Effect of fluid and particle inertia on the rotation of an oblate spheroidal particle suspended in linear shear flow.

This work describes the inertial effects on the rotational behavior of an oblate spheroidal particle confined between two parallel opposite moving walls, which generate a linear shear flow. Numerical results are obtained using the lattice Boltzmann method with an external boundary force. The rotation of the particle depends on the particle Reynolds number, Re(p)=Gd(2)ν(-1) (G is the shear rate, d is the particle diameter, ν is the kinematic viscosity), and the Stokes number, St=αRe(p) (α is the solid-to-fluid density ratio), which are dimensionless quantities connected to fluid and particle inertia, respectively. The results show that two inertial effects give rise to different stable rotational states. For a neutrally buoyant particle (St=Re(p)) at low Re(p), particle inertia was found to dominate, eventually leading to a rotation about the particle's symmetry axis. The symmetry axis is in this case parallel to the vorticity direction; a rotational state called log-rolling. At high Re(p), fluid inertia will dominate and the particle will remain in a steady state, where the particle symmetry axis is perpendicular to the vorticity direction and has a constant angle ϕ(c) to the flow direction. The sequence of transitions between these dynamical states were found to be dependent on density ratio α, particle aspect ratio r(p), and domain size. More specifically, the present study reveals that an inclined rolling state (particle rotates around its symmetry axis, which is not aligned in the vorticity direction) appears through a pitchfork bifurcation due to the influence of periodic boundary conditions when simulated in a small domain. Furthermore, it is also found that a tumbling motion, where the particle symmetry axis rotates in the flow-gradient plane, can be a stable motion for particles with high r(p) and low α.

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

We numerically analyze 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 Re(a). As Re(a)→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 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 analyze 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 λ(c)≈0.137 below which tumbling is stable (as well as log rolling). The simulation results show a bifurcation line in the λ-Re(a) plane that reaches λ≈0.1275 at the smallest shear Reynolds number (Re(a)=1) at which we could simulate with the lattice Boltzmann code, in qualitative agreement with the analytical results.