Linear Instability of Planar Shear Banded Flow

We study the linear stability of planar shear banded flow with respect to perturbations with wave vector in the plane of the banding interface, within the nonlocal Johnson-Segalman model. We find that perturbations grow in time, over a range of wave vectors, rendering the interface linearly unstable. Results for the unstable eigenfunction are used to discuss the nature of the instability. We also comment on the stability of phase separated domains to shear flow in model $H$.

Figure 1

Underlying constitutive curve; steady state flow curve. $a=0.3$, $\eta =0.05$. Banding occurs on the plateau.

Figure 2

A 1D banded profile, with spatial gradients restricted to the flow-gradient direction, $y$. $\overline{\dot{\gamma }}=2.0$, towards the left of the plateau in Fig. 1. $l=0.01$, $N_{\mathrm{base}}=800$. Note the total shear stress $T_{xy}$ is uniform across the cell.

Figure 3

Real part of the eigenvalue of the most unstable mode. $a=0.3$, $\eta =0.05$, $\overline{\dot{\gamma }}=2.0$, Reynolds number $\rho /\eta =0$. The data for $l=0.01$ correspond to the base profile in Fig. 2. Symbols: data. Solid lines: cubic splines.

Figure 4

Peak of the dispersion relation, i.e., $\mathrm{Re}{\omega }_{\max }$ at $d\mathrm{Re}{\omega }_{\max }/d{q}_x=0$. Parameters as for Fig. 3. Limits of the banding regime shown by vertical lines. Symbols: data. Dotted lines: cubic splines, as a guide to the eye.

Figure 5

Perturbation to flow field $s_1\mathrm{Re}\mathbf{v}(y)e^{i{q}_{x}x}$ (arrows), and contour lines of perturbed normal stress ${\tilde{\Sigma }}_{xx}(x,y)={\Sigma }_{xx}(y)+s_2\mathrm{Re}{\sigma }_{xx}(y)e^{i{q}_{x}x}$ (dotted lines), corresponding to the eigenvalue of Fig. 3 with $l=0.01$, $q_x=2.0$. Contours downwards: 0.45, 0.60, 0.75, 0.90, 1.05, 1.20, 1.35 (middle value shown thicker). Arbitrary scale factors $s_1=1.5$ and $s_2=0.3$.

Figure 6

Real part of the most unstable eigenvalue. $a=0.3$, $\eta =0.05$, $\overline{\dot{\gamma }}=2.0$, Reynolds number $\rho /\eta =0.01$ (negligible), $l=0.01$. Contours are $-0.45,-0.40,\dots$ (dotted line), $0.00$ (dashed line), and $\dots 0.25,0.30$ (solid line).