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A paradox of hovering insects in two-dimensional space

Published online by Cambridge University Press:  25 December 2008

MAKOTO IIMA
MAKOTO IIMA*
Affiliation:
Research Institute for Electronic Science, Hokkaido University, N12W6, Sapporo, 060-0812, Japan
*
†Email address for correspondence: makoto@nsc.es.hokudai.ac.jp
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Abstract

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.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 617 , 25 December 2008 , pp. 207 - 229
Copyright
Copyright © Cambridge University Press 2008

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