a1 Department of Mathematics and Lawrence Berkeley National Laboratory University of California, Berkeley, CA 94721, USA
a2 Hatsopoulos Microfluids Laboratory, Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
Motivated by the propulsion mechanisms adopted by gastropods, annelids and other invertebrates, we consider shape optimization of a flexible sheet that moves by propagating deformation waves along its body. The self-propelled sheet is separated from a rigid substrate by a thin layer of viscous Newtonian fluid. We use a lubrication approximation to model the dynamics and derive the relevant Euler–Lagrange equations to simultaneously optimize swimming speed, efficiency and fluid loss. We find that as the parameters controlling these quantities approach critical values, the optimal solutions become singular in a self-similar fashion and sometimes leave the realm of validity of the lubrication model. We explore these singular limits by computing higher-order corrections to the zeroth order theory and find that wave profiles that develop cusp-like singularities are appropriately penalized, yielding non-singular optimal solutions. These corrections are themselves validated by comparison with finite element solutions of the full Stokes equations, and, to the extent possible, using recent rigorous a priori error bounds.
(Received January 30 2006)
(Revised December 21 2007)