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

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 [mu] solely in terms of a stresslet component and a quadrupole component exerted on a single freely suspended sphere. In the calculation of [mu], 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 [mu] in terms of H/(2a) and (lambda) over tilde =./a = 1, where H is the gap between walls, a is the sphere radius and lambda is the wall slip length using the Navier slip boundary condition. As compared with the no-slip case, the intrinsic viscosity strongly depends on (lambda) over tilde for given H/(2a), especially for small H/(2a). For example, in the very confined case H/(2a) = 2 (a lower bound found for practical validity of single-wall models) and for (lambda) over tilde = 1, 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 [mu] in the entire range 2 <= H/(2a) = 100 and (lambda) over tilde = 1.