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

Abstract

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 ( 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 as functions of the particle volume fraction ( ), the restitution coefficient ( ) and the Stokes number ( ). 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 ( ) 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 ( ) including the bistable regime that occurs at . For highly dissipative particles ( ) 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 ( ) 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.