Numerical analysis of the angular motion of a neutrally buoyant spheroid in shear flow at small Reynolds numbers

Abstract

We numerically analyse the rotation of a neutrally buoyant spheroid in a shear flow at small shear Reynolds number. Using direct numerical stability analysis of the coupled nonlinear particle-flow problem we compute the linear stability of the log-rolling orbit at small shear Reynolds number, ${\mathrm{Re}}_a$. As ${\mathrm{Re}}_a\to 0$ and as the box size of the system tends to infinity we find good agreement between the numerical results and earlier analytical predictions valid to linear order in ${\mathrm{Re}}_a$ for the case of an unbounded shear. The numerical stability analysis indicates that there are substantial finite-size corrections to the analytical results obtained for the unbounded system. We also compare the analytical results to results of lattice-Boltzmann simulations to analyse the stability of the tumbling orbit at shear Reynolds numbers of order unity. Theory for an unbounded system at infinitesimal shear Reynolds number predicts a bifurcation of the tumbling orbit at aspect ratio ${\lambda }_{\mathrm{c}}\approx 0.137$ below which tumbling is stable (as well as log rolling). The simulation results show a bifurcation line in the $\lambda$-${\mathrm{Re}}_a$ plane that reaches $\lambda \approx 0.1275$ at the smallest shear Reynolds number (${\mathrm{Re}}_a=1$) at which we could simulate with the lattice-Boltzmann code, in qualitative agreement with the analytical results.

updated version, 11 pages, 4 figures, 3 tables

References in corpus (5)

• Rotation Rate of Rods in Turbulent Fluid Flow
• Shape-dependence of particle rotation in isotropic turbulence
• Rotation of a spheroid in a simple shear at small Reynolds number
• Effect of weak fluid inertia upon Jeffery orbits
• The role of inertia for the rotation of a nearly spherical particle in a general linear flow

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• Settling of an asymmetric dumbbell in a quiescent fluid
• Angular dynamics of small crystals in viscous flow
• Alignment statistics of rods with the Lagrangian stretching direction in a channel flow
• Intrinsic viscosity of a suspension of weakly Brownian ellipsoids in shear
• Angular velocity of a spheroid log rolling in a simple shear at small Reynolds number
• Inertial torque on a squirmer
• Spinning and tumbling of micron-sized triangles in a micro-channel shear flow