A paradox of hovering insects in two-dimensional space

A paradox concerning the flight of insects in two-dimensional space is identified: insects maintaining their bodies in a particular position (hovering) cannot, on average, generate hydrodynamic force if the induced flow is temporally periodic and converges to rest at infinity. This paradox is derived by using the far-field representation of periodic flow and the generalized Blasius formula, an exact formula for a force that acts on a moving body, based on the incompressible Navier-Stokes equations. Using this formula, the time-averaged force can be calculated solely in terms of the time-averaged far-field flow. A straightforward calculation represents the averaged force acting on an insect under a uniform flow, -< V >, determined by the balance between the hydrodynamic force and an external force such as gravity. The averaged force converges to zero in the limit < V > --> 0, which implies that insects in two-dimensional space cannot hover under any finite external force if the direction of the uniform flow has a component parallel to the external force. This paradox provides insight into the effect of the singular behaviour of the flow around hovering insects: the far-field wake covers the whole space. On the basis of these assumptions, the relationship between this paradox and real insects that actually achieve hovering is discussed.