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

Angular velocity of a spheroid log rolling in a simple shear at small Reynolds number