Mean boundary velocity determines Taylor-sheet swimming speed

A Taylor expansion of the no-slip boundary condition about the flat sheet makes the second-order mean tangential velocity $⟨u_2(0)⟩=-⟨y_1{\partial }_y{u}_1(0)⟩$. The mean mode of Stokes flow is linear in height when no pressure gradient is imposed. In an unbounded fluid, bounded velocity excludes mean shear; in a confined fluid, the force-free condition excludes it. The remaining mean velocity is uniform and equals the swimming speed in the sheet frame. Thus the leading speed can be found from the first-order field without solving the oscillatory second-order field.

The second-order mean no-slip boundary condition correction contains products of first-order deformation modes and velocity gradients. Distinct integer wavenumbers have zero averaged mixed products. Their separate mean longitudinal and transverse contributions therefore add, even though mixed products create nonzero oscillatory flow modes.

When longitudinal and transverse sheet waves share unit phase speed but have different wavenumbers $k_{\parallel },k_{\perp }$, their second-order swimming coefficient is $(k_{\perp }^2{b}^2-k_{\parallel }^2{a}^2)/2$. Thus their leading propulsion cancels at $b/a=k_{\parallel }/k_{\perp }$. This statement concerns the small-amplitude leading order.