This is a 2D cross-section (the real domain of the problem — a wing leading edge is, to first order, a 2D flow past a cylinder of radius R). Air follows the closed-form potential-flow field around a circle of unit radius:
u_x = 1 − (x²−y²)/(x²+y²)²
u_y = −2xy/(x²+y²)² (lengths in units of R, speed in units of V∞)
Droplets don't follow the air exactly — inertia lets them cross curving streamlines. Each droplet relaxes toward the local air velocity over a Stokes relaxation time τ = ρ_water·d²/(18·μ_air), giving the Stokes (inertia) number St = τ·V∞/R.
Collection efficiency β(φ) is computed here by the same limiting-trajectory method real trajectory codes (LEWICE, DROPLET) use: thousands of droplets are seeded at evenly-spaced offsets y₀ far upstream and integrated to impact. Mass conservation along a streamtube means the far-field spacing dy₀ that lands in a surface arc ds must satisfy LWC·V·dy₀ = LWC·V·β·ds, so β(φ) = dy₀/ds is recovered directly by counting how many seeded droplets land in each angular bin — no closed-form fit, no empirical curve.
β(φ) = (# droplets landing in bin)·Δy₀ / (R·Δφ)
The local water catch rate is ṁ(φ) = LWC·V∞·β(φ). Whether that catch freezes on contact (rime) or partly runs back as a film before freezing (glaze) is set by a real Messinger surface energy balance at the 0 °C ice/water interface: the latent heat released by the frozen fraction n must supply both the heat lost to forced convection and the heat needed to warm the supercooled droplets up to 0 °C:
n(φ)·ṁ(φ)·L_f = h·(−T_amb) + ṁ(φ)·c_water·(−T_amb)
h = Nu·k_air / D, Nu from the Churchill–Bernstein correlation for a cylinder in cross-flow
Rime ice (n → 1) is opaque, rounded and hugs the wetted band. Glaze ice (n → 0) is glassy, and unfrozen runback water is deposited as extra thickness just beyond the wetted edge — producing the double "horn" shape. Because n is evaluated per angular bin (using the local ṁ(φ), not a single global number), the rime/glaze transition is visibly sharper near the stagnation point than at the shoulders — exactly as in real icing.