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Revisiting ignited–quenched transition and the non-Newtonian rheology of a sheared dilute gas–solid suspension

Published online by Cambridge University Press:  03 November 2017

Saikat Saha  and
Meheboob Alam Open the ORCID record for Meheboob Alam [Opens in a new window]
Saikat Saha
Affiliation:
Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur P.O., Bangalore 560064, India
Meheboob Alam*
Affiliation:
Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur P.O., Bangalore 560064, India
*
†Email address for correspondence: meheboob@jncasr.ac.in
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Abstract

The hydrodynamics and rheology of a sheared dilute gas–solid suspension, consisting of inelastic hard spheres suspended in a gas, are analysed using an anisotropic Maxwellian as the single particle distribution function. For the simple shear flow, the closed-form solutions for granular temperature and three invariants of the second-moment tensor are obtained as functions of the Stokes number ( $St$ ), the mean density ( $\unicode[STIX]{x1D708}$ ) and the restitution coefficient ( $e$ ). Multiple states of high and low temperatures are found when the Stokes number is small, thus recovering the ‘ignited’ and ‘quenched’ states, respectively, of Tsao & Koch (J. Fluid Mech., vol. 296, 1995, pp. 211–246). The phase diagram is constructed in the three-dimensional ( $\unicode[STIX]{x1D708},St,e$ )-space that delineates the regions of ignited and quenched states and their coexistence. The particle-phase shear viscosity and the normal-stress differences are analysed, along with related scaling relations on the quenched and ignited states. At any $e$ , the shear viscosity undergoes a discontinuous jump with increasing shear rate at the ‘quenched–ignited’ transition. The first ( ${\mathcal{N}}_{1}$ ) and second ( ${\mathcal{N}}_{2}$ ) normal-stress differences also undergo similar first-order transitions: (i)  ${\mathcal{N}}_{1}$ jumps from large to small positive values and (ii)  ${\mathcal{N}}_{2}$ from positive to negative values with increasing $St$ , with the sign change of ${\mathcal{N}}_{2}$ identified with the system making a transition from the quenched to ignited states. The superior prediction of the present theory over the standard Grad’s method and the Burnett-order Chapman–Enskog solution is demonstrated via comparisons of transport coefficients with simulation data for a range of Stokes number and restitution coefficient.

JFM classification

Rarefied Gas Flow: Kinetic theory Non-Newtonian Flows: Rheology Complex Fluids: Suspensions

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Journal of Fluid Mechanics , Volume 833 , 25 December 2017 , pp. 206 - 246
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© 2017 Cambridge University Press 

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