78 Progress of Theoretical Physics Supplement No. 195, 2012 Non-Modal Stability and Optimal Perturbations in Unbounded Granular Shear Flow: Three-Dimensionality and Particle Spin Meheboob Alam Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur PO, Bangalore 560064, India The effects of three-dimensional perturbations and particle rotations are analyzed on the non-modal stability characteristics of an unbounded granular shear flow for which the stability problem is solved as an initial value problem. A kinetic-theory constitutive model is used that incorporates the spin degrees of freedom along with certain micro-polar effects of granular materials. The singular values of the underlying non-normal linear operator play a central role in the non-modal analysis in contrast to the standard modal analysis where the eigenvalues determine the asymptotic (in)stability of the flow. For linearly stable flows, it is shown that the perturbation energy can be amplified by a few orders of magnitude at short times before decaying in the asymptotic time limit. Optimal perturbations, that correspond to maximum energy growth over all possible initial conditions, are found to be two-dimensional in a smooth granular fluid. The effect of particle rotation has been assessed by varying the tangential restitution coefficient (β) for smooth particles (β = -1) to perfectly rough particles (β = 1), with significant enhancement of maximum energy for rough particles. Since the non-modal mechanism can significantly amplify perturbation energy, this provides a viable alternate route for pattern formation in a sheared granular fluid. §1. Introduction The stability analyses of granular shear flows have attracted much attention re- cently. 1)–16) The major motivation of these works has been to understand certain dynamical features of shear flows (particle clustering, shear-band formation, vortical structures, etc. 5), 17)–19) ) as well as to uncover the scalings of the underlying hydrody- namic modes. 20)–22) Explaining the dynamical characteristics of granular fluids using continuum equations is a stringent test of the adopted constitutive model. In the context of solar and planetary physics, the unbounded granular shear flow appears generically in astrophysical disks and in a variety of ring systems. 23)–27) For example, the inner parts of the Saturn’s ring (around the planet Saturn) rotate more rapidly than the outer parts (Kepler’s third law) and this differential rotation is equivalent to an “uniform shear flow” superimposed on a mean rotation field, resulting in the well-known “Keplerian” shear in the ring system with respect to an observer moving with mean rotation. One of the long-standing problem in planetary science is the origin and stability of Saturn’s ring. 23), 26), 28)–31) Typically, certain simplified forms of granular hydrodynamics equations and constitutive relations 23), 32)–37) are used to analyze the stability problem of Saturn’s ring. 23), 29)–31), 33) So far, the standard modal stability of Saturn’s ring has been carried out (see Schmidt et al. 31) for a comprehensive recent review on related issues). In the modal/asymptotic stability analysis, the eigenvalues of the linear stability operator determines the long-time asymptotic behavior of the system: if the real part of any eigenvalue is positive, the the flow is stable and unstable otherwise. Downloaded from https://academic.oup.com/ptps/article-abstract/doi/10.1143/PTPS.195.78/1864556 by guest on 18 June 2020