Axial resistance matrix of a slender helix ( $(G _{z} F _{z}) = (A B B D) (Ω U)$ )

b

L

, tangent angle

ϕ

to the horizontal plane, and local drag coefficient

C

, resistive-force theory gives

A = \frac{C L}{2} (1 + cos^{2} ϕ), B = - \frac{C L b}{2} sin ϕ cos ϕ, D = \frac{C L b ^{2}}{2} (1 + sin^{2} ϕ) .

The sign of

B

selects the handedness convention. With the force exerted on the fluid taken positive,

A D - B^{2} = C^{2} L^{2} b^{2} /2 > 0

. Translation-rotation coupling supplies propulsion for a rotating helical microswimmer with a spherical head.

Table of contents

Axial coupling of an elliptic helix ( $F_{z} /Ω = G_{z} / W = - c A B b \int_{- π}^{π} (s^{'})^{- 1} d θ$ )

For a slender centreline

X (θ) = (A cos θ, B sin θ, b θ)

,

- π \leq θ \leq π

, let

s^{'} (θ) = (A^{2} sin^{2} θ + B^{2} cos^{2} θ + b^{2})^{1/2}

and use local force density

f = c (2 I - tt) v

,

t = d X / d s

,

c = 2 π μ / ln (L / R)

. Rotation with angular speed

Ω

gives

t \cdot v = Ω A B / s^{'}

, hence

F_{z} = - c Ω A B b \int_{- π}^{π} \frac{d θ}{s ^{'} ( θ )} .

Axial translation with speed

W

gives exactly the same coefficient for the torque:

G_{z} = - c W A B b \int d θ / s^{'}

. Indeed

(X \times t)_{z} = A B / s^{'}

. Both forces and torques here are exerted on the fluid. In the circular case

A = B = a

, this reduces to

F_{z} /Ω = G_{z} / W = - 2 π c a^{2} b / a^{2} + b^{2}

. The equality is also a consequence of the Lorentz reciprocal theorem, since the local resistance tensor is symmetric.

Determinant of helical resistance in resistive-force theory ( $A D - B^{2} = ℓ^{2} a^{2} ξ_{∥} ξ_{⊥}$ )

Local resistive-force theory for a helix gives

A D - B^{2} = ℓ^{2} a^{2} ξ_{∥} ξ_{⊥}

. Positive local drag therefore guarantees a positive-definite axial resistance matrix and positive viscous power loss. The identity also keeps the denominator in counterrotating two-helix balances positive.

Handedness reversal of helical hydrodynamic resistance ( $B \mapsto - B, A, D unchanged$ )

Mirroring the handedness of a helix reverses its axial translation-rotation coupling coefficients while preserving its pure translation and rotation resistances. With

t = cos θ e_{z} - sin θ e_{φ}

for a left-handed helix and the force-on-body convention,

B = C = ℓ a (ξ_{⊥} - ξ_{∥}) sin θ cos θ

.

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