Oscillating rigid rod in resistive-force theory

A rigid rod with moving centre and changing orientation has material velocity $\dot{\mathbf{X}}+s\dot{\theta }\mathbf{n}$. Its centred rotational velocity is odd in arclength and has no resultant force, while its orientation-dependent resistance tensor couples translation to transverse force. Resistive-force theory uses per-length coefficients $c_{\parallel },c_{\perp }$.

The leading mean viscous dissipation of a translating and rotating rigid rod is $c_{\perp }{ϵ}^2{\omega }^2(L{a}^2+L^3/12)/2$ for equal dimensionless translation and angle amplitudes. Both terms are quadratic in amplitude; rotational material velocity cannot be omitted simply because the angle is small.

For $x=ϵa\cos \omega t$ and a tangent $(\sin \theta ,\cos \theta )$ with $\theta =ϵ\cos (\omega t+\varphi )$, the leading force on the fluid is $⟨F_y⟩=-(c_{\perp }-c_{\parallel })L{ϵ}^{2}a\omega \sin \varphi /2$. The sign changes under reversal of the tilt convention. Quadrature strokes maximize its magnitude; reciprocal strokes give zero, consistent with kinematic reversibility of Stokes flow.