Spotlight #73 — Fluid Physics & Astrophysics: Vortex Rings, Stellar Interiors & Exoplanet Transits

I. Fluid Dynamics: From Vortex Rings to Rarefied Gas

A vortex ring is one of the most durable structures in fluid dynamics. Unlike a blob of dye that diffuses away within seconds, a vortex ring can travel metres through still air, maintaining its identity for tens of ring-radii of travel. It is self-propelling: the ring induces a velocity field that advects its own core forward at speed

V_ring = (Gamma / 4*pi*R) * (ln(8R/a) - 1/4)

  Gamma : circulation (m^2/s)  — integral of vorticity over cross-section
  R     : ring radius (m)
  a     : core radius (m, a << R for a thin-core ring)
  The Biot–Savart law for a circular filament gives this leading-order result.

The logarithmic term reflects the long-range induction from the ring's own far field. Thin rings (a/R → 0) travel fastest because the self-induction diverges as the core shrinks, yet viscosity always thickens the core over time, gradually slowing the ring. This is why a dolphin's bubble ring eventually stalls and rises rather than shooting forward indefinitely.

Leapfrogging

Launch two co-axial rings of the same sign in tandem and a remarkable dance begins: the trailing ring accelerates (it travels through the converging induced flow of the leader), threads through the leading ring, becomes the new leader, and the cycle repeats. This leapfrogging is periodic and stable when the rings are identical, but breaks down into chaos when they differ slightly in circulation or radius — a beautiful route to turbulence observable in a tank of water with dye injection.

The Taylor–Green vortex is the theorist's favourite test case for turbulence. The initial velocity field is a simple trigonometric pattern — smooth, periodic, analytically known — yet within a few turnover times it develops all the hallmarks of fully developed turbulence: a broad inertial range in the energy spectrum, intense vortex tubes, and intermittent dissipation events. The reason is that the Navier–Stokes equations are nonlinear: any smooth initial condition at high Reynolds number will cascade energy from large scales to small scales until viscous dissipation finally absorbs it.

Initial condition (3D, periodic box [0, 2*pi]^3):
  u = V_0 * sin(x) * cos(y) * cos(z)
  v = -V_0 * cos(x) * sin(y) * cos(z)
  w = 0

Kolmogorov energy spectrum (inertial range):
  E(k) ~ C * epsilon^(2/3) * k^(-5/3)

  epsilon : mean energy dissipation rate (m^2/s^3)
  k       : wavenumber (rad/m)
  C       : Kolmogorov constant ~ 1.5
  Valid for L^-1 << k << eta^-1  (L = integral scale, eta = Kolmogorov microscale)

The −5/3 slope of the energy spectrum is Kolmogorov's 1941 prediction, and it holds remarkably well in experiments and simulations across many orders of magnitude in Reynolds number. The simulation plots this spectrum in real time, letting you measure the slope and watch it steepen from −2 (the initial large-scale regime) toward −5/3 as turbulence develops, and then flatten again as energy accumulates at the dissipation scale.

A rising bubble is governed by the balance between buoyancy, drag, surface tension, and inertia. For small bubbles, surface tension holds the shape spherical and Stokes drag dominates. For large bubbles, inertia wins: the bubble deforms into an oblate spheroid, then a spherical cap, and — in the right density ratio regime — eventually pinches off its rear to form a toroidal bubble ring. The dimensionless numbers that control this phase diagram are the Eötvös number Eo = gΔρR²/σ (ratio of buoyancy to surface tension) and the Morton number Mo = gμ&sup4;Δρ/(ρ²σ³).

The Navier–Stokes equations assume the fluid is a continuous medium — a valid approximation when the mean free path λ is much smaller than the flow length scale L. The ratio Kn = λ/L, the Knudsen number, tells you when this breaks down. At Kn > 0.01, slip velocity at walls becomes measurable; at Kn > 0.1, the entire velocity profile changes character; at Kn > 10, molecules travel from wall to wall without any collisions (free-molecular flow). This regime is ubiquitous in spacecraft aerodynamics, MEMS devices, and the upper atmosphere above 80 km.

