Rheology of Dilute Inertial Suspensions

In inertial suspensions, inertia becomes important on the scale of the disperse particulate phase (the ‘micro-scale’). From the rheological standpoint, the interest is in suspensions of neutrally buoyant particles, with micro-scale inertial effects characterized by the particle Reynolds number. Dilute inertial suspensions differ fundamentally from their Stokesian counterparts, in possessing a non-Newtonian rheology and a finite microstructural relaxation time, even in the absence of interparticle interactions. We discuss the role of micro-scale inertia for dilute suspensions of both spherical and anisotropic particles. The discussion on inertial suspensions is preceded by one on Stokesian suspensions, the emphasis being on the rheological indeterminacy arising from either an indeterminate pair-distribution function (spherical particles), or an indeterminate single-particle orientation distribution (spheroidal particles). The effect of inertia is accordingly classified into: (1) a ‘direct effect’ where inertial contributions to rheological properties explicitly involve the particle Reynolds number, and become vanishingly small when the Reynolds number goes to zero; and (2) an ‘indirect effect’ where inertia determines the leading order viscosity, even for vanishingly small Reynolds numbers, due to the Stokesian indeterminacy. As part of the indirect effect, we discuss the tumbling-spinning transition for inertial suspensions of thin oblate spheroids, and the role of stochastic orientation fluctuations in leading to a transition from a shear-thickening to a shearthinning rheology, with changing aspect ratio, for sufficiently long times; and hysteretic behavior for shorter times. While the emphasis is mainly on theoretical calculations for small particle Reynolds numbers, where applicable we discuss simulations that provide insight into inertial effects at finite particle Reynolds number. Although micro-scale inertia is predicted to have profound rheological consequences, measurements of the same pose problems. Traditional rheometric devices are prone to instabilities and secondary flows arising from macro-scale inertia, and we therefore end with a discussion of recent suspension-transition experiments (a non-rheometric setting), that would allow for an inference of the underlying inertial rheology.