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

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, ${\rm Re}_a$. As ${\rm Re}_a \to 0$ 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 ${\rm Re}_a$ 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 $\lambda_{\rm c} \approx 0.137$ below which tumbling is stable (as well as log rolling). The simulation results show a bifurcation line in the $\lambda$-${\rm Re}_a$ plane that reaches $\lambda \approx0.1275$ at the smallest shear Reynolds number (${\rm Re}_a=1$) at which we could simulate with the lattice-Boltzmann code, in qualitative agreement with the analytical results.