Vorticity structuring and velocity rolls triggered by gradient shear bands

We suggest a mechanism by which vorticity structuring and velocity rolls can form in complex fluids, triggered by the linear instability of one-dimensional gradient shear banded flow. We support this with a numerical study of the diffusive Johnson-Segalman model. In the steady vorticity structured state, the thickness of the interface between the bands remains finite in the limit of zero stress diffusivity, presenting a possible challenge to the accepted theory of shear banding.

Figure 1

(a) Homogeneous constitutive curve and steady state flow curve for 1D planar gradient banded flow in the DJS model at $a=0.3,\eta =0.05$. Inset: schematic arrangement of the bands in curved Couette flow. (b) Schematic constitutive curve giving shear thickening vorticity banding. (c) A shear thinning curve allowing gradient or vorticity banding in 1D.

Figure 2

(a) Dispersion relation for perturbations about a 1D banded state for $l=0.00375$, 0.00250, 0.00125. $a=0.3$, $\eta =0.05$, $\rho ∕\eta =0.02$. Inset: Linear dynamics of 2D code, starting from a flat interface. Solid lines: weight in modes, $q_z{L}_z∕2\pi =1,2$; dashed: analytical prediction. (b) Peak ${\omega }^{*}$ ($q^{*}$ in inset) in dispersion relation vs $\overline{\dot{\gamma }}$ across the plateau of Fig. 1a. Vertical dashed lines denote the edges of the stress plateau.

Figure 3

Steady state at $a=0.3$, $\eta =0.05$, $\overline{\dot{\gamma }}=2.0$, $l=0.00375$, $L_z=4.0$. Left: greyscale of ${\Sigma }_{xx}$ in the $y\text{−}z$ plane. Middle: $(y-z)$ velocity vectors, showing roll states. Right: vorticity banding of viscoelastic shear stress.

Figure 4

Selected stress: 1D initial state and 2D steady state. $L_z=2.0$, $a=0.3$, $\eta =0.05$, $\overline{\dot{\gamma }}=2.0$. At $l=0$, the stress varies erratically about an average that depends on the grid. Inset: evolution to steady state at $l=0.00375$.

Figure 5

Profile ${\Sigma }_{xx}$ normal to the interface; $l=0.00375$ (solid lines), $l=0.00250$ (dashed lines), $0.00125$ (dot-dashed lines). $a=0.3$, $\eta =0.05$, $\overline{\dot{\gamma }}=2.0$, $L_z=2.0$. (a),(b) 1D state with interfacial thickness $d\propto l$. (c),(d); (e),(f) 2D state at the $z$ coordinate where the interface has maximal $+$ and $-y$ displacements. The thickness $d$ appears independent of $l$, but note the bump of thickness $O\left(l\right)$. The offset $y_0\left(l\right)$ is chosen to center the profile at the same location in each case.