Plumera: Rayleigh–Taylor Instability
Mechanics, Chaos & Fluids arc, page 99 of 193. Back to The Pixelated Physics Universe
Who they are
Plumera holds heavy fluid above light, and gravity pulls it into fingers: spikes fall, bubbles rise.
Plumera is a character of the Mechanics, Chaos & Fluids arc: drawn in pixels, ruled by one piece of real physics, and never allowed to break it.
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. Lord Rayleigh analysed it in 1882, and G. I. Taylor followed in 1950.
$$ \begin{gathered} \sigma = \sqrt{A\,g\,k} \\ A = \frac{\rho_h - \rho_l}{\rho_h + \rho_l} \\ \text{ripple} \propto e^{\sigma t} \end{gathered} $$
| Symbol | Meaning | SI unit |
|---|---|---|
| $\sigma$ | growth rate of the ripple | 1/s |
| $t$ | time | s |
| $A$ | Atwood number (0 to 1) | 1 |
| $g$ | acceleration (gravity, or the frame's acceleration from heavy into light) | m/s² |
| $k$ | wavenumber of the ripple, 2π/λ | 1/m |
| $\rho_h,\ \rho_l$ | densities of the heavy and light fluids | kg/m³ |
Honest caveat: The formula ignores viscosity and surface tension.
Sources
- Rayleigh (1882) Investigation of the Character of the Equilibrium of an Incompressible Heavy Fluid of Variable Density, Proc. Lond. Math. Soc. 14, 170
- Taylor (1950) The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I, Proc. R. Soc. Lond. A 201, 192
- Sharp (1984) An overview of Rayleigh-Taylor instability, Physica D 12, 3
Crossovers
Plumera meets Navia ($1M Fluid Problem). Plumera's fingers grow from the same Navier–Stokes equations Navia guards. Her linear instability is well understood; whether the full 3D equations can ever blow up is the million-dollar question.
Plumera meets Stratalat (Layered Latte Convection). Plumera puts heavy fluid on top of light, and it overturns. Stratalat's latte layers form when heat and dissolved sugar diffuse at different rates, which makes stable layers rather than overturning.
Wear the lore
Concept keywords: Rayleigh–Taylor Instability, Plumera, Mechanics, Chaos & Fluids, physics lore, Pixelated Physics