Elastic turbulence in highly entangled polymers and wormlike micelles

Abstract

We show theoretically that an initially homogeneous planar Couette flow of a concentrated polymeric fluid is linearly unstable to the growth of two-dimensional (2D) perturbations, within two widely used constitutive models: the Johnson-Segalman model and the Rolie-Poly model. We perform direct nonlinear simulations of both models in 2D to show that this instability leads to a state of elastic turbulence comprising several narrow shear bands that dynamically coalesce, split and interact. Importantly, we show that this 2D instability arises not only in fluids that have a non-monotonic constitutive curve, and therefore show shear banding in 1D calculations, but also in shear thinning fluids with a monotonic constitutive curve, for which an initially homogeneous base state is stable in 1D. For the former category, the high shear branch of the constitutive curve is unstable to 2D instability in both models, so that the high shear band may be turbulent. In the Rolie-Poly model, the low shear branch is also likewise unstable. Our work provides the first simulation evidence for elastic turbulence in highly entangled polymeric fluids. It also potentially explains rheo-chaotic states seen experimentally in shear banding wormlike micelles. We additionally demonstrate elastic turbulence within both models in the planar Poiseuille geometry.

Figures (3)

• Figure 1: Constitutive curves and stability maps in $\textrm{pCf}$ of the JS model with $\delta=10^{-2}$, $\varepsilon=10^{-3}$ and of the RP model with $\delta=10^{-4}$, $\varepsilon=10^{-3}$. (a) Constitutive curves of the JS model for several values of $a$. (b) Constitutive curves of the RP model for several values of $Z$. (c) Stability map of the JS model in the $a\textrm{-}W$ plane. (c) Stability map of the RP model in the $Z\textrm{-}W$ plane. Colours show linear instabilities with $0<\omega^*=\omega_{1D}$ (red), $0<\omega_{1D}<\omega^*$ (green) and $\omega_{1D}<0<\omega^*$ (blue). Hatching for the JS model in c) shows regions where the PDI is the most unstable mode. The PDI does not arise for the RP model in d). The dotted line in d) additionally shows the neutral curve when $\delta=10^{-3}$. Dispersion relations and most unstable eigenmodes at symbols along the horizontal dashed lines in (c) at $a=0.5$ and in (d) at $Z=1000$ are shown in Fig. S1 in the Supplementary Material.
• Figure 2: (a) Flow curves obtained by ramping down the shear rate within the JS model in $\textrm{pCf}$: in 2D (solid blue line), 1D (solid orange line) and 0D (black dashed lines). In each case the initial condition comprises a state of homogeneous shear at $W=20$ (red circle). In the 2D simulations, steady stress values are shown by crosses and fluctuating stress values by closed (resp. open) circles for standard deviations larger (resp. smaller) then $0.01$, with vertical red bars showing $\pm1$ standard deviation. Linear stability regimes of the 0D curve are marked by colours as in Fig. \ref{['linear_stability1']}a. $a=0.5$, $\delta=10^{-2}$, $\varepsilon=10^{-3}$. (b) Snapshots of the trace $\Sigma_{xx}+\Sigma_{yy}+\Sigma_{zz}$ of the polymer stress for states marked $(i)-(vi)$ in (a). (c) Snapshot of the trace of the polymer stress in the RP model in $\textrm{pCf}$ for $W=20$ and $Z=50$, $\delta=10^{-3}$, $\varepsilon=10^{-3}$. (d) Counterpart for the RP model in $\textrm{pPf}$ for $G=-1$. Timeseries and frequency spectra of the stress signals are shown in the Supplementary Material supp.
• Figure 3: The trace $\Sigma_{xx}+\Sigma_{yy} +\Sigma_{zz}$ in Poiseuille-Couette flow of the JS model with parameters $G=-0.5$, $W=20$, $\delta=10^{-2}$, $a=0.5$ and $\varepsilon=10^{-3}$.