= Axial resistance matrix of a slender helix
{title2=$\binom{F_z}{G_z}=\begin{pmatrix}A&B\\B&D\end{pmatrix}\binom{U}{\Omega}$}
For radius $b$, arc length $L$, tangent angle $\phi$ to the horizontal plane, and local drag coefficient $C$, <resistive-force theory> gives
$$
A=\frac{CL}{2}(1+\cos^2\phi),\quad
B=-\frac{CLb}{2}\sin\phi\cos\phi,\quad
D=\frac{CLb^2}{2}(1+\sin^2\phi).
$$
The sign of $B$ selects the handedness convention. With the force exerted on the fluid taken positive, $AD-B^2=C^2L^2b^2/2>0$. Translation-rotation coupling supplies propulsion for a rotating <helical microswimmer with a spherical head>.
Back to article page
