Instability-induced ordering, universal unfolding and the role of gravity in granular Couette flow

Abstract

Linear stability theory and bifurcation analysis are used to investigate the role of gravity in shear-band formation in granular Couette flow, considering a kinetictheory rheological model. We show that the only possible state, at low shear rates, corresponds to a 'plug' near the bottom wall, in which the particles are densely packed and the shear rate is close to zero, and a uniformly sheared dilute region above it. The origin of such plugged states is shown to be tied to the spontaneous symmetry-breaking instabilities of the gravity-free uniform shear flow, leading to the formation of ordered bands of alternating dilute and dense regions in the transverse direction, via an infinite hierarchy of pitchfork bifurcations. Gravity plays the role of an 'imperfection', thus destroying the 'perfect' bifurcation structure of uniform shear. The present bifurcation problem admits universal unfolding of pitchfork bifurcations which subsequently leads to the formation of a sequence of a countably infinite number of 'isolas', with the solution structures being a modulated version of their gravity-free counterpart. While the solution with a plug near the bottom wall looks remarkably similar to the shear-banding phenomenon in dense slow granular Couette flows, a 'floating' plug near the top wall is also a solution of these equations at high shear rates. A two-dimensional linear stability analysis suggests that these floating plugged states are unstable to long-wave travelling disturbances.The unique solution having a bottom plug can also be unstable to long waves, but remains stable at sufficiently low shear rates. The implications and realizability of the present results are discussed in the light of shear-cell experiments under 'microgravity' conditions.

Collision statistics in sheared inelastic hard spheres

“…In low density systems with large numbers of particles, clustering occurs [27]. This leads to a local variation of the strain rate in the system, and, consequently, the system will not be homogeneously sheared.…”

“…Due to the computational limitations, the wall separation is typically of the order of a few particle diameters, and wall effects dominate the simulation results. For large system sizes, shear instability is observed [27].…”

Shearbanding and inhomogeneous states in granular fluid

“…(1) the mass density (ρ =ρ p φ , with φ being the volume fraction of particles), (2)(3)(4) three components of hydrodynamic velocity [u = (u, v, w) Tr , with u, v and w being the velocity components along x, y and z-directions, respec-tively] and (5) the granular temperature (T ), These equations are made dimensionless by using the gap between two walls (h) and the inverse of the overall shear rate (h/Ū w ) as reference length and time scales, respectively. The dimensionless forms of hydrodynamic equations and the transport coefficients can be found in Ref.…”

“…The nonlinear "equilibrium" solutions, A = A e , correspond to stationary solutions of Landau equation (4). This equation always admits one 'zero' solution (A e = 0) which represents the base state of USF.…”

“…The shear-banding phenomena has been studied in earth-bound shear-cell experiments of dense granular materials [2] as well as in theory and simulations of both dense and dilute sheared systems [4]. On theoretical fronts, the gradient banding phenomena found in simulations of granular plane shear flow have been explained in terms of the nonlinear saturation of instabilities of the underlying hydrodynamic equations via an order-parameter (Landau) equation [5] in diluteto-moderately dense flows.…”