• Flagella Scallop Theorem pixel-art physics T-shirt, white tee, front view

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Flagella

The Corkscrew Escape Physics

\[F = A v + B\Omega, \qquad N = B v + D\Omega\]
$29.99 USD
$29.99 USD
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Size guide S–5XL · inches

Flagella swims where water feels like honey, and she gets there with a corkscrew.

The Physics

For a bacterium, viscosity wins completely: the Reynolds number is about 10⁻⁵, against about 10⁴ for a human swimmer. Purcell's scallop theorem says that a stroke that looks the same run backwards gets you nowhere. A rotating helix breaks that symmetry, because turning a corkscrew is linked to pushing it.

F = A v + B Ω, N = B v + D Ω

  • F: thrust force (N)
  • N: torque (N·m)
  • v: swimming speed (m/s)
  • Ω: rotation rate (rad/s)
  • A, B, D: drag coefficients set by the helix shape and viscosity

E. M. Purcell set out the scallop theorem in 1977 (Am. J. Phys.); Berg and Anderson showed bacteria swim by rotating their flagellar bundles (Nature, 1973).

On the shirt

Front: Flagella, the pixel-art character, with the title. Back: Purcell's propulsion matrix, the Reynolds numbers, the scallop theorem and the 1973 and 1977 sources. Printed on a unisex tee.

Shipping calculated at checkout by destinationMade to order, ships in 2–5 business days

Details

Size & fitS–5XL

Unisex heavy cotton (Gildan 5000), classic fit.

SizeWidthLengthSleeve
S182815.1
M202916.5
L223018
XL243119.5
2XL263221
3XL283322.4
4XL303423.7
5XL323525

Ter vergelijking: maat M is 20 in (50.80 cm) breed, ongeveer 3.143 × 1034 Plancklengtes (lP = 1.616255 × 10−35 m). Een meetlint is sneller.

Measurements in inches, ±1.5 in tolerance.

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Returns & replacements21 days

Every item is printed to order, so we can’t accept returns or exchanges for change of mind, or if you ordered the wrong size or colour.

Print defect, misprint, damage or the wrong item? We’ll send a free replacement, or a refund if you prefer. Report it within 21 days of delivery with a photo; no need to send it back.

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The physics

I swim where water feels like syrup. For a bacterium the Reynolds number is about 10⁻⁵; for you in a pool it is about 10⁴. At my scale inertia is nothing and viscosity is everything.

In flow that slow, time runs both ways: replay a stroke backwards and you undo it. Edward Purcell called this the scallop theorem in 1977, since a scallop that just opens and closes gets nowhere. A turning corkscrew never replays itself. In Purcell's propulsion matrix, F = Av + BΩ and N = Bv + DΩ, the off-diagonal B couples spin to thrust. Keep turning, keep going.

Howard Berg and Robert Anderson showed in 1973 that bacteria really do swim by rotating their flagellar filaments, driven by a motor at the base.

Equations

\[F = A v + B\Omega, \qquad N = B v + D\Omega\]
\[Re:\ \text{human swimmer} \sim 10^{4}, \quad E.\ coli \sim 10^{-5}\]
SymbolMeaningUnit
\(F\) axial force on the helix \(\mathrm{N}\)
\(N\) axial torque \(\mathrm{N·m}\)
\(v\) axial velocity \(\mathrm{m/s}\)
\(\Omega\) angular velocity \(\mathrm{rad/s}\)
\(A\) propulsion-matrix coefficient \(\mathrm{N·s/m}\)
\(B\) propulsion-matrix coupling coefficient \(\mathrm{N·s}\)
\(D\) propulsion-matrix coefficient \(\mathrm{N·m·s}\)
\(Re\) Reynolds number \(\mathrm{1}\)

Sources

  1. Purcell (1977) Life at low Reynolds number, Am. J. Phys. 45, 3
  2. Berg & Anderson (1973) Bacteria Swim by Rotating their Flagellar Filaments, Nature 245, 380

Field notes

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