A Chapman–Enskog formalism for inertial suspensions

We present a Chapman–Enskog formalism to analyze the microstructure and rheology of Brownian suspensions subjected to external flows in the limit of weak particle inertia. The Fokker–Planck equation for the phase-space probability density in an inertial suspension is solved in a perturbative fashion for small values of the Stokes number, St. The latter is a dimensionless measure of particle inertia defined as the ratio of the inertial relaxation time of an individual particle to the flow time-scale. While restricted to the limit when the inertial relaxation time is much smaller than the flow time-scale, the procedure allows for an arbitrary ratio of the configurational and flow time-scales, and in addition, places no restriction on the particle concentration. The perturbation is performed about the inertialess limit; it entails expanding the solution in a series of Hermite polynomials involving the fluctuation velocity, the difference between the actual velocity of a particle and the velocity of a (fictitious) inertialess particle at the same location, and solving the resulting configuration-space equations for the expansion coefficients. The analysis yields the form of the O(St) corrections to the Smoluchowski equation that characterize, to first order, the effect of particle-phase inertia on the spatial microstructure of a suspension. The nature and relevance of these and higher-order corrections is discussed.