Landau-type Order Parameter Equation for Shear Banding in Granular Couette Flow

We show that a Landau-type “order-parameter” equation describes the onset of shear-band formation in granular plane Couette flow wherein the flow undergoes an ordering transition into alternate layers of dense and dilute regions of low and high shear rates, respectively, parallel to the flow direction. Even though the linear theory predicts the stability of the homogeneous shear solution in dilute flows, our analytical bifurcation theory suggests that there is a subcritical finite-amplitude instability that is likely to lead to shear-band formation in dilute flows, which is in agreement with previous numerical simulations.

Figure 1

(a) Phase diagram in the ($H$, $\varphi$) plane, showing the neutral stability contour (red line) and the contours of vanishing first Landau coefficient (blue lines) for a restitution coefficient of 0.95. The inset shows a typical density eigenfunction within the “linearly” unstable region. (b) Shear-band formation in simulations of dilute granular PCF: $\varphi =0.05$, $e=0.6$ and $N=20000$. (adapted from M.-L. Tan’s thesis 1995). Lees-Edwards boundary condition has been used with the top boundary moving to the right and the bottom boundary to the left with the same speed. (c) Variations of $a^{(0)}$ and $a^{(2)}$ with density at $H=200$ for $e=0.95$. The critical density is defined as the one at which the linear growth rate is zero, ${\varphi }_c=\varphi (a^{(0)}=0)\approx 0.157$ at $H=200$.

Figure 2

(a) Bifurcation diagram in the amplitude-vs-density plane for $e=0.95$ (red) and 0.6 (blue) at $H=200$. Inset shows a bifurcation diagram at $H=100$ and $e=0.95$. (b) Finite-amplitude solutions for the density and the shear rate at $\underline{\varphi }=0.15$, $H=200$ and $e=0.6$. Note that the density has been multiplied by a factor of 6.