High-frequency viscosity of a dilute suspension of elongated particles in a linear shear flow between two walls

A general expression for the effective viscosity of a dilute suspension of arbitrary-shaped particles in linear shear flow between two parallel walls is derived in terms of the induced stresslets on particles. This formula is applied to $N$-bead rods and to prolate spheroids with the same length, aspect ratio and volume. The effective viscosity of non-Brownian particles in a periodic shear flow is considered here. The oscillating frequency is high enough for the particle orientation and centre-of-mass distribution to be practically frozen, yet small enough for the flow to be quasi-steady. It is known that for spheres, the intrinsic viscosity [{\μ}] increases monotonically when the distance H between the walls is decreased. The dependence is more complex for both types of elongated particles. Three regimes are theoretically predicted here: (i) a 'weakly confined' regime (for H>l, where l is the particle length), where [{\μ}] is slightly larger for smaller H; (ii) a 'semi-confined' regime, when H becomes smaller than l, where [{\μ}] rapidly decreases since the geometric constraints eliminate particle orientations corresponding to the largest stresslets; (iii) a 'strongly confined' regime when H becomes smaller than 2-3 particle widths d, where [{\μ}] rapidly increases owing to the strong hydrodynamic coupling with the walls. In addition, for sufficiently slender particles (with aspect ratio larger than 5-6) there is a domain of narrow gaps for which the intrinsic viscosity is smaller than that in unbounded fluid.