• Plumera Rayleigh–Taylor Instability pixel-art physics T-shirt, black tee, front view

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Plumera

Rayleigh–Taylor Instability Physics

\[\sigma = \sqrt{A\,g\,k}\]
$29.99 USD
$29.99 USD
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Size guide S–5XL · inches

Plumera holds heavy fluid above light, and gravity pulls it into fingers: spikes fall, bubbles rise.

The Physics

When a heavy fluid sits on top of a light one in gravity, any ripple on the interface grows. In the linear theory, a ripple of wavenumber k grows exponentially at rate σ, depending on the density contrast through the Atwood number. Spikes of heavy fluid fall and bubbles of light fluid rise.

σ = √(A g k), A = (ρ_h − ρ_l)/(ρ_h + ρ_l)

  • σ: growth rate (1/s)
  • A: Atwood number (dimensionless)
  • g: gravitational acceleration (m/s²)
  • k: wavenumber (1/m)
  • ρ_h, ρ_l: heavy and light densities (kg/m³)

Lord Rayleigh analysed it in 1882, and G. I. Taylor followed in 1950.

Honest caveat: The formula ignores viscosity and surface tension.

On the shirt

Front: Plumera, the pixel-art character, with the title. Back: the growth rate, the Atwood number, spikes and bubbles and the 1882 and 1950 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

Para hacerse una idea: la talla M mide 20 in (50.80 cm) de ancho, unas 3.143 × 1034 longitudes de Planck (lP = 1.616255 × 10−35 m). Una cinta métrica es más rápida.

Measurements in inches, ±1.5 in tolerance.

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Shipping calculated at checkout by destination. Printed to order for you.

Delivery times & where we ship

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.

Full refund & replacement policy

The physics

I hold heavy fluid above light, and gravity pulls it into fingers: spikes fall, bubbles rise.

Put a heavy fluid on top of a light one and any ripple on the boundary grows. In the linear theory a ripple of wavenumber k grows as e^(σt), with σ = √(A g k), where the Atwood number A measures the density contrast. Short ripples grow fastest. Soon the heavy fluid falls in spikes, the light fluid rises in bubbles, and the tips roll up into mushroom caps.

Lord Rayleigh analysed it in 1882 and G. I. Taylor extended it to accelerated fluids in 1950. You can see it in supernova remnants, in fusion capsules, and in cream poured into coffee. The simple formula ignores viscosity and surface tension, which damp the shortest ripples.

Equations

\[\sigma = \sqrt{A\,g\,k}\]
\[A = \frac{\rho_h - \rho_l}{\rho_h + \rho_l}\]
\[\text{ripple} \propto e^{\sigma t}\]
SymbolMeaningUnit
\(\sigma\) growth rate of the ripple \(\mathrm{1/s}\)
\(t\) time \(\mathrm{s}\)
\(A\) Atwood number (0 to 1) \(\mathrm{1}\)
\(g\) acceleration (gravity, or the frame's acceleration from heavy into light) \(\mathrm{m/s²}\)
\(k\) wavenumber of the ripple, 2π/λ \(\mathrm{1/m}\)
\(\rho_h,\ \rho_l\) densities of the heavy and light fluids \(\mathrm{kg/m³}\)

Sources

  1. Rayleigh (1882) Investigation of the Character of the Equilibrium of an Incompressible Heavy Fluid of Variable Density, Proc. Lond. Math. Soc. 14, 170
  2. Taylor (1950) The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I, Proc. R. Soc. Lond. A 201, 192
  3. Sharp (1984) An overview of Rayleigh-Taylor instability, Physica D 12, 3

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