Inertial Swimming as a Singular Perturbation

The swimming of a sheet, originally treated by G.I. Taylor (1951) for the case of Stokes flow, is considered at moderate and high Reynolds numbers using matched asymptotic expansions. It is shown that for propagating waves with frequency ω, wavenumber k, and amplitude b, the swimming speed must be deduced from a dual expansion in powers of the small parameters bkR^1/2 and R^−1/2, where R = ω/νk² is the Reynolds number. The result of Tuck (1968) for the leading term of the swimming velocity is recovered, and higher-order results are given. For the case of a planar, stretching sheet, the expansion is in powers of bk and R^−1/2 and a limit for large R is obtained as a boundary layer. We contrast these results with the inviscid case, where no swimming is possible. We also consider briefly the application of these ideas to “recoil swimming”, wherein the movements of the center of mass and center of volume of a body allow swimming at both finite and infinite Reynolds numbers.