Modeling of macroscopic stresses in a dilute suspension of small weakly inertial particles

Abstract

In this paper we derive asymptotically the macroscopic bulk stress of a suspension of small inertial particles in an incompressible Newtonian fluid. We apply the general asymptotic framework to the special case of ellipsoidal particles and show the resulting modification due to inertia on the well-known particle-stresses based on the theory by Batchelor and Jeffery.

Motion of several slender rigid filaments in a Stokes flow

“…This coupling between translation/rotation velocities and force/torque is typical of the case of rigid bodies with shape anisotropies. It is usually called the Jeffery effect, see [20,23,43].…”

“…Let v a divergence-free vector field in W 1,8 p 1ďiďN S i q. We define µ Ci rvs as the vector measure, supported on C i , defined by (43) µ Ci rvs, φ :" I Ci rv, φs, for any φ P C c pR 3 ; R 3 q.…”

Mean field surrogate model of a dilute and chaotic particle suspension

“…In [7] we derived a macroscopic model for the behavior of a dilute particle suspension, which accounts for small inertial particle effects. The model is based on the asymptotic approach for a single inertia-free particle by Junk and Illner [5], where the suspended bodies are characterized by the size ratio 1.…”

“…It also adresses small hydrodynamic interaction via the terms associated to U, H. An important observation is that the latter appear in (1a) of order O( 3/2 ) compared to the one of the particle extra stress tensor Σ that is of O( 5/2 ). This indicates that their omission in [7] due to the ergodicity assumption neglects a major contribution to the overall dynamics in the non-homogeneous case.…”

“…Because of the assumption of ergodicity, which is inherent in [1], it is unclear whether the description is also reasonable for non-homogeneous particle distributions in space. By applying the kinetic approach of [2] to the asymptotic particle model, we come up with a novel suspension model which covers the one in [7], is valid for non-homogeneous particle distributions and poses an approximation to the full suspension problem.…”