Characterization of superimposed instabilities in the planar extensional flow of viscoelastic fluids

We simulate the isothermal, incompressible, time-dependent flow of Boger fluids, using the FENE-CR constitutive model, in the optimized shape cross-slot extensional rheometer [Haward et al., Phys. Rev. Lett. 109, 128301 (2012)]. We uncover a family of predominantly elastic instabilities resulting from the interaction of stationary asymmetric and time-dependent flow transitions. The superposition of these instabilities, with varying degrees of relative amplitude, produces five distinct flow regimes which are classified by their elasticity, suggesting that the dynamic system is situated in the vicinity of a triple point in the inertia-elasticity-viscosity parameter space. A detailed characterization of the various associated modes of instability is provided. Spectral analysis of the first component of the velocity vector suggests that the flow regimes featuring loss of birefringence strand integrity become chaotic and locally turbulent near the center of the cross-slot, above a certain flow rate threshold.

Figure 1

(a) Geometry of the optimized shape cross-slot extensional rheometer (OSCER device). The half-width $H$ is used as the characteristic length scale. (b) Detail of the computational mesh. The deformation of computational cells is the result of the application of an optimization algorithm to the mesh of a planar cross-slot, described in Ref. [18]. (c) Zoomed-out view of the mesh, showing the inlet and outlet channels. Cells in these channels are progressively larger in the streamwise direction towards the inlets and outlets.

Figure 2

Illustrative results for the very low elasticity regime. The example given is the fluid aPS7-350 ppm at $\mathrm{W}{\mathrm{i}}_{\mathrm{nom}}=10.4$ and $\mathrm{E}{\mathrm{l}}_{\mathrm{nom}}=1.6$. The initial, transient segment of the simulation was discarded, and time was reset to zero. (a), (b) Space-time diagrams of the normalized first normal stress difference, respectively, along the horizontal and vertical centerlines, with the evolving position of the stagnation point overlaid on the former. (c) Continuous line: time series of the first velocity component $u_x$ at the center of the OSCER device, normalized by the characteristic velocity $U_0$; dotted line: time series of the normalized first normal stress difference sampled at the same location. (d) Phase-plane plot constructed with the time series shown in the previous panel.

Figure 3

Illustrative results for the low elasticity regime. The example given is the fluid aPS7-1400 ppm at $\mathrm{W}{\mathrm{i}}_{\mathrm{nom}}=3.4$ and $\mathrm{E}{\mathrm{l}}_{\mathrm{nom}}=3.2$. The initial, transient segment of the simulation was discarded, and time was reset to zero. (a), (b) Space-time diagrams of the normalized first normal stress difference, respectively, along the horizontal and vertical centerlines, with the evolving position of the stagnation point overlaid on the former. (c) Continuous line: time series of the first velocity component $u_x$ at the center of OSCER device, normalized by the characteristic velocity $U_0$; dotted line: time series of the normalized first normal stress difference sampled at the same location. (d) Phase-plane plot constructed with the time series shown in the previous panel.

Figure 4

Illustrative results for the intermediate elasticity regime. The example given is the fluid aPS10-700 ppm at $\mathrm{W}{\mathrm{i}}_{\mathrm{nom}}=7.6$ and $\mathrm{E}{\mathrm{l}}_{\mathrm{nom}}=5.6$. The initial, transient segment of the simulation was discarded, and time was reset to zero. (a), (b) Space-time diagrams of the normalized first normal stress difference, respectively, along the horizontal and vertical centerlines, with the evolving position of the stagnation point overlaid on the former. (c) Continuous line: time series of the first velocity component $u_x$ at the center of OSCER device, normalized by the characteristic velocity $U_0$; dotted line: time series of the normalized first normal stress difference sampled at the same location. (d) Phase-plane plot constructed with the time series shown in the previous panel.

Figure 5

Depictions of the flow field at specific times, represented as contour plots of the normalized first normal stress difference $N_1^{*}=N_1/\phantom{N_1^{*}=N_1\left({\eta }_0{U}_0/\phantom{{\eta }_0{U}_{0}2H}2H\right)}\left({\eta }_0{U}_0/\phantom{{\eta }_0{U}_{0}2H}2H\right)$ with superimposed streamlines. (a) Snapshot depicting the flow of aPS7-350 ppm, at $\mathrm{W}{\mathrm{i}}_{\mathrm{nom}}=10.4$ and $\mathrm{E}{\mathrm{l}}_{\mathrm{nom}}=1.6$, taken at $t/{\lambda }_0\approx 7.1$ of the corresponding time series in Fig. 2. (b) Snapshot depicting the flow of aPS10-700 ppm, at $\mathrm{W}{\mathrm{i}}_{\mathrm{nom}}=7.6$ and $\mathrm{E}{\mathrm{l}}_{\mathrm{nom}}=5.6$, taken at $t/{\lambda }_0\approx 6.8$ of the corresponding time series in Fig. 4. (c) Snapshot depicting the flow of aPS16-1400 ppm, at $\mathrm{W}{\mathrm{i}}_{\mathrm{nom}}=2.0$ and $\mathrm{E}{\mathrm{l}}_{\mathrm{nom}}=32.2$, zoomed-in along the $y$-axis direction, taken at $t/{\lambda }_0\approx 22.4$ of the corresponding time series in Fig. 7. (d) Stationary asymmetric flow field, aPS16-350 ppm, $\mathrm{W}{\mathrm{i}}_{\mathrm{nom}}=3.4, \mathrm{E}{\mathrm{l}}_{\mathrm{nom}}=13.7$.

