Abstract
This chapter deals with the study of linear transport around the uniform or simple shear flow state. The analysis is made from a perturbation solution of the Boltzmann kinetic equation through first-order in the deviations of the hydrodynamic fields with respect to their values in the (unperturbed) non-Newtonian shear flow state. Given that the reference state (zeroth-order approximation in the Chapman–Enskog-like expansion) applies to arbitrary shear rates, the successive approximations in perturbation expansion retain all the hydrodynamic orders in the shear rate. As expected, due to the anisotropy in velocity space induced in the system by the shear flow, mass, momentum, and heat fluxes are given in terms of tensorial transport coefficients instead of the conventional scalar Navier–Stokes transport coefficients. The study is carried out for monocomponent granular gases and binary granular mixtures in the tracer limit.
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Chapman, S., Cowling, T.G.: The Mathematical Theory of Nonuniform Gases. Cambridge University Press, Cambridge (1970)
Lutsko, J.F.: Chapman-Enskog expansion about nonequilibrium states with application to the sheared granular fluid. Phys. Rev. E 73, 021302 (2006)
Walton, O.R., Braun, R.L.: Viscosity and temperature calculations for shearing assemblies of inelastic, frictional disks. J. Rheol. 30, 949–980 (1986)
Campbell, C.S., Brennen, C.: Computer simulation of granular shear flows. J. Fluid Mech. 151, 167–188 (1985)
Hopkins, M.H., Louge, M.Y.: Inelastic microstructure in rapid granular flows of smooth disks. Phys. Fluids A 3, 47–56 (1991)
Goldhirsch, I., Tan, M.L.: The single-particle distribution function for rapid granular shear flows of smooth inelastic disks. Phys. Fluids 8, 1752–1763 (1996)
Alam, M., Luding, S.: Rheology of bidisperse granular mixtures via event-driven simulations. J. Fluid Mech. 476, 69–103 (2003)
Savage, S.B.: Instability of unbounded uniform granular shear flow. J. Fluid Mech. 241, 109–123 (1992)
Babic, M.: On the stability of rapid granular flows. J. Fluid Mech. 254, 127–150 (1993)
Alam, M., Nott, P.R.: The influence of friction on the stability of unbounded granular shear flow. J. Fluid Mech. 343, 267–301 (1997)
Alam, M., Nott, P.R.: Stability of plane Couette flow of a granular material. J. Fluid Mech. 377, 99–136 (1998)
Kumaran, V.: Asymptotic solution of the Boltzmann equation for the shear flow of smooth inelastic disks. Physica A 275, 483–504 (2000)
Kumaran, V.: Anomalous behaviour of hydrodynamic modes in the two dimensional shear flow of a granular material. Physica A 284, 246–264 (2000)
Kumaran, V.: Hydrodynamic modes of a sheared granular flow from the Boltzmann and Navier-Stokes equations. Phys. Fluids 13, 2258–2268 (2001)
Garzó, V.: Transport coefficients for an inelastic gas around uniform shear flow: linear stability analysis. Phys. Rev. E 73, 021304 (2006)
Lee, M., Dufty, J.W.: Transport far from equilibrium: uniform shear flow. Phys. Rev. E 56, 1733–1745 (1997)
Garzó, V., Santos, A.: Kinetic Theory of Gases in Shear Flows. Nonlinear Transport. Kluwer Academic Publishers, Dordrecht (2003)
Astillero, A., Santos, A.: Aging to non-Newtonian hydrodynamics in a granular gas. Europhys. Lett. 78, 24002 (2007)
Jenkins, J.T., Richman, M.W.: Plane simple shear of smooth inelastic circular disks: the anisotropy of the second moment in the dilute and dense limits. J. Fluid Mech. 192, 313–328 (1988)
Saha, S., Alam, M.: Non-Newtonian stress, collisional dissipation and heat flux in the shear flow of inelastic disks: a reduction via Grad’s moment method. J. Fluid Mech. 757, 251–296 (2014)
Garzó, V., Santos, A.: Hydrodynamics of inelastic Maxwell models. Math. Model. Nat. Phenom. 6, 37–76 (2011)
Tij, M., Tahiri, E., Montanero, J.M., Garzó, V., Santos, A., Dufty, J.W.: Nonlinear Couette flow in a low density granular gas. J. Stat. Phys. 103, 1035–1068 (2001)
Lees, A.W., Edwards, S.F.: The computer study of transport processes under extreme conditions. J. Phys. C 5, 1921–1929 (1972)
Résibois, P., de Leener, M.: Classical Kinetic Theory of Fluids. Wiley, New York (1977)
Natarajan, V.V.R., Hunt, M.L., Taylor, E.D.: Local measurements of velocity fluctuations and diffusion coefficients for a granular material flow. J. Fluid Mech. 304, 1–25 (1995)
Menon, N., Durian, D.J.: Diffusing-wave spectroscopy of dynamics in a three-dimensional granular flow. Science 275, 1920–1922 (1997)
Zik, O., Stavans, J.: Self-diffusion in granular flows. Europhys. Lett. 16, 255–258 (1991)
Savage, S.B., Dai, R.: Studies of granular shear flows. Wall slip velocities, “layering” and self-diffusion. Mech. Mater. 16, 225–238 (1993)
Zamankhan, P., Polashenski Jr., W., Tafreshi, H.V., Manesh, A.S., Sarkomaa, P.J.: Shear-induced particle diffusion in inelastic hard sphere fluids. Phys. Rev. E 58, R5237–R5240 (1998)
Campbell, C.S.: Self-diffusion in granular shear flows. J. Fluid Mech. 348, 85–101 (1997)
Garzó, V.: Tracer diffusion in granular shear flows. Phys. Rev. E 66, 021308 (2002)
Garzó, V.: Mass transport of an impurity in a strongly sheared granular gas. J. Stat. Mech. P02012 (2007)
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Garzó, V. (2019). Transport Around Steady Simple Shear Flow in Dilute Granular Gases. In: Granular Gaseous Flows. Soft and Biological Matter. Springer, Cham. https://doi.org/10.1007/978-3-030-04444-2_8
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DOI: https://doi.org/10.1007/978-3-030-04444-2_8
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