  We analyse the angular velocity of a small neutrally buoyant spheroid log
rolling in a simple shear. When the effect of fluid inertia is negligible the
angular velocity $\omega$ equals half the fluid vorticity. We compute by
singular perturbation theory how weak fluid inertia reduces the angular
velocity in an unbounded shear, and how this reduction depends upon the shape
of the spheroid (on its aspect ratio). In addition we determine the angular
velocity by direct numerical simulations. The results are in excellent
agreement with the theory at small but not too small values of the shear
Reynolds number, for all aspect ratios considered. For the special case of a
sphere we find $\omega/s = -1/2+0.0540\, \mbox{Re}_\mbox{s}^{3/2}$ where $s$ is
the shear rate, and $\mbox{Re}_\mbox{s}$ is the shear Reynolds number. This
result differs from that derived by Lin et al. [J. Fluid Mech. 44 (1970) 1] who
obtained a numerical coefficient roughly three times larger.

Fluid Dynamics