| According to the linear stability analysis of kinetic-theory continuum models, the rapid granular plane Couette flow can become unstable due to traveling and stationary waves, having modulations in both streamwise (x) and gradient (y) directions: the `dominant' stationary and travelling instabilities (having the largest growth rate) appear when the streamwise wavenumber (kx) is of order one, i.e. kx=O(1); in addition there are long-wave (kx∼ 0) stationary and travelling instabilities, having much smaller growth rates than their dominant counterparts at kx=O(1), that result from the evolution of the unstable shear-banding mode (kx=0) with the streamwise wavenumber. In this work a weakly nonlinear stability theory is developed to understand the effect of nonlinearities on various linear instability modes as well as to unveil the underlying bifurcation scenario in a two-dimensional granular plane Couette flow. The relevant order parameter equation for the most unstable two-dimensional disturbance has been derived using the amplitude expansion method of our previous work on the shear-banding instability Shukla and Alam, Phys. Rev. Lett. 103, 068001 (2009); Shukla and Alam, J. Fluid Mech. (2010, accepted). Along with the linear eigenvalue problem, the mean-flow distortion, the second harmonic, the distortion to the fundamental mode and the first Landau coefficient are calculated numerically using the spectral collocation method. Two types of bifurcations, Hopf and pitchfork, that result from travelling and stationary linear instabilities, respectively, are analysed using the first Landau coefficient. It is shown that the subcritical finite amplitude instability can appear in the linearly stable regime for a range of parameters. The present bifurcation theory shows that the flow is subcritically unstable to disturbances of long wave-lengths (kx∼ 0) in the dilute limit, and both the supercritical and subcritical states are possible at moderate densities for the dominant stationary and traveling instabilities for which kx=O(1). Finite-amplitude patterns of solid fraction, velocity and granular temperature for all types of instabilities are contrasted with their linear counterparts. We show that the granular plane Couette flow is prone to a plethora of resonances, and the evidence for two types of such modal resonances is demonstrated: (i) a `mean-flow resonance' which occurs due to the interaction of the streamwise-independent shear-banding modes (kx=0) and the mean flow distortions, and (ii) an exact `2:1 resonance' that results from the interaction of two stationary waves with their wave-number ratio being 2:1. The extension of the present theory to deal with such resonantly interacting modes will be discussed. |
|