Nonlinear Dynamics of an Interface between Shear Bands

We study numerically the nonlinear dynamics of a shear banding interface in two-dimensional planar shear flow, within the nonlocal Johnson-Segalman model. Consistent with a recent linear stability analysis, we find that an initially flat interface is unstable with respect to small undulations for a sufficiently small ratio of the interfacial width $\ell$ to cell length $L_x$. The instability saturates in finite amplitude interfacial fluctuations. For decreasing $\ell /L_x$ these undergo a nonequilibrium transition from simple traveling interfacial waves with constant average wall stress, to periodically rippling waves with a periodic stress response. When multiple shear bands are present we find erratic interfacial dynamics and a stress response suggesting low dimensional chaos.

Figure 1

(a) Growth rate $\omega (q_x)$ of perturbations about an initially 1D banded state. (○, solid line): predictions of linear stability [16] for $\ell =0.005,0.01,0.015$. (□): early-time dynamics of the full 2D numerics for $\ell =0.005$, $L_x=4,6,8$; $\ell =0.01$, $L_x=2,4,6,8$; $\ell =0.015$, $L_x=2,4,8$. (b) Power spectrum $P(q_x)$ in the traveling wave regime for $\ell =0.005$, $L_x=1$ (○); $\ell =0.01$, $L_x=2$ (□); $\ell =0.015$, $L_x=2,4,6$ (△).

Figure 2

“Phase diagram” for $\overline{\dot{\gamma }}=2.0$, showing the different nonlinear regimes for IC1 (accurate to spacing between numbers) and the numbers of linearly unstable modes [Ref. 16 ].

Figure 3

Gray scale of order parameters for traveling wave in the ($x,y$) plane for $\ell =0.015$, $L_x=6$, and upper wall velocity $V\equiv \overline{\dot{\gamma }}L=2$ to the right.

Figure 4

Rippling wave at $\ell =0.005$, $L_x=4$, $\overline{\dot{\gamma }}=2$. Top: gray scale of ${\Sigma }_{xx}(x,y)$. Upper wall moves to the right. White line in (d): interface height. Middle: average wall stress. Inset: $N_y=800$, $(N_x,dt)=(200,0.0003)$, solid line; (400, 0.0001), dotted line; (400, 0.0003), dashed line; (500, 0.0002), dot-dashed line. Bottom: gray scale of interface height in $(x-ct,t)$ plane, with convective motion subtracted out. Time $t=150\dots 160$ upwards.

Figure 5

Multiple bands after start-up from rest for $\ell =0.005$, $L_x=2$, $\overline{\dot{\gamma }}=2$. Top: gray scale of ${\Sigma }_{xx}(x,y)$ at time $t=120$, reconstructed from the first 15 Fourier modes. Bottom: shear stress $T_{xy}(t)$.