= Resistive-force theory
{wiki}
= RFT
{c}
{synonym}
<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> is
$$
\mathbf f=-\xi_\parallel(\mathbf v\cdot\mathbf t)\mathbf t
-\xi_\perp\left[\mathbf v-(\mathbf v\cdot\mathbf t)\mathbf t\right].
$$
The 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 $\xi_\perp/\xi_\parallel\simeq2$.
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 gives
$$
f_x=(\xi_\perp-\xi_\parallel)V\sin\theta\cos\theta.
$$
A 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.
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