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

摘要

The hydrodynamics and rheology of a sheared dilute gas-solid suspension,consisting of inelastic hard-spheres suspended in a gas, are analysed usinganisotropic Maxwellian as the single particle distribution function. Theclosed-form solutions for granular temperature and three invariants of thesecond-moment tensor are obtained as functions of the Stokes number ($St$), themean density ($\nu$) and the restitution coefficient ($e$). Multiple states ofhigh and low temperatures are found when the Stokes number is small, thusrecovering the "ignited" and "quenched" states, respectively, of Tsao \& Koch(J. Fluid Mech.,1995). The phase diagram is constructed in thethree-dimensional ($\nu, St, e$)-space that delineates the regions of ignitedand quenched states and their coexistence. Analytical expressions for theparticle-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 shearrate (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) ${\mathcalN}_2$ from positive to negative values with increasing $St$, with thesign-change of ${\mathcal N}_2$ identified with the system making a transitionfrom the quenched to ignited states. The superior prediction of the presenttheory over the standard Grad's method and the Chapman-Enskog solution isdemonstrated via comparisons of transport coefficients with simulation data fora range of Stokes number and restitution coefficient.