Effective viscosity of a dilute homogeneous suspension of spheres in Poiseuille flow between parallel slip walls

Abstract

For flows in microchannels, a slip on the walls may be efficient in reducing viscous dissipation. A related issue, addressed in this article, is to decrease the effective viscosity of a dilute monodisperse suspension of spheres in Poiseuille flow by using two parallel slip walls. Extending the approach developed for no-slip walls in Feuillebois et al. (J. Fluid Mech., vol. 800, 2016, pp. 111-139), a formal expression is obtained for the suspension intrinsic viscosity solely in terms of a stresslet component and a quadrupole component exerted on a single freely suspended sphere. In the calculation of, the hydrodynamic interactions between a sphere and the slip walls are approximated using either the nearest wall model or the wall-superposition model. Both the stresslet and quadrupole are derived and accurately calculated using bipolar coordinates. Results are presented for in terms of and, where is the gap between walls, is the sphere radius and is the wall slip length using the Navier slip boundary condition. As compared with the no-slip case, the intrinsic viscosity strongly depends on for given, especially for small. For example, in the very confined case (a lower bound found for practical validity of single-wall models) and for, the intrinsic viscosity is three times smaller than for a suspension bounded by no-slip walls and five times smaller than for an unbounded suspension (Einstein, Ann. Phys., vol. 19, 1906, pp. 289-306). We also provide a handy formula fitting our results for in the entire range and.