Persson & Yang Leak Rate Simulator

As the zoom parameter ζ steps from 1 to its Nyquist limit, progressively finer roughness wavelengths are resolved. At each frame Persson's theory predicts the real contact area fraction A/A₀. The percolation threshold — where A/A₀ ≈ 0.4 — marks the first frame at which a connected gap path spans the seal width. Black = contact  |  White = gap.

Binary contact map  ·  ζ = 1 → Nyquist  ·  Persson & Yang (2008)  ·  click to enlarge

Filtered surface mesh (left) & binary contact map (right)  ·  click to enlarge

Critical constriction wavelength λ_c, gap thickness u_c, and leak rate Q̇ versus squeezing pressure at the percolation onset (A/A₀ = 40%), for two independent sweeps. Click any plot to enlarge.

Varying roughness σ  (H = 0.8)

Fig. 6 — Critical wavelength λ_c vs squeezing pressure. Higher roughness σ shifts the percolation onset to shorter wavelengths at a given pressure — the seal must be loaded harder to close off the dominant gap channels.

Fig. 7 — Critical gap thickness u_c vs pressure. Rougher surfaces allow a larger mean opening before the contact area drops to the percolation threshold — more roughness means more gap volume at onset.

Fig. 8 — Leak rate Q̇ vs squeezing pressure. The pressure dependence reflects the u_c³ term in the Poiseuille flow equation — a 10× change in gap thickness means 1000× change in flow rate.

Varying Hurst exponent H  (σ = 2 μm)

Fig. 9 — λ_c vs pressure, varying Hurst exponent H. H controls the fractal dimension of the roughness spectrum — higher H (smoother long-range structure) shifts the critical wavelength curve downward.

Fig. 10 — u_c vs pressure. The Hurst exponent governs how energy is distributed across length scales in the PSD — lower H means more fine-scale roughness and a larger gap at onset.

Fig. 11 — Q̇ vs pressure. Surfaces with lower H (rougher fine-scale structure) produce higher leak rates at the same macroscopic squeezing pressure — and the spread across curves widens dramatically at low pressures.

The theory was applied to a rubber seal (E = 10 MPa) squeezed against a rough substrate. Each surface roughness level n adds a fixed increment to the RMS roughness σ at the Hertz contact patch, reducing the contact area fraction and opening a percolation path.

All three parameter sweeps on a common Q̇ (m³/s) axis. Left: surface roughness. Center: squeezing force. Right: fluid pressure difference. Full seal contact length scaling applied throughout.

Scenario dashboard — seven operating conditions from baseline roughness (sealed, no percolation path) through the worst case (roughness level 3 + low squeezing force + high ΔP). Green zone: contact area above percolation threshold. Red zone: connected gap, active leakage.

Surface roughness sweep (squeezing force = 13 N, ΔP = 35 kPa). At baseline roughness (n = 0) the seal holds — one roughness level is enough to open a percolation path. Each additional level raises σ and dramatically increases Q̇.

Squeezing force sweep (ΔP = 35 kPa, roughness levels 1–3). Higher squeezing force increases Hertz contact pressure and closes the gap — a practical mitigation for rougher surfaces. Nominal load at 13 N marked.

Pressure sweep (squeezing force = 13 N, roughness levels 1–3). Leak rate scales linearly with ΔP (Poiseuille regime). Nominal 35 kPa marked — note how high roughness at high pressure approaches the dashboard's worst-case scenarios.

Adjust squeezing force, fluid pressure difference, and surface roughness — the Persson solver runs in real time and updates Q̇ in m³/s and mL/min. Built with Python + Streamlit, served directly from this server.

Full tabulated output for all three sweeps and the scenario dashboard, in SI units.

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