Nonlinear resistive-force balance for a travelling filament ($\mathbf{w}=c\mathbf{t}-(V-U)\mathbf{i}$)

For an inextensible planar filament with tailward tangent $\mathbf{t}=(X_s,Y_s)$, a travelling shape has material velocity $(V-U)\mathbf{i}-c\mathbf{t}$ in stationary fluid. Thus the relative fluid velocity is the displayed vector. With $\mathbf{n}=(-Y_s,X_s)$ and $q=V-U>0$, a local inertial normal drag $K_N|w_n|w_n\mathbf{n}$ has axial component $-K_N{q}^2|Y_s{|}^3$. A positive tangential slip $w_t=c-q{X}_s$ gives the opposite axial force. A prescribed tangential law $K_T{w}_t^{3/2}h(s)\mathbf{t}$ therefore gives the force-free balance $K_N{q}^2\int |Y_s{|}^{3}ds=K_T\int (c-q{X}_s{)}^{3/2}{X}_{s}h(s)ds$. The weight $h(s)$ is part of the constitutive approximation and must be justified separately from its speed exponent.