The 3D version of this simulator integrates real per-bead Coulomb repulsion, spring tension and drag over an N-body chain — a direct particle simulation of the whipping jet. This 2D counterpart uses a genuinely different, independently-computed method: a closed-form force-balance ODE for how fast each jet element accelerates in the field, plus a linear-stability dispersion relation (the standard technique used in the electrospinning literature, e.g. Hohman et al. 2001) to predict the whipping wavelength and growth rate analytically, instead of letting bending emerge from thousands of pairwise particle forces.
Force balance per jet element (exact, solved analytically):
dv/dt = k_E·E − k_D·μ·v , E = V/d
⇒ v(t) = v_term + (v₀ − v_term)·e^(−k_D·μ·t), v_term = k_E·E / (k_D·μ)
Mass conservation along the jet (continuity, A·v = const):
stretch S(t) = v(t)/v₀ ⇒ fiber diameter ∝ 1/√S
Whipping (linear stability, long-wave dispersion relation):
σ(k)² = A·k² − B·k⁴, A ∝ E/R (destabilizing), B ∝ μ·R² (stabilizing)
k* = √(A/2B) (fastest-growing wavenumber)
σ* = A / (2√B) (its growth rate)
Every jet element's velocity comes straight from that first-order ODE's exact analytic solution — no per-frame numerical force integration is needed for the axial motion. Its shrinking radius follows directly from mass conservation once the velocity history is known. The lateral whip trace is drawn as a wave of wavelength 2π/k* whose amplitude grows as e^(σ*·t) along each element's own residence time, saturating smoothly as it approaches the electrode gap — the same qualitative behaviour as the 3D jet's chaotic bending, reached by an entirely different, analytic route.
- Voltage — raises the field E = V/d, which raises the terminal velocity v_term (more stretch, thinner fiber) and raises the destabilizing coefficient A (faster, shorter-wavelength whipping).
- Flow rate — sets the initial ejection speed v₀; more flow means more volume to draw down for the same stretch, so fibers land thicker.
- Gap distance — trades off two effects baked into the ODE: a longer gap lowers E (less stretching drive) but gives each element more time to approach v_term (more stretching time) — the model runs both effects through the same closed-form v(t), it does not assume one wins.
- Viscosity / concentration — raises the drag coefficient k_D·μ, which lowers v_term (thicker fiber) while also raising the stabilizing coefficient B (slower, longer-wavelength, gentler whipping) — both consequences of the same drag term, not independent knobs.
Stated plainly: the two models are not expected to output identical diameters for the same slider values — they are independent derivations (one a discrete N-body simulation, one a continuum force-balance + linear-stability model) of the same real electrospinning physics, and are numerically verified separately against their own governing equations.