Low-Reynolds Swimmer 3D

Life without inertia

For a swimmer of length L and speed U in a fluid of density ρ and viscosity μ the Reynolds number is Re = ρ U L / μ. A sperm cell (L ≈ 50 µm, U ≈ 50 µm/s in water) has Re ≈ 10^-3: viscous forces beat inertial ones by a factor of a thousand. The flow obeys the Stokes equations, which have no time derivative. Consequences: motion stops the instant the forces stop (the readout “coasting distance” is about 10^-5 µm, less than an atom), and the shape sequence matters, not the speed at which it is traversed.

The scallop theorem (Purcell, 1977)

Because Stokes flow is linear and time-independent, a swimmer that changes shape through a reciprocal sequence (a sequence that retraces itself in reverse, such as a scallop opening and closing, or a tail beating as a standing wave) ends each cycle where it started. Net propulsion needs a non-reciprocal beat such as a travelling wave. The orange twin in the simulation beats as a pure standing wave y = A(s) sin(ks) sin(ωt) and goes nowhere; the blue cell uses y = A(s) sin(ks − ωt).

Resistive force theory

A slender filament moving with velocity v feels a force per unit length f = −ct vt t − cn vn n where t and n are the tangent and normal and the Gray–Hancock coefficients are ct = 2πμ/(ln(2λ/r) − 1/2), cn = 4πμ/(ln(2λ/r) + 1/2) with wavelength λ and filament radius r. Dragging sideways costs about K = cn/ct ≈ 1.7 times more than dragging along. The spherical head adds the Stokes drag 6πμa and the rotational drag 8πμa3. Writing each node's velocity as v = U + Ω × r + ∂r/∂t (body motion plus shape change), the conditions ΣF = 0 and ΣT = 0 are a 6 × 6 linear system for the swimming velocity U and spin Ω, solved at every time step; the cell pose is then integrated.

Small-amplitude result (an independent check)

For an infinite sinusoid of amplitude b and wave number k the force balance per wavelength gives U/V = (K − 1)(bk)2/2 to lowest order, where V = ω/k is the wave speed. The third graph compares this with the exact numerical balance (no expansion) as a function of bk. Real cells have bk of order 0.5–1, where the formula over-predicts slightly.

Speed does not depend on viscosity

Every force in the problem is proportional to μ, so μ cancels from the force balance: thickening the fluid changes the power (P ∝ μ) but not the speed. Try the “thick fluid” preset. (Real sperm slow in mucus because the beat itself changes; and the fluid becomes non-Newtonian.)

Four in-page numerical checks

1. Exact force balance on an infinite wave versus (K − 1)(bk)2/2. 2. Scallop theorem: net displacement over four cycles for the standing-wave beat versus the travelling wave. 3. Total force and torque residuals of the solved state (about 10^-16) and Re < 0.01. The helical beat in the 3D page uses the same solver with a corkscrew wave and also rotates the cell about its long axis.

Honest simplifications

Resistive force theory neglects hydrodynamic interactions between parts of the filament (errors of tens of percent); the head is a sphere; the beat shape is prescribed (the real wave is produced by dynein motors and bending mechanics) and is only approximately inextensible; there are no walls (real sperm swim along surfaces and are attracted to them); the viscosity is Newtonian; the tracer beads do not move with the flow.