Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 355 / 2

In resistive-force theory, the force per unit length is local and linear in the filament velocity relative to the fluid:The distinct parallel and perpendicular drag coefficients encode the anisotropic resistance of a slender filament. Neglecting nonlocal hydrodynamic interactions makes this a leading logarithmic approximation to slender-body theory.

Choose the handedness for which the helix tangent is $\mathbf{t}=\cos \theta {\mathbf{e}}_z+\sin \theta {\mathbf{e}}_{\varphi }$. Its local rigid velocity is $\mathbf{u}=\tilde{U}{\mathbf{e}}_z+R\tilde{\Omega }{\mathbf{e}}_{\varphi }$, so $\mathbf{u}\cdot \mathbf{t}=\tilde{U}\cos \theta +R\tilde{\Omega }\sin \theta$. Integrating the axial force and torque from the drag law gives the symmetric hydrodynamic resistance matrix in the question, withReversing helical handedness reverses the sign of $B$ but leaves $A$ and $D$ unchanged.

Force and torque freedom of the cell--helix pair giveWritingsolution of this linear system yieldsThe body counter-rotates relative to the motor, and the swimming direction reverses with the helix handedness through $B$.

The helix exerts on the fluid the torqueOverall torque balance also gives $N=-D_0\Omega$. HenceFor the usual choice $\omega >0$, this is positive when the resistance matrix is positive definite.

Substitution into $ϵ=A_0{U}^2/(N\omega )$ givesWhen $B^2\ll AD$ and $D\ll D_0$,After scaling body size so that $A_0\mapsto \lambda A_0$, differentiation with respect to $\lambda A_0$ shows that the optimum impedance match is $\lambda A_0=A$. Therefore