Effect of hydrodynamic slip on the rotational dynamics of a thin Brownian platelet in shear flow

Abstract

The classical theory by Jeffery predicts that, in the absence of Brownian fluctuations, a thin rigid platelet rotates continuously in a shear flow, performing periodic orbits. However, a stable orientation is possible if the surface of the platelet displays a hydrodynamic slip length$\lambda$comparable to or larger than the thickness of the platelet. In this article, by solving the Fokker–Plank equation for the orientation distribution function and corroborating the analysis with boundary integral simulations, we quantify a threshold Péclet number,${Pe}_{c}$, above which such alignment occurs. We found that for${Pe}$smaller than${Pe}_{c}$, but larger than a second threshold, a regime emerges where Brownian fluctuations are strong enough to break the platelet's alignment and induce rotations, but with a period of rotation that depends on the value of$\lambda$. For${Pe}$below this second threshold, slip has a negligible effect on the orientational dynamics. We use these thresholds to classify the dynamics of graphene-like nanoplatelets for realistic values of$\lambda$and apply our results to the quantification of the orientational contribution to the effective viscosity of a dilute suspension of nanoplatelets with slip. We find a non-monotonic variation of this term, with a minimum occurring when the slip length is comparable to the thickness of the particle.