Figure 6

Illustrative results for the high elasticity regime. The example given is the fluid aPS10-1400 ppm at $\mathrm{W}{\mathrm{i}}_{\mathrm{nom}}=3.6$ and $\mathrm{E}{\mathrm{l}}_{\mathrm{nom}}=8.1$. The initial, transient segment of the simulation was discarded, and time was reset to zero. (a), (b) Space-time diagrams of the normalized first normal stress difference, respectively, along the horizontal and vertical centerlines, with the evolving position of the stagnation point overlaid on the former. (c) Continuous line: time series of the first velocity component $u_x$ at the center of OSCER device, normalized by the characteristic velocity $U_0$; dotted line: time series of the normalized first normal stress difference sampled at the same location. (d) Phase-plane plot constructed with the time series shown in the previous panel.

Figure 7

Illustrative results for the very high elasticity regime. The example given is the fluid aPS16-1400 ppm at $\mathrm{W}{\mathrm{i}}_{\mathrm{nom}}=2.0$ and $\mathrm{E}{\mathrm{l}}_{\mathrm{nom}}=32.2$. The initial, transient segment of the simulation was discarded, and time was reset to zero. (a), (b) Space-time diagrams of the normalized first normal stress difference, respectively, along the horizontal and vertical centerlines, with the evolving position of the stagnation point overlaid on the former. (c) Continuous line: time series of the first velocity component $u_x$ at the center of OSCER device, normalized by the characteristic velocity $U_0$; dotted line: time series of the normalized first normal stress difference sampled at the same location. (d) Phase-plane plot constructed with the time series shown in the previous panel.

Figure 8

Stability diagram showing the evolution of each fluid in the Wi-Re parameter space. Empty circles represent stationary symmetric flow, whereas filled squares indicate a stationary asymmetric flow field configuration. Each of the five time-dependent regimes identified in this work is represented as follows: filled circles, very low elasticity regime; filled diamonds, low elasticity regime; filled triangles, intermediate elasticity regime; empty triangles, high elasticity regime; empty diamonds, very high elasticity regime.

Figure 9

Histograms showing the probability distribution of the normalized first normal stress difference, $N_1^{*}=N_1/\phantom{N_1^{*}=N_1\left({\eta }_0{U}_0/\phantom{{\eta }_0{U}_{0}2H}2H\right)}\left({\eta }_0{U}_0/\phantom{{\eta }_0{U}_{0}2H}2H\right)$, at the center of the OSCER device. For each regime, represented by one of the fluids, results are shown for the highest stable flow rate (blue) and a relevant unstable flow (red), also depicted in Figs. 2, 3, 4, 5, 6, 7. Data are organized into twenty equally spaced bins of $N_1^{*}$ between 0 and 800.

Figure 10

Power spectra obtained from time series of the normalized first velocity component $u_x/U_0$ sampled at coordinates $\left(x,y\right)=\left(H,0\right)$, for the very low elasticity regime. The degree of vertical shifting of the PSD is indicated in the legend of each panel. The frequency axis is linear, and the power density axis is logarithmic.

Figure 11

Power spectra obtained from time series of the normalized first velocity component $u_x/U_0$ sampled at coordinates $\left(x,y\right)=\left(H,0\right)$, for the intermediate elasticity regime. The degree of vertical shifting of the PSD is indicated in the legend. The frequency axis is linear, and the power density axis is logarithmic.

Figure 12

Power spectra obtained from time series of the normalized first velocity component $u_x/U_0$ sampled at coordinates $\left(x,y\right)=\left(H,0\right)$, for the low elasticity regime. The degree of vertical shifting of the PSD is indicated in the legend. Both the frequency and power density axes are logarithmic.

Figure 13

Power spectra obtained from time series of the normalized first velocity component $u_x/U_0$ sampled at coordinates $\left(x,y\right)=\left(H,0\right)$, for the high elasticity regime. The degree of vertical shifting of the PSD is indicated in the legend. In panel (a.1) the frequency axis is linear and the power density axis is logarithmic, whereas in panel (a.2) both axes are logarithmic.

Figure 14

Power spectra obtained from time series of the normalized first velocity component $u_x/U_0$ sampled at coordinates $\left(x,y\right)=\left(H,0\right)$, for the very high elasticity regime. The degree of vertical shifting of the PSD is indicated in the legend of each panel. In panels (b.1) and (c.1) the frequency axis is linear, and the power density axis is logarithmic, whereas in panels (a), (b.2) and (c.2) both axes are logarithmic.

Figure 15

Turbulence intensity along the downstream centerline of the OSCER device for the highest simulated flow rate for each of the test fluids. (a) Low elasticity regime and (b) very high elasticity regime. A combined linear/log scale is used in the $x/H$ axis.

Figure 16

Principal frequency $f_{max}$ (a) multiplied by the zero shear-rate relaxation time ${\lambda }_0$ or (b) multiplied by the nominal effective relaxation time ${\lambda }_{\mathrm{eff},\mathrm{nom}}$, plotted against the dimensionless flow rate expressed as the nominal Weissenberg number $\mathrm{W}{\mathrm{i}}_{\mathrm{nom}}$. ${\lambda }_{\mathrm{eff},\mathrm{nom}}$ is calculated at the center of the OSCER device for a hypothetical stationary extensional flow with characteristic strain rate $\dot{\varepsilon }=1.5U_0/\phantom{1.5U_{0}15H}15H$. Only the flows with a clearly defined maximum power density peak are represented, usually in the form of the principal component of a harmonic series.