic instability with 0.3 wt % PEO solution ($\mathrm{El}=10.4$, $\mathrm{Re}=15$). See also the video corresponding to (f) contained in the Supplemental Material [78].

Figure 4

Experimental $\mu$-PIV images in the $x=0$ plane made using water seeded with fluorescent microparticles. Panels (a)–(d) show results for progressive increases in Re from below the transition, while panels (e)–(h) show the results for progressively decreasing Re from above the transition. (a),(e) $\mathrm{Re}=37.1$; (b),(f) $\mathrm{Re}=39.2$; (c),(g) $\mathrm{Re}=40.0$; (d),(h) $\mathrm{Re}=41.2$. The color scale indicates the local value of the normalized axial vorticity.

Figure 5

A comparison between experimental measurements and numerical simulations of the dimensionless vorticity (${\omega }_{x}w/U$) over the $x=0$ plane for fluids of various elasticity number El. Top panels show experimental $\mu$-PIV images obtained at the same value of the dimensionless order parameter $ϵ=0.15$ [Eq. (8)]: (a) $c=0\mathrm{wt}\%$ PEO, $\mathrm{Re}=47.2$; (b) $c=0.001\mathrm{wt}\%$ PEO, $\mathrm{Re}=37.0$; (c) $c=0.003\mathrm{wt}\%$ PEO, $\mathrm{Re}=28.3$. Bottom panels show converged solutions obtained from constant El numerical simulations with the FENE-MCR model and $L^2=5000$ at similar Reynolds numbers to the experiments: (d) $\beta =1$ (Newtonian), $\mathrm{Re}=47.0$; (e) $\beta =0.99$, $\mathrm{Re}=40.0$; (f) $\beta =0.95$, $\mathrm{Re}=26.2$. The color scale indicates the dimensionless vorticity. Superimposed streamline projections exhibit the directionality of the secondary flow in the cross section.

Figure 6

Experimental dimensionless vorticity fields in the $x=0$ plane at $ϵ=0.15$ for (a) $c=0.01\mathrm{wt}\%$ PEO in water ($\mathrm{El}=0.011$, $\mathrm{Re}=19.7$) and (b) $c=0.03\mathrm{wt}\%$ PEO in water ($\mathrm{El}=0.078$, $\mathrm{Re}=17.2$). Under these conditions the flow exhibited mild unsteadiness and the images shown are averaged over 15 individual fields. The color scale indicates the dimensionless vorticity (${\omega }_{x}w/U$). Superimposed streamlines exhibit the directionality of the flow. See also the videos contained in the Supplemental Material [78].

Figure 7

Order parameter $\psi$ as a function of Re for solutions of PEO in water, with the viscosity ratio spanning $0.24<\beta <1$ for the experimental fluids. Closed symbols indicate data obtained with increasing flow rates and open symbols indicate data obtained with decreasing flow rates. Data are fitted with the Landau model [Eq. (8), solid lines]. Numerical data are obtained with the FENE-MCR model with $L^2=5000$ and constant El.

Figure 8

A comparison between (a) experimental measurements with a $c=0.0001\mathrm{wt}\%$ solution of PEO in 8 wt % PEG ($\beta =0.97$, $\mathrm{El}=0.00083$) and (b) numerical simulations with $L^2=5000$ and $\beta =0.97$ at constant $\mathrm{El}=0.00083$. The color scale indicates the dimensionless vorticity ${\omega }_{x}w/U$ in the $x=0$ plane. Projected streamlines show the directionality of the secondary flow in the cross section.

Figure 9

Order parameter $\psi$ as a function of Re for solutions of PEO in 8 wt % PEG, with the viscosity ratio spanning $0.91<\beta <1$ for the experimental fluids. Closed symbols indicate data obtained with increasing flow rates and open symbols indicate data obtained with decreasing flow rates. Data are fitted with the Landau model [Eq. (8), solid lines]. Numerical data (decreasing Re ramp only) are obtained with the FENE-MCR model with $L^2=5000$ and constant El.

Figure 10

Lower critical Reynolds number ${\mathrm{Re}}_c^{*}$ as a function of the elasticity number El for inertia-dominated flow instabilities of low concentration PEO solutions in a cross-slot device with $\alpha =1$. For experimental data, closed symbols represent steady instabilities, while half-closed symbols represent unsteady instabilities. For numerical results, open symbols represent constant ${\mathrm{Wi}}_{\mathrm{eff}}$, while closed symbols represent constant El simulations.

Figure 11

The ratio of the parameters $g$ and $k$ as a function of El. The dashed line indicates $g=0$, where the transition would be tricritical. Above the line the transition is supercritical; below the line the transition is subcritical with hysteresis.

Figure 12

${\mathrm{Wi}}_{\mathrm{eff},c}^{*}$ as a function of El. For experiments, closed symbols represent steady spiral instabilities, half-closed symbols represent unsteady spiral vortex instabilities, and open symbols represent elasticity-dominated asymmetries. For numerical results, open symbols represent constant ${\mathrm{Wi}}_{\mathrm{eff}}$, while closed symbols represent constant El simulations. The dashed line marks $\mathrm{El}=1$. Power-law fit through the data is ${\mathrm{Wi}}_{\mathrm{eff},c}^{*}=6{\mathrm{El}}^{0.8}$. (Note that error bars on the two lowest El experimental data points extend beyond the boundaries of the plot.)

Figure 13

Stability diagram in dimensionless ${\mathrm{Wi}}_{\mathrm{eff}}-\mathrm{Re}$ parameter space. A stable region is indicated below the solid line, drawn through the experimental data as a guide to the eye. For experiments, closed symbols represent steady spiral vortex instabilities, half-closed symbols represent unsteady spiral vortex instabilities, and open symbols represent elasticity-dominated asymmetries. For numerical results, open symbols represent constant ${\mathrm{Wi}}_{\mathrm{eff}}$, while closed symbols represent constant El simulations.

Figure 14

Contours of the normalized first normal stress difference $N_1/({\eta }_{0}U/w)$ (indicated by the color scale) with superimposed streamlines along the center planes of the cross slot. Constant elasticity number simulations using the FENE-MCR model with $\mathrm{El}=0.0042$, $\beta =0.90$ and (a) ${\mathrm{Wi}}_{\mathrm{eff}}=0.02$, $\mathrm{Re}=4.76$ (symmetric), (b) ${\mathrm{Wi}}_{\mathrm{eff}}=0.08$, $\mathrm{Re}=19.05$ (symmetric), and (c) ${\mathrm{Wi}}_{\mathrm{eff}}=0.125$, $\mathrm{Re}=29.76$ (asymmetric).