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.