 Power of a periodic planar filament

The leading viscous dissipation due to a small planar filament stroke is $\dot{W}={\xi }_{\perp }{T}^{-1}{\int }_0^T{\int }_0^{\lambda }{y}_t^{2}dxdt$. It follows from the work against the local resistive-force theory force. The transverse velocity controls the leading power; the quadratic longitudinal velocity contributes only at higher order.