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

<jats:p>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 (<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112017007224_inline1" /><jats:tex-math>$St$</jats:tex-math></jats:alternatives></jats:inline-formula>), the mean density (<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112017007224_inline2" /><jats:tex-math>$\unicode[STIX]{x1D708}$</jats:tex-math></jats:alternatives></jats:inline-formula>) and the restitution coefficient (<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112017007224_inline3" /><jats:tex-math>$e$</jats:tex-math></jats:alternatives></jats:inline-formula>). 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 (<jats:italic>J. Fluid Mech.</jats:italic>, vol. 296, 1995, pp. 211–246). The phase diagram is constructed in the three-dimensional (<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112017007224_inline4" /><jats:tex-math>$\unicode[STIX]{x1D708},St,e$</jats:tex-math></jats:alternatives></jats:inline-formula>)-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 <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112017007224_inline5" /><jats:tex-math>$e$</jats:tex-math></jats:alternatives></jats:inline-formula>, the shear viscosity undergoes a discontinuous jump with increasing shear rate at the ‘quenched–ignited’ transition. The first (<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112017007224_inline6" /><jats:tex-math>${\mathcal{N}}_{1}$</jats:tex-math></jats:alternatives></jats:inline-formula>) and second (<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112017007224_inline7" /><jats:tex-math>${\mathcal{N}}_{2}$</jats:tex-math></jats:alternatives></jats:inline-formula>) normal-stress differences also undergo similar first-order transitions: (i) <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112017007224_inline8" /><jats:tex-math>${\mathcal{N}}_{1}$</jats:tex-math></jats:alternatives></jats:inline-formula> jumps from large to small positive values and (ii) <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112017007224_inline9" /><jats:tex-math>${\mathcal{N}}_{2}$</jats:tex-math></jats:alternatives></jats:inline-formula> from positive to negative values with increasing <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112017007224_inline10" /><jats:tex-math>$St$</jats:tex-math></jats:alternatives></jats:inline-formula>, with the sign change of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112017007224_inline11" /><jats:tex-math>${\mathcal{N}}_{2}$</jats:tex-math></jats:alternatives></jats:inline-formula> 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.</jats:p>