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. This paradox provides insight into the effect of the singular behavior of the flow around hovering insects: the far-field wake covers the whole space.\\[4pt] Reference:\\[0pt] M. Iima, {\it J. Fluid Mech.}, (2008), {\bf 617}, 207--229.