| Wed June 8th 2011 16:00 HR Z203 | Seminar | Nonlinear Stability, Bifurcation and Mode Interaction in Granular Plane Couette Flow Priyanka Shukla |
Details: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 twodimensional granular plane Couette flow. The relevant order parameter equation, the LandauStuart equation, for the most unstable two dimensional disturbance has been derived using the center manifold method and the amplitude expansion method. Along with the linear eigenvalue problem, the meanflow distortion, the second harmonic, the distortion to the fundamental mode and the first Landau coefficient are calculated analytically (for shearbanding instability) and numerically using the spectral collocation method. It is found that the flow is subcritically unstable for the shearbanding modes in the dilute regime although the flow is linearly stable in this regime. 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 wavelengths 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 wave number is of O(1). Finiteamplitude patterns of solid fraction, velocity and granular temperature for all types of instabilities are contrasted with their linear counter parts. We show that the granular plane Couette flow is prone to a plethora of resonances, and the evidence for two types of modal resonances is demonstrated: (i) a ‘meanflow resonance’ which occurs due to the interaction of the streamwiseindependent shear banding modes (zero wavenumber modes) and the mean flow distortions, and (ii) an exact ‘2 : 1 resonance’ tha | ||
