Early Stage Kinetics in a Unified Model of Shear-Induced Demixing and Mechanical Shear Banding Instabilities

We present a unified model of shear-induced demixing and “mechanical” shear banding instabilities in polymeric and surfactant solutions, by combining a simple flow instability with a two-fluid approach to concentration fluctuations. Within this model, we calculate the “spinodal” limit of stability of initially homogeneous shear states to demixing/banding, and predict the selected length and time scales at which inhomogeneity first emerges after a shear start-up “quench” into the unstable region, finding qualitative agreement with experiment. Our analysis is the counterpart, for this driven phase transition, of the Cahn-Hilliard calculation for unsheared fluid-fluid demixing.

Figure 1

Schematic constitutive curve showing flow instability.

Figure 2

Lines: Intrinsic constitutive curves for $\varphi =0.11$, 0.091, 0.072, 0.053, 0.034, and 0.015 (downwards). Symbols: Spinodals for the uncoupled limit $\zeta \to \infty$ at fixed $\mathcal{D}\propto f^{\prime \prime }/\zeta (\varphi )$(○); coupled model with $\mathcal{D}(\varphi =0.11)$ from DLS (□), and artificially reduced $\mathcal{D}$ (◇, △).

Figure 3

Instability occurring as the homogeneous start-up flow evolves towards the intrinsic unstable constitutive curve.

Figure 4

Start-up dispersion relations, ${\omega }_k$. (a) Uncoupled, $\dot{\gamma }=7.0$, ${\tau }_{\mathrm{i}\mathrm{n}\mathrm{s}\mathrm{t}}\approx 0.9$, no selected $k^{*}$. (b) $\mathcal{D}=2.6\times {10}^{-4}$, $\dot{\gamma }=4.0$, ${\tau }_{\mathrm{i}\mathrm{n}\mathrm{s}\mathrm{t}}\approx 2.5$, $\log (k^{*}/\pi )\approx 1.8$. (c) $\mathcal{D}=2.6\times {10}^{-9}$, $\dot{\gamma }=0.01$, ${\tau }_{\mathrm{i}\mathrm{n}\mathrm{s}\mathrm{t}}\gg 20$, $\log (k^{*}/\pi )\approx 1.5$.