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 (M = < CC >, 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 M as functions of the particle volume fraction (nu), 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, nu, 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, nu) including the bistable regime that occurs at St similar to O(5). For highly dissipative particles (e << 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 >> 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.