 Resistive-force theory

Resistive-force theory approximates the local viscous force on a moving slender elastic filament by separate drag coefficients along and across its tangent. Let $\mathbf{t}$ be the unit tangent and $\mathbf{v}$ the material velocity relative to the ambient fluid. The force per unit length on the elastic filament isThe positive parallel and perpendicular drag coefficients of a slender filament are defined per unit length; for a sufficiently slender isolated body, their leading logarithmic ratio is ${\xi }_{\perp }/{\xi }_{\parallel }\simeq 2$.

The directional difference can generate propulsion. For example, take a straight element with $\mathbf{t}=(\cos \theta ,\sin \theta )$ and impose transverse velocity $\mathbf{v}=V{\mathbf{e}}_y$. Direct substitution givesA transverse motion of a tilted element therefore has an axial force component. In a propagating flagellum deformation, the correlation of tilt and transverse velocity can produce nonzero mean thrust. For a freely swimming body, the unknown translation and rotation follow by imposing zero total external force and torque, including the resistance of the head if present.

This is a local Stokes flow approximation. It neglects the nonlocal hydrodynamic interactions retained by slender-body theory, so nearby boundary conditions, tightly spaced elastic filaments and detailed end effects can require a more complete model.