The DSMC (Direct Simulation Monte Carlo) method, developed by Graeme Bird in the 1960s, simulates a representative sample of molecules, computing their trajectories and stochastic collisions at each time step. It converges to the Boltzmann equation rather than Navier–Stokes, and captures effects like velocity distribution functions departing from the Maxwell–Boltzmann equilibrium — effects that no continuum method can reproduce.

Most real fluids are neither purely viscous (like water) nor purely elastic (like rubber): they are viscoelastic. Mucus, blood, molten polymers, biological gels, and Earth's mantle all flow like liquids at long timescales and rebound like solids at short timescales. The Maxwell model captures this with a single relaxation time τ:

Maxwell constitutive equation:
  dσ/dt + σ/τ = G * dε/dt

  σ : stress (Pa)       ε : strain
  G : elastic modulus (Pa)  τ = η/G : relaxation time (s)
  η : dynamic viscosity (Pa·s)

At timescales t << τ  : solid-like (elastic storage)
At timescales t >> τ  : liquid-like (viscous flow)

Wave speed in a Maxwell fluid (frequency ω):
  c(ω) = sqrt(G/ρ) * |ωτ| / sqrt(1 + ω^2 τ^2)
  → 0 at low frequency (no shear waves in a liquid)
  → sqrt(G/ρ) at high frequency (shear wave speed of solid)

II. Astrophysics: Stellar Interiors, White Dwarfs & Exoplanet Transits

A star is a self-gravitating ball of gas in hydrostatic equilibrium: gravity pulls inward, pressure gradient pushes outward, and nuclear reactions at the centre supply the energy that maintains the temperature gradient. Four coupled differential equations describe the entire structure:

Stellar structure equations:
  dM/dr = 4π r^2 ρ                   (mass continuity)
  dP/dr = -G M(r) ρ / r^2              (hydrostatic equilibrium)
  dL/dr = 4π r^2 ρ ε(r,T,X)  (energy generation)
  dT/dr = -(3 κ ρ L) / (64π σ_SB r^2 T^3)   (radiative transport)
         or   -(1 - 1/γ) T/P * dP/dr   (adiabatic, if convective)

  ρ : density   κ : opacity   ε : nuclear energy generation rate
  X   : hydrogen mass fraction   γ : adiabatic index
  σ_SB : Stefan–Boltzmann constant

The switch between radiative and convective transport is governed by the Schwarzschild criterion: convection sets in wherever the actual temperature gradient steepens beyond the adiabatic gradient. Low-mass stars like the Sun have convective envelopes surrounding a radiative core; massive stars above roughly 1.5 M⊙ flip this arrangement, with convective cores and radiative envelopes. The simulation shows both cases and lets you trace the boundary in real time as you adjust stellar mass.

The pp-chain and CNO cycle

Below about 1.5 solar masses the proton–proton chain dominates nuclear burning: four protons fuse into one helium-4 nucleus, releasing 26.73 MeV (the mass defect converted by E = mc²). Above 1.5 M⊙, the CNO cycle takes over, using carbon, nitrogen, and oxygen as catalysts. Because the CNO rate scales roughly as T^20 versus T^4 for the pp-chain, CNO stars are far more temperature-sensitive: a slight increase in central temperature produces a huge increase in luminosity, explaining why the main sequence steepens sharply above 1.5 M⊙ on the Hertzsprung–Russell diagram.

A white dwarf has no nuclear energy source. It is simply a hot ember — a degenerate carbon–oxygen core roughly the size of Earth — radiating stored thermal energy into space. Because electron degeneracy pressure supports it against gravity regardless of temperature, cooling does not trigger contraction: the star just dims. The classical Mestel cooling law gives the luminosity as a function of age:

Mestel cooling law (simplified):
  L/L_sun ~ 10^5 * (M/M_sun)^(5/7) * (mu_e / mu_I)^(2/7) * (t / 1 Gyr)^(-7/5)

  mu_e : mean molecular weight per electron
  mu_I : mean molecular weight per ion
  t    : cooling age (Gyr)

  A 0.6 M_sun CO white dwarf reaches L ~ 10^-4 L_sun at ~5 Gyr
  and L ~ 10^-5 L_sun at ~10 Gyr  (the faintest nearby white dwarfs)

The cooling law is a powerful cosmochronometer: the coolest white dwarfs in a stellar population give a lower bound on its age. But there is a wrinkle. When the core temperature drops below about 10&sup6; K, the Coulomb coupling parameter Γ (ratio of electrostatic to thermal energy) exceeds 175 and the ion lattice crystallises. The phase transition releases latent heat, temporarily halting the cooling and producing a pile-up of white dwarfs at a specific luminosity — a feature clearly detected by Gaia DR2 in the Hertzsprung–Russell diagram of 260,000 nearby white dwarfs.

When a planet passes in front of its host star, it blocks a tiny fraction of the starlight: the transit depth δ = (R_p/R_*)², where R_p is the planetary radius and R_* is the stellar radius. For an Earth-Sun analogue, δ ~ (6371/696000)² ~ 84 ppm — a dimming of 0.0084%, achievable with space-based photometry from Kepler or TESS but invisible from the ground. For a hot Jupiter (R_p ~ 1.2 R_Jup, R_* ~ 1 R_Sun), δ ~ 1.3%, readily measured with a modest telescope.

Key transit observables:
  Transit depth:  δ = (R_p / R_*)^2
  Transit duration:  T_14 = (P / π) * arcsin(sqrt((R_* + R_p)^2 - b^2 R_*^2) / a)
  Impact parameter: b = (a/R_*) * cos(i)    (i = orbital inclination)

Limb darkening (quadratic law):
  I(μ) / I(1) = 1 - u_1*(1-μ) - u_2*(1-μ)^2
  μ = cos(θ)  θ = angle from disc centre
  u_1, u_2 : tabulated from stellar atmosphere models (e.g. ATLAS9)

  Limb darkening rounds the flat-bottomed transit into a curved dip
  and must be accounted for to recover R_p precisely.

The simulation reproduces the four-contact transit model: first external contact (T1), first internal contact (T2), second internal contact (T3), and second external contact (T4). Between T2 and T3 the planet is fully on the stellar disc and the flux decrement is approximately constant at depth δ; the ingress (T1–T2) and egress (T3–T4) slopes encode the planet-to-star radius ratio and the transit speed. Limb darkening — the fact that the stellar limb is cooler and dimmer than the centre — rounds the flat bottom into a curved bowl, an effect especially pronounced in blue optical bandpasses.

What transits reveal beyond radius

A single transit light curve gives you the planetary radius (from depth), the orbital inclination (from duration), and the limb-darkening profile (from the in-transit curvature). Combine with radial velocity measurements and you get the planet's mass and hence its bulk density — the key to distinguishing rocky super-Earths from water worlds from mini-Neptunes. Transmission spectroscopy, measuring how transit depth varies with wavelength, probes the planetary atmosphere: molecular absorption features of water, carbon dioxide, methane, or sodium inflate the effective radius at specific wavelengths and imprint themselves on the wavelength-dependent depth. It is the primary tool for characterising exoplanet atmospheres with JWST.

Try All Eight Simulations

Closing Thought

Fluid dynamics and astrophysics share more common ground than is often acknowledged. The same Navier–Stokes equations that govern a smoke ring also describe convection in the solar interior. The same Rayleigh–Taylor instability that breaks up a rising bubble also seeds structure in a supernova remnant. The same viscoelastic rheology that governs polymer flow also controls how Earth's mantle responds to glacial loading over millennia. Wave 110 places these eight simulations side by side so you can see those threads of connection and pull on them yourself — no installation, no dependencies, just physics running in your browser.