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Transient and asymptotic stability of granular shear flow

Published online by Cambridge University Press:  26 April 2006

P. J. Schmid  and
H. K. Kytömaa
P. J. Schmid
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Current address: Department of Applied Mathematics, FS-20, University of Washington, Seattle, WA 98195, USA.
H. K. Kytömaa
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
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Abstract

The linear stability of granular material in an unbounded uniform shear flow is considered. Linearized equations of motion derived from kinetic theories are used to arrive at a linear initial-value problem for the perturbation quantities. Two cases are investigated: (a) wavelike disturbances with time constant wavenumber vector, and (b) disturbances that will change their wave structure in time owing to a shear-induced tilting of the wavenumber vector. In both cases, the stability analysis is based on the solution operator whose norm represents the maximum possible amplification of initial perturbations. Significant transient growth is observed which has its origin in the non-normality of the involved linear operator. For case (a), regions of asymptotic instability are found in the two-dimensional wavenumber plane, whereas case (b) is found to be asymptotically stable for all physically meaningful parameter combinations. Transient linear stability phenomena may provide a viable and fast mechanism to trigger finite-amplitude effects, and therefore constitute an important part of pattern formation in rapid particulate flows.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 264 , 10 April 1994 , pp. 255 - 275
Copyright
© 1994 Cambridge University Press

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References

Anderson, T. B. & Jackson, R. 1968 Fluid mechanical description of fluidized beds. I & EC Fundamentals, pp. 12–21.
Babić, M. 1992 Particle clustering: an instability of rapid granular flows. In Advances in Micromechanics of Granular Materials (ed. H. H. Shen, M. Satake, M. Mehrabadi, C. S. Chang & C. S. Campbell). Elsevier.
Babić, M. 1993 On the stability of rapid granular flows. J. Fluid Mech. 254, 127–150.Google Scholar
Bagnold, R. A. 1954 Experiments on a gravity free dispersion of large solid spheres in a Newtonian fluid. Proc. R. Soc. Lond. 225, 49–63.Google Scholar
Barnes, H. A. 1989 Shear-thickening in suspension of non-aggregating solid particles dispersed in Newtonian liquids. J. Rheol. 33, 329–366.Google Scholar
Batchelor, G. K. 1988 A new theory of the instability of a uniform fluidized bed. J. Fluid Mech. 193, 75–110.Google Scholar
Bellman, R. 1960 Introduction to Matrix Analysis. McGraw-Hill.
Butler, K. M. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids A 4, 1637–1650.Google Scholar
Campbell, C. S. 1990 Rapid granular flows. Ann. Rev. Fluid Mech. 22, 57–92.Google Scholar
Campbell, C. S. & Brennen, C. E. 1985 Computer simulations of granular shear flows. J. Fluid Mech. 151, 167–188.Google Scholar
Farrell, B. F. & Ioannou, P. J. 1993 Optimal excitation of three-dimensional perturbations in viscous constant shear flow. Phys. Fluids A 5, 1390–1400.Google Scholar
Foscolo, P. U. & Gibilaro, L. G. 1987 Fluid dynamic stability of fluidized suspensions: the particle bed model. Chem. Engng. Sci. 42, 1489–1500.Google Scholar
Gantmacher, F. R. 1989 Matrix Theory. Chelsea.
Haff, P. K. 1983 Grain flow as a fluid-mechanical phenomenon. J. Fluid Mech. 134, 401–430.Google Scholar
Hill, C. D. & Bedford, A. 1979 Stability of the equations for particulate sediment. Phys. Fluids 22, 1252–1254.Google Scholar
Hopkins, M. A. & Louge, M. Y. 1991 Inelastic microstructure in rapid granular flows of smooth disks. Phys. Fluids A 3, 47–57.Google Scholar
Homsy, G. M., El-Kaissy, M. M. & Didwania, A. 1980 Instability waves and the origin of bubbles in fluidized beds – II, Comparison with theory. Intl J. Multiphase Flow 6, 305–318.Google Scholar
Jackson, R. 1985 Hydrodynamic stability of fluid-particle systems. In Fluidization (ed. J. F. Davidson, R. Clift & D. Harrison), Chap. 2. Academic Press.
Jenkins, J. T. & Savage, S. B. 1983 A theory for the rapid flow of identical, smooth, nearly elastic particles. J. Fluid Mech. 130, 187–202.Google Scholar
Lun, C. K., Savage, S. B., Jeffrey, D. J. & Chepurniy, N. 1984 Kinetic theories for granular flow: inelastic particles in Couette flow and slightly inelastic particles in a general flow field. J. Fluid Mech. 140, 223–256.Google Scholar
McNamara, S. 1993 Hydrodynamic modes of a uniform granular medium. Phys. Fluids A 5, 3056–3070.Google Scholar
Magnus, W. 1954 On the exponential solution of differential equations for a linear operator. Commun. Pure Appl. Maths 7, 649–673.Google Scholar
Mello, T. M., Diamond, P. H. & Levine, H. 1991 Hydrodynamic modes of a granular shear flow. Phys. Fluids A 3, 2067–2075.Google Scholar
Phillips, O. M. 1969 Shear flow turbulence. Ann. Rev. Fluid Mech. 1, 245–264.Google Scholar
Prosperetti, A. & Jones, A. V. 1987 The linear stability of general two-phase flow models–II Intl J. Multiphase Flow 13, 161–171.Google Scholar
Reddy, S. C. & Henningson, D. S. 1993 Energy growth in viscous channel flows. J. Fluid Mech. 252, 209–238.Google Scholar
Reddy, S. C., Schmid, P. J. & Henningson, D. S. 1993 Pseudospectra of the Orr–Sommerfeld operator. SIAM J. Appl. Maths. 53, 15–45.Google Scholar
Savage, S. B. 1992 Instability of ‘unbounded’ uniform granular shear flow. J. Fluid Mech. 241, 109–123.Google Scholar
Trefethen, L. N., Trefethen, A. E., Reddy, S. C. & Driscoll, T. 1993 Hydrodynamic stability without eigenvalues. Science 261, 578–584.Google Scholar
Walton, O. R., Kim, H. & Rosato, A. 1991 Micro-structure and stress differences in shearing flows. In Mechanics and Computing in 1990′s and Beyond (ed. H. Adeli & R. L. Sierakowski), vol. 2, pp. 1249–1253. ASCE.