Nonlinear stability, bifurcation and mode interaction in granular plane couette flow

Abstract

This thesis starts with a brief overview of patterns in rapid granular flows driven by vibration, gravity and shear, e.g. standing wave patterns and convection rolls in a vibrated granular system, density waves in gravity driven granular Poiseuille flow, fingering in chute flow, shearbanding in shear flow, etc. The pattern forming order parameter models such as coupled complex GinzburgLandau model, Swift-Hohenberg model and continuous coupled map model have been described in chapter 1. The continuum theory of granular fluid and Navier Stokes order constitutive models for the inelastic hard-sphere and hard-disk fluids have been detailed in chapter 2. In chapter 3 a general weakly nonlinear stability analysis using amplitude expansion method has been described. A spectral based numerical scheme for solving weakly nonlinear equations and solvability condition have been developed in chapter 3. In the first problem of present thesis, a weakly nonlinear theory, in terms of the well-known Landau equation, has been developed to describe the nonlinear saturation of shear-banding instability in rapid granular plane Couette flow. The shear-banding instability corresponds to streamwise-independent perturbations (d/dx(-) = 0 and d/dy{-) ^ 0, where x and y refer to flow and gradient directions, respectively) of the underlying steady uniform shear flow which degenerates into alternate layers of dense and dilute regions of low and high shear-rates, respectively, along the gradient direction. The nonlinear stability of this shear-banding instability is analyzed using two perturbation methods, the center manifold reduction method (chapter 4) and the amplitude expansion method (chapter 5); the resulting nonlinear problem has been reduced to a sequence of linear problems for the fundamental mode, its higher-order harmonics and distortions, and the base-flow distortions of various order. The first Landau coefficient, which is the leading nonlinear correction in the Landau equation at cubic order in the amplitude of perturbation, derived from the present method exactly matches with the same obtained from the center manifold reduction technique. The nonlinear modes are found to follow certain symmetries of the base flow and the fundamental mode. These symmetries helped to identify analytical solutions for the base-flow distortion and the second harmonic, leading to an exact calculation of the first Landau coefficient. The present analytical solutions are further used to validate an spectral-based numerical method for nonlinear stability calculation. The regimes of supercritical and subcritical bifurcations for the shear-banding instability are identified, leading to the prediction that the lower branch of the neutral stability contour in the (H, 0°)-plane, where H is the scaled Couette gap (the ratio between the Couette gap and the particle diameter) and cffi is the mean density or the volume fraction of particles, is sub-critically unstable. Our results suggest that there is a subcritical finite amplitude instabifity for dilute fiows even though the dilute flow is stable according to the linear theory which agrees with previous numerical simulation. Bifurcation diagrams are presented, and the predicted finite-amplitude solutions, representing shear-localization and density segregation, are discussed in the light of previous molecular dynamics simulations of plane shear fiow. Our analysis suggests that there is a sequence of transitions among three types of pitchfork bifurcations with increasing mean density: from (i) the bifurcation from infinity in the Boltzmann limit to (ii) subcritical bifurcation at moderate densities to (iii) supercritical bifurcation at a larger density to (iv) subcritical bifurcation in the dense limit and finally again to (v) supercritical bifurcation near the close packing density. It is shown that the appearance of subcritical bifurcation in the dense limit depends on the choice of the contact radial distribution function and the constitutive relations. The critical mean density at any transition from one bifurcation-type to another is exactly calculated from our analytical bifurcation theory. The scalings of the first Landau coefficient, the equilibrium amplitude and the pha;Se diagram, in terms of mode number and inelasticity, are demonstrated. The granular plane Couette flow serves as a paradigm that supports all three possible types of pitchfork bifurcations, with the mean density {(p^) being the single control parameter that dictates the nature of bifurcation.