This is not a flattened view of the 3D bead-chain jet. It's an independent
linear-stability field model: instead of tracking discrete charged beads in 3D space,
it evolves a scalar lateral-displacement amplitude field η(s,t) directly on a 1D
spatial grid running from the nozzle (s=0) to the collector (s=1), drawn as a scrolling
spacetime "waterfall" (horizontal = position along the jet, vertical = time).
drag·∂η/∂t = −A·∂²η/∂s² − B·∂⁴η/∂s⁴ − γη³ − v·∂η/∂s + noise(s)
A = k_A·(charge)²·field − k_T·stiffness (destabilising Coulomb term)
B = k_B·stiffness (stabilising bending/elastic term)
Fourier analysis of this PDE gives a closed-form dispersion relation
σ(k) = (A·k² − B·k⁴)/drag — the same competition between destabilising
self-repulsion and stabilising elasticity that drives the 3D whipping instability,
but expressed as a growth-rate spectrum rather than individual particle
trajectories. It predicts a single fastest-growing wavenumber
k* = √(A/2B) with peak growth rate σ_max = A²/(4B·drag) — plotted live on the right
as the σ(k) curve, and used to derive the dominant whip wavelength shown in the
readouts. When A ≤ 0 (stiff, low-charge, low-field solutions) every mode decays:
no instability at all, matching the real onset threshold.
- Electric field / charge density — raise A, feeding the instability faster and shifting k* to shorter wavelengths.
- Elastic modulus — raises both B (bending resistance) and the tension term subtracted from A, suppressing growth.
- Flow rate — sets the downstream advection speed of the field and the baseline (unstretched) jet radius used for the fiber-diameter estimate.
The instantaneous jet-profile trace (top right) is η(s) at the current instant — a
genuine spatial snapshot of the whip envelope, not a projected 3D curve. The cubic
−γη³ term is a standard Landau/Ginzburg saturation nonlinearity that caps the linear
growth once the whip becomes finite-amplitude, just as real elastic and geometric
nonlinearities cap the 3D bead chain's whip radius.