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

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 ( [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 ( [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 ( N₁ ) and second ( N₂ ) normal-stress differences also undergo similar first-order transitions: (i) N₁ jumps from large to small positive values and (ii) N₂ from positive to negative values with increasing $St$ , with the sign change of N₂ 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.