In the Stokes limit, the trajectories of neutrally buoyant torque-free non-Brownian spheroids in ambient planar linear flows are well known. These flows form a one-parameter family, with the velocity gradient tensor given by
$\unicode[STIX]{x1D735}\boldsymbol{u}^{\infty \dagger }=\dot{\unicode[STIX]{x1D6FE}}(\mathbf{1}_{x}^{\prime }\mathbf{1}_{y}^{\prime }+\unicode[STIX]{x1D706}\mathbf{1}_{y}^{\prime }\mathbf{1}_{x}^{\prime })$
. The parameter
$\unicode[STIX]{x1D706}$
is related to the ratio of the vorticity to the extension (given by
$(1-\unicode[STIX]{x1D706})/(1+\unicode[STIX]{x1D706})$
), and ranges from
$-1$
to 1, with
$\unicode[STIX]{x1D706}=1\,,0$
and
$-1$
being planar extensional flow, simple shear flow and solid-body rotation respectively. The unit vectors
$\mathbf{1}_{x}^{\prime }$
and
$\mathbf{1}_{y}^{\prime }$
are unit vectors along the flow and gradient axes of the simple shear flow (
$\unicode[STIX]{x1D706}=0$
). The trajectories, as described by a unit vector along the spheroid symmetry axis, are closed orbits for
$\unicode[STIX]{x1D706}<\unicode[STIX]{x1D706}_{crit}$
, where
$\unicode[STIX]{x1D706}_{crit}=\unicode[STIX]{x1D705}^{2}(1/\unicode[STIX]{x1D705}^{2})$
for an oblate (a prolate) spheroid of aspect ratio
$\unicode[STIX]{x1D705}$
. We investigate analytically the orientation dynamics of such a spheroid in the presence of weak inertial effects. The inertial corrections to the angular velocities at
$O(Re)$
and
$O(St)$
, where
$Re$
and
$St$
are the Reynolds (
$Re=\unicode[STIX]{x1D70C}_{f}\dot{\unicode[STIX]{x1D6FE}}L^{2}/\unicode[STIX]{x1D707}$
) and Stokes numbers (
$St=\unicode[STIX]{x1D70C}_{p}\dot{\unicode[STIX]{x1D6FE}}L^{2}/\unicode[STIX]{x1D707}$
) respectively, are derived using a reciprocal theorem formulation. Here,
$L$
is the semimajor axis of the spheroid,
$\unicode[STIX]{x1D707}$
is the viscosity of the suspending fluid,
$\dot{\unicode[STIX]{x1D6FE}}$
is the shear rate, and
$\unicode[STIX]{x1D70C}_{p}$
and
$\unicode[STIX]{x1D70C}_{f}$
are the particle and fluid densities respectively. A spheroidal harmonics formalism is then used to evaluate the reciprocal theorem integrals and obtain closed-form expressions for the inertial corrections. The detailed examination of these corrections is restricted to the aforementioned Stokesian closed-orbit regime (
$\unicode[STIX]{x1D706}<\unicode[STIX]{x1D706}_{crit}$
). Here, even weak inertia, for asymptotically long times, of
$O(1/(\dot{\unicode[STIX]{x1D6FE}}Re))$
or
$O(1/(\dot{\unicode[STIX]{x1D6FE}}St))$
, will affect the leading-order orientation distribution on account of the indeterminate nature of the distribution across orbits in the Stokes limit. For
$\unicode[STIX]{x1D706}<\unicode[STIX]{x1D706}_{crit}$
, inertia results in a drift across the closed orbits in Stokes flow, and this orbital drift is characterized using a multiple time scale analysis. The orbits stabilized by the inertial drift, at
$O(Re)$
and
$O(St)$
, are identified in the
$\unicode[STIX]{x1D706}{-}\unicode[STIX]{x1D705}$
plane. For the majority of (
$\unicode[STIX]{x1D706},\unicode[STIX]{x1D705}$
) combinations, the stabilized orbit is either one confined to the plane of symmetry (the flow-gradient plane) of the ambient flow (the tumbling orbit) or one where the spheroid is aligned with the ambient vorticity vector (the spinning orbit). However, for some (
$\unicode[STIX]{x1D706},\unicode[STIX]{x1D705}$
) combinations, depending on the initial orientation, the orbit stabilized can be either the spinning or the tumbling orbit, since both orbits have non-trivial basins of attraction, separated by a pair of unstable (repelling) limit cycles, on the unit sphere of orientations. A stochastic orientation decorrelation mechanism in the form of rotary Brownian motion, characterized by a Péclet number,
$Pe_{r}$
(
$Pe_{r}=\dot{\unicode[STIX]{x1D6FE}}/D_{r}$
, where
$D_{r}$
is the rotary Brownian diffusivity), is included to eliminate the aforementioned dependence on the initial orientation distribution for certain (
$\unicode[STIX]{x1D706}$
,
$\unicode[STIX]{x1D705}$
) combinations. The unique steady-state orientation distribution determined by the combined effect of Brownian motion and inertia is obtained by solving a closed-orbit-averaged drift–diffusion equation. The steady-state orientation dynamics of an inertial spheroid in a planar linear flow, in the presence of weak thermal orientation fluctuations, has similarities to the thermodynamic description of a one-component system. Thus, we identify a tumbling–spinning transition in a
$C{-}\unicode[STIX]{x1D705}{-}Re\,Pe_{r}$
space. Here,
$C$
is the orbital coordinate that acts as a label for the closed orbits in the Stokes limit. This transition implies hysteretic orientation dynamics in certain regions in the
$C$
–
$\unicode[STIX]{x1D705}$
–
$Re\,Pe_{r}$
space, although the hysteretic volume shrinks rapidly on either side of simple shear flow. In the hysteretic region, one requires exceedingly large times to achieve the unique steady-state distribution (underlying the thermodynamic interpretation), and for durations relevant to experiments, the system may instead attain an initial-condition-dependent metastable distribution.