Laminar skin-friction scaling along a slender swimmer (${\tau }_w\sim \rho \sqrt{\nu }{u}_t^{3/2}{s}^{-1/2}$)

A steady attached laminar boundary layer with slip speed $u_t$ and downstream distance $s$ has thickness $\delta \sim \sqrt{\nu s/u_t}$. The wall shear is ${\tau }_w\sim \mu u_t/\delta$, giving the displayed scaling, where $\mu =\rho \nu$ is dynamic viscosity. Multiplying by a nearly constant wetted perimeter gives local drag per unit length proportional to $u_t^{3/2}{s}^{-1/2}$, and integration gives drag proportional to $u_t^{3/2}\sqrt{L}$. A force density weighted by $s^{1/2}$ instead produces $L^{3/2}$; it can be a separately prescribed phenomenological model but is not this local laminar shear law. Strong curvature, unsteadiness, separation or turbulent transition can invalidate the estimate.