Nonlinear stability, bifurcation and vortical patterns in three-dimensional granular plane Couette flow

Abstract

The effects of three-dimensional (3D) perturbations, having wave-like modulations along both the streamwise and spanwise/vorticity directions, on the nonlinear states of five types of linear instability modes, the nature of their bifurcations and the resulting nonlinear patterns are analysed for granular plane Couette flow using an order-parameter theory which is an extension of our previous work on two-dimensional (2D) perturbations (Shukla & Alam, J. Fluid Mech., vol. 672, 2011b, pp. 147-195). The differential equations for modal amplitudes (the fundamental mode, the mean-flow distortion, the second harmonic and the distortion of the fundamental mode), up to cubic-order in perturbation amplitude, are solved using a spectral-based numerical technique, yielding an estimate of the first Landau coefficient that accounts for the leading-order nonlinear effect on finite-amplitude perturbations. In the near-critical regime of flows, we found evidence of mean-flow resonance, characterized by the divergence of the first Landau coefficient, that occurs due to the interaction/resonance between a linear instability mode and a mean-flow mode. The nonlinear solutions are found to appear via both pitchfork and Hopf bifurcations from the underlying linear instability modes, leading to supercritical nonlinear states of stationary and travelling wave solutions. The subcritical travelling wave solutions have also been uncovered in the linearly stable regimes of flow. It is shown that multiple nonlinear states of both stationary and travelling waves can coexist for a given parameter combination of mean density and Couette gap. The 3D nonlinear solutions persist for a range of spanwise wavenumbers up to k z = O(1) that originate from 2D instabilities which occur beyond a moderate value of the mean density. For purely 3D instabilities in dilute flows (having no analogue in 2D flows), the supercritical finite-amplitude solutions persist for a much larger range of spanwise wavenumber up to k z = O(10). For all instabilities, the vortical motion on the cross-stream plane has been characterized in terms of the fixed/critical points of the underlying flow field: saddles, nodes (sources and sinks) and vortices have been identified. While the cross-stream velocity field for supercritical solutions in dilute flows contains nodes and saddles, the subcritical solutions are dominated by large-scale vortices in the background of saddle-node-type motions. The latter type of flow pattern also persists at moderate densities in the form of supercritical nonlinear solutions that originate from the dominant 2D instability modes for which the vortex appears to be driven by two nearby saddles. The location † Email address for correspondence: meheboob@jncasr.ac.in 350 M. Alam and P. Shukla of this vortex is found to be correlated with the local maxima of the streamwise vorticity.

Pattern transition, microstructure, and dynamics in a two-dimensional vibrofluidized granular bed

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Simulation of cohesive fine powders under a plane shear

“…Let us consider cohesive powders under a plane shear. So far there exist many studies for one or two effects of the shear, an attractive force, and an inelastic collision [44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60][61][62], but we only know one example for the study of the jamming transition to include all three effects [63]. On the other hand, when the Lennard-Jones (LJ) molecules are quenched below the coexistence curve of gas-liquid phases [44][45][46][47][48][49], a phase ordering process proceeds after the nucleation takes place [50][51][52].…”

Shearbanding and inhomogeneous states in granular fluid

“…The dimensionless forms of hydrodynamic equations and the transport coefficients can be found in Ref. [8].…”

“…Under the assumption of small but finite-size perturbations, the above set of PDEs (1) is reduced to a set of linear differential equations [5,8]:…”

“…(2) are obtained from the solvability condition of the inhomogeneous equation for the distortion to the fundamental mode, X [1;n] , at O(A n , n = 3, 5,...). They satisfy the well-known Landau equation [5,8]:…”