Anisotropic particle in viscous shear flow : navier slip, reciprocal symmetry, and jeffery orbit

Abstract

The hydrodynamic reciprocal theorem for Stokes flows is generalized to incorporate the Navier slip boundary condition, which can be derived from Onsager's variational principle of least energy dissipation. The hydrodynamic reciprocal relations and the Jeffery orbit, both of which arise from the motion of a slippery anisotropic particle in a simple viscous shear flow, are investigated theoretically and numerically using the fluid particle dynamics method [Phys. Rev. Lett. 85, 1338 (2000)]. For a slippery elliptical particle in a linear shear flow, the hydrodynamic reciprocal relations between the rotational torque and the shear stress are studied and related to the Jeffery orbit, showing that the boundary slip can effectively enhance the anisotropy of the particle. Physically, by replacing the no-slip boundary condition with the Navier slip condition at the particle surface, the cross coupling between the rotational torque and the shear stress is enhanced, as manifested through a dimensionless parameter in both of the hydrodynamic reciprocal relations and the Jeffery orbit. In addition, simulations for a circular particle patterned with portions of no-slip and Navier slip are carried out, showing that the particle possesses an effective anisotropy and follows the Jeffery orbit as well. This effective anisotropy can be tuned by changing the ratio of no-slip portion to slip potion. The connection of the present work to nematic liquid crystals' constitutive relations is discussed.

A Numerical Study of Fluid-Particle Interaction with Slip Boundary Condition

“…12 shows the fitted curves obtained with different slip lengths s = 0.0025, 0.005, 0.01, 0.02, and 0.03. As s increases, the cosine curves shift with an enhanced amplitude, which are in accordance with the results in Zhang et al [30].…”

“…Bretherton [6] demonstrated the general validity of the Jeffery orbit for particles with rotational symmetry. Recently, Zhang et al [30] studied the slip effect on Jeffery orbit using a continuum approach, in which the solid particle was treated as a fluid phase with fictitious high density and viscosity. In this section, we perform a direct simulation for the concerned problem by coupling the fluid dynamics and the motion of particle through hydrodynamic forces and torques on the particle boundary.…”

“…Combining the above techniques, we have an efficient algorithm to fully decouple the calculations of particle motion, mapping of variables, and fluid dynamics. We then simulate the rotation of an ellipsoidal particle in a simple shear flow and compare with the analytic results from the theory of Jeffery orbit and the results obtained in [30].…”

“…In the past decade, two categories of numerical simulations have been developed, including the continuum approach and the direct numerical simulation (DNS) approach. In the continuum approach, solid particles and fluids are viewed as interpenetrating mixtures with different viscosities that governed by conservation laws [21,27,29,30]. Despite its efficiency and flexibility, the continuum approach suffers from false response from the inside viscous material which is used to mimic the rigid objects, causing inaccurate hydrodynamical effects.…”

“…In [30], Zhang et al investigated the effects of boundary slip on the orientational motion of an anisotropic particle in a simple shear flow. A fluid-particle system is numerically solved using the fluid particle dynamics (FPD) method [27].…”