The back-reaction of an incompressible two-dimensional flow of constant strain to the accelerated motion of a straight cylinder is calculated in the general case that the flow is inhomogeneous, nonstationary, and with nonvanishing (although constant) vorticity. The resulting enhanced inertia of the cylinder is basically given by its relative acceleration with respect to the background flow. Further force terms that appear because of the nonstationary and inhomogeneity of the background flow are the following: a force identical to that experienced by the mass elements of the unperturbed flow at the position of the axis of the cylinder; the customary lift force because of the circulation around the body; finally, a force term that appears only with nonvanishing relative speed between the body and the background flow and which is related to the energy which has to be imparted by the body to the surrounding fluid in order to adapt the perturbation to the new local velocity. The results have application to the dynamics of magnetic flux tubes in the convection zone and atmosphere of the Sun as well as to other astrophysical problems.