Burnett-order constitutive relations, second moment anisotropy and co-existing states in sheared dense gas–solid suspensions

The Burnett- and super-Burnett-order constitutive relations are derived for homogeneously sheared gas–solid suspensions by considering the co-existence of ignited and quenched states and the anisotropy of the second moment of velocity fluctuations ( $\unicode[STIX]{x1D648}=\langle \boldsymbol{C}\boldsymbol{C}\rangle ,C$ is the fluctuation or peculiar velocity) – this analytical work extends our previous works on dilute (Saha & Alam, J. Fluid Mech. , vol. 833, 2017, pp. 206–246) and dense (Alam et al. , J. Fluid Mech. , vol. 870, 2019, pp. 1175–1193) gas–solid suspensions. For the combined ignited–quenched theory at finite densities, the second-moment balance equation, truncated at the Burnett order, is solved analytically, yielding expressions for four invariants of $\unicode[STIX]{x1D648}$ as functions of the particle volume fraction ( $\unicode[STIX]{x1D708}$ ), the restitution coefficient ( $e$ ) and the Stokes number ( $St$ ). The phase boundaries, demarcating the regions of (i) ignited, (ii) quenched and (iii) co-existing ignited–quenched states, are identified via an ordering analysis, and it is shown that the incorporation of excluded-volume effects significantly improves the predictions of critical parameters for the ‘quenched-to-ignited’ transition. The Burnett-order expressions for the particle-phase shear viscosity, pressure and two normal-stress differences are provided, with their Stokes-number dependence being implicit via the anisotropy parameters. The roles of ( $St,\unicode[STIX]{x1D708},e$ ) on the granular temperature, the second-moment anisotropy and the nonlinear transport coefficients are analysed using the present theory, yielding quantitative agreements with particle-level simulations over a wide range of ( $St,\unicode[STIX]{x1D708}$ ) including the bistable regime that occurs at $St\sim O(5)$ . For highly dissipative particles ( $e\ll 1$ ) that become increasingly important at large Stokes numbers, it is shown that the Burnett-order solution is not adequate and further higher-order solutions are required for a quantitative agreement of transport coefficients over the whole range of control parameters. The latter is accomplished by developing an approximate super-super-Burnett-order theory for the ignited state ( $St\gg 1$ ) of sheared dense gas–solid suspensions in the second part of this paper. An extremum principle based on viscous dissipation and dynamic friction is discussed to identify ignited–quenched transition.