Dew on a Spiderweb

What is modelled

An orb web hangs between two plants on a clear night. The air cools toward its dew point; once it is saturated, water vapour condenses on the silk. Each tiny drop is cooled below the air by radiating heat to the cold sky, so it grows; the water film on the thread then breaks into regularly spaced beads (the Plateau-Rayleigh instability) and neighbouring beads coalesce into bigger ones. In the morning, warming and sunlight reverse the process.

Equations

• Air and humidity: the temperature follows Ta(t) = T0 + (trend) t with the vapour content fixed until saturation (Magnus formula es = 6.112 exp(17.62 T/(243.12 + T)) hPa); after that RH = 100 % and the dew point follows the air.
• Radiative cooling: Q = ε σ Ta4 (1 − εsky) − as S with the clear-sky emissivity εsky = 0.787 + 0.764 ln(Td/273.15), increased by clouds as (1 + 0.22 c2.75).
• Drop energy balance: conduction from the air ka ΔT / r plus latent heat L ρw dr/dt equals the radiative loss, so ΔT = (