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 anisotropic Maxwellian as the single particle distribution function. 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 ($\nu$) 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.,1995). The phase diagram is constructed in the three-dimensional ($\nu, St, e$)-space that delineates the regions of ignited and quenched states and their coexistence. Analytical expressions for the particle-phase shear viscosity and the normal stress differences are obtained, 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 (i.e.~ discontinuous shear-thickening) 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 Chapman-Enskog solution is demonstrated via comparisons of transport coefficients with simulation data for a range of Stokes number and restitution coefficient.