A helical nanorobot with an embedded magnetic moment m is placed in a uniform field B that rotates at frequency f in the plane perpendicular to the swim axis. Magnetic torque forces the body to co-rotate; because the body is chiral (a corkscrew), rotation-to-translation coupling from viscous drag converts that spin into net forward swimming — the same low-Reynolds-number mechanism bacterial flagella use, and the one exploited by real magnetically-actuated micro/nanorobots proposed for targeted, wireless drug delivery (including astronaut healthcare research for long-duration spaceflight, where gravity-driven circulation can't be relied on).
Resistive-force theory (slender-rod drag coefficients):
c⊥ = 4πη / (ln(2Λ/a) + 0.5) c∥ = 2πη / (ln(2Λ/a) − 0.5)
Propulsion (efficiency) factor, pitch angle θ:
k(θ) = sinθ·cosθ·(c⊥ − c∥) / (c⊥cos²θ + c∥sin²θ) — peaks ≈ 40–50°
Swim speed (synchronous regime):
U = k(θ) · R_helix · Ω, Ω = 2πf
Step-out frequency (magnetic torque = max viscous drag torque):
Ω_c = m·B / (8πη·a_eff³), f_c = Ω_c / 2π
above f_c the body can no longer keep pace with the field and
desynchronises — average rotation, and speed, collapse toward
⟨Ω⟩ ≈ Ω_c² / Ω_field
- Frequency & field strength — raise f toward fc and speed grows linearly; push past it and the robot slips, tumbling instead of advancing.
- Pitch angle θ — the geometric trade-off between how much drag anisotropy a turn of helix has and how steep the thread is; too shallow or too steep and propulsion falls off even though rotation is identical.
- Viscosity — higher η raises the drag torque the field must overcome, lowering fc, which is why the same robot that swims fine in water can stall out in a more viscous fluid.
- Chirality — flips which way the same rotation sense propels the body, exactly as a left- vs right-handed screw threads in opposite directions for the same twist.
Note the swim-speed formula above has no η in it: at low Reynolds number, synchronous corkscrew speed depends only on frequency and geometry, not viscosity — viscosity only decides whether that synchronous regime is reachable at all. Reynolds number stays far below 1 throughout, which is exactly why a reciprocal (non-chiral) shape couldn't swim here at all — Purcell's "scallop theorem."