  We compute the leading-order inertial corrections to the instantaneous force
acting on a rigid body moving with a time-dependent slip velocity in a linear
flow field, assuming that the variation of the undisturbed flow at the body
scale is much larger than the slip velocity between the body and the fluid.
Motivated by applications to turbulent particle-laden flows, we seek a
formulation allowing this force to be determined for an arbitrarily-shaped body
moving in a general linear flow. We express the equations governing the flow
disturbance in a non-orthogonal coordinate system moving with the undisturbed
flow and solve the problem using matched asymptotic expansions. The use of the
co-moving coordinates enables the leading-order inertial corrections to the
force to be obtained at any time in an arbitrary linear flow field. We then
specialize this approach to compute the time-dependent force components for a
sphere moving in three canonical flows: solid body rotation, planar elongation,
and uniform shear. We discuss the behaviour and physical origin of the
different force components in the short-time and quasi-steady limits. Last, we
illustrate the influence of time-dependent and quasi-steady inertial effects by
examining the sedimentation of prolate and oblate spheroids in a pure shear
flow.
