Viscorotational shear instability of Keplerian granular flows

The linear stability of viscous Keplerian flow around a gravitating center is studied using the rheological granular fluid model. The linear rheological instability triggered by the interplay of the shear rheology and Keplerian differential rotation of incompressible dense granular fluids is found. Instability sets in in granular fluids, where the viscosity parameter grows faster than the square of the local shear rate (strain rate) at constant pressure. Found instability can play a crucial role in the dynamics of dense planetary rings and granular flows in protoplanetary disks.

Figure 1

Normalized growth rate of axisymmetric perturbations in granular fluids under the influence of rheological viscous stress $\mathrm{Im}\left[\omega (k_x,k_z)\right]/{\mathrm{\Omega }}_0$ for different values of nondimensional pressure $g_p={\mathrm{\Omega }}_0{G}_P$ and shear $g_s={\mathrm{\Omega }}_0{G}_S/\nu$ rheology parameters: (a) $g_P=-0.1,g_S=0$; (b) $g_P=0,g_S=1$; (c) $g_P=-0.1,g_S=1$; and (d) $g_P=0.1,g_S=1$.

Figure 2

Normalized growth rate of nonaxisymmetric perturbations in the $k_x-k_y$ plane: $\mathrm{Im}\left[\omega (k_x,k_y)\right]/{\mathrm{\Omega }}_0,k_z/k_{\nu }=0.01$ and (a) $g_P=-0.2,g_S=0$; (b) $g_P=-0.2,g_S=1.5$. Horizontal arrows indicate wave-number drift due to the background shear.

Figure 3

Evolution of the energy of nonaxisymmetric perturbations with $g_p=0,g_s=1.5,k_x\left(0\right)/k_{\nu }=-0.8,k_z/k_{\nu }=0.01$, and different values of azimuthal wave number $k_y$. Modes with higher $k_y$ undergo faster shearing deformation having less time to grow due to the viscorotational mechanism.