Skip to main content Accessibility help
×
Home

Unified theory for a sheared gas–solid suspension: from rapid granular suspension to its small-Stokes-number limit

  • M. Alam (a1), S. Saha (a1) and R. Gupta (a1)

Abstract

A non-perturbative nonlinear theory for moderately dense gas–solid suspensions is outlined within the framework of the Boltzmann–Enskog equation by extending the work of Saha & Alam (J. Fluid Mech., vol. 833, 2017, pp. 206–246). A linear Stokes’ drag law is adopted for gas–particle interactions, and the viscous dissipation due to hydrodynamic interactions is incorporated in the second-moment equation via a density-corrected Stokes number. For the homogeneous shear flow, the present theory provides a unified treatment of dilute to dense suspensions of highly inelastic particles, encompassing the high-Stokes-number rapid granular regime ( $St\rightarrow \infty$ ) and its small-Stokes-number counterpart, with quantitative agreement for all transport coefficients. It is shown that the predictions of the shear viscosity and normal-stress differences based on existing theories deteriorate markedly with increasing density as well as with decreasing Stokes number and restitution coefficient.

Copyright

Corresponding author

Email address for correspondence: meheboob@jncasr.ac.in

References

Hide All
Alam, M. & Saha, S. 2017 Normal stress differences and beyond-Navier–Stokes hydrodynamics. EPJ Conf. Proc. 140, 11014.10.1051/epjconf/201714011014
Araki, S. & Tremaine, S. 1986 The dynamics of dense particle disks. Icarus 65, 83109.
Batchelor, G. K. 1970 The stress in a suspension of force-free particles. J. Fluid Mech. 41, 545577.10.1017/S0022112070000745
Bird, G. A. 1994 Direct Simulation Monte Carlo of Gas Flows. Oxford University Press.
Boyer, F., Pouliquen, O. & Guazzelli, E. 2011 Dense suspensions in rotating-rod flows: normal stresses and particle migration. J. Fluid Mech. 686, 525.
Brady, J. F. & Bossis, G. 1988 Stokesian dynamics. Annu. Rev. Fluid Mech. 20, 111157.
Brey, J. J., Dufty, J. W. & Santos, A. 1999 Kinetic models for granular flow. J. Stat. Phys. 97, 281303.
Chapman, S. & Cowling, T. G. 1970 The Mathematical Theory for Non-Uniform Gases. Cambridge University Press.
Garzo, V., Tenneti, S., Subramaniam, S. & Hrenya, C. 2012 Enskog kinetic theory for monodisperse gas–solid flows. J. Fluid Mech. 712, 129168.
Goldreich, P. & Tremaine, S. 1978 The velocity dispersion in Saturn’s rings. Icarus 34, 227239.
Grad, H. 1949 On the kinetic theory of rarefied gases. Commun. Pure Appl. Maths 2, 331407.
Guazzelli, E. & Morris, J. F. 2011 A Physical Introduction to Suspension Dynamics. Cambridge University Press.
Guazzelli, E. & Pouliquen, O. 2018 Rheology of dense granular suspensions. J. Fluid Mech. 852, P1.
Gupta, R. & Alam, M. 2017 Hydrodynamics, wall-slip, and normal-stress differences in rarefied granular Poiseuille flow. Phys. Rev. E 95, 022903.
Hinch, E. J. 1977 An averaged-equation approach to particle interactions in a fluid suspension. J. Fluid Mech. 83, 695720.
Hockling, L. M. 1973 The effect of slip on the motion of a sphere close to a wall and of two adjacent spheres. J. Engng Maths 7, 207223.
Jackson, R. 2000 Dynamics of Fluidized Particles. Cambridge University Press.
Jaynes, E. T. 1957 Information theory and statistical mechanics. Phys. Rev. 106, 620630.
Jenkins, J. T. & Richman, M. W. 1985 Grad’s 13-moment system for a dense gas of inelastic spheres. Arch. Rat. Mech. Anal. 87, 355377.
Jenkins, J. T. & Richman, M. W. 1988 Plane simple shear of smooth inelastic circular disks. J. Fluid Mech. 192, 313328.
Koch, D. L. 1990 Kinetic theory for a monodisperse gas–solid suspension. Phys. Fluids A 2, 17111723.
Koch, D. L. & Hill, R. J. 2001 Inertial effects in gas–solid suspension and porous-media flows. Annu. Rev. Fluid Mech. 33, 619647.
Koch, D. L. & Sangani, A. S. 1999 Particle pressure and marginal stability limits for a homogeneous monodisperse gas-fluidized bed: kinetic theory and numerical simulations. J. Fluid Mech. 400, 229263.
Kong, B., Fox, R. O., Feng, H., Capecelatro, P. R. & Desjardins, O. 2017 Euler–Euler anisotropic Gaussian mesoscale simulation of homogeneous cluster-induced gas–particle turbulence. AIChE J. 63 (7), 26302643.
Lhuillier, D. 2009 Migration of rigid particles in non-Brownian viscous suspensions. Phys. Fluids 21, 023302.
Lun, C. K. K. & Savage, S. B. 2003 Kinetic theory for inertia flows of dilute turbulent gas–solids mixtures. In Granular Gas Dynamics (ed. Pöschel, T. & Brilliantov, N. V.), p. 263. Springer.
Montanero, J. M. & Santos, A. 1997 Viscometric effects in a dense hard-sphere fluid. Physica A 240, 229238.
Nott, P. R. & Brady, J. F. 1994 Pressure-driven flow of suspensions: simulation and theory. J. Fluid Mech. 275, 157199.
Nott, P. R., Guazzelli, E. & Pouliquen, O. 2011 The suspension balance model revisited. Phys. Fluids 23, 043304.
Parmentier, J-F. & Simonin, O. 2012 Transition models from the quenched to ignited states for flows of inertial particles suspended in a simple sheared viscous fluid. J. Fluid Mech. 711, 147160.
Richman, M. W. 1989 The source of second moment in dilute granular flows of highly inelastic spheres. J. Rheol. 33, 12931306.
Saha, S. & Alam, M. 2014 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, 251296.
Saha, S. & Alam, M. 2016 Normal stress differences, their origin and constitutive relations for a sheared granular fluid. J. Fluid Mech. 795, 549580.
Saha, S. & Alam, M. 2017 Revisiting ignited-quenched transition and the non-Newtonian rheology of a sheared dilute gas–solid suspension. J. Fluid Mech. 833, 206246.
Sangani, A. S., Mo, G., Tsao, H.-K. & Koch, D. L. 1996 Simple shear flows of dense gas–solid suspensions at finite Stokes numbers. J. Fluid Mech. 313, 309341.
Savage, S. B. & Jeffrey, D. J. 1981 The stress tensor in a granular flow at high shear rates. J. Fluid Mech. 110, 255272.
Sela, N. & Goldhirsch, I. 1998 Hydrodynamic equations for rapid flows of smooth inelastic spheres, to Burnett order. J. Fluid Mech. 361, 4174.
Shukhman, G. 1984 Collisional dynamics of particles in Saturn’s rings. Sov. Astron. 28, 547584.
Tsao, H.-K. & Koch, D. L. 1995 Simple shear flows of dilute gas–solid suspensions. J. Fluid Mech. 296, 211246.
Verberg, R. & Koch, D. L. 2006 Rheology of particle suspensions with low to moderate fluid inertia at finite particle inertia. Phys. Fluids 18, 083303.
Vié, A., Doisneau, F. & Massot, M. 2015 On the anisotropic Gaussian velocity closure for inertial-particle laden flow. Commun. Comput. Phys. 17, 146.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed