 Axial resistance matrix of a slender helix

For radius $b$, arc length $L$, tangent angle $\varphi$ to the horizontal plane, and local drag coefficient $C$, resistive-force theory givesThe sign of $B$ selects the handedness convention. With the force exerted on the fluid taken positive, $AD-B^2=C^2{L}^2{b}^2/2>0$. Translation-rotation coupling supplies propulsion for a rotating helical microswimmer with a spherical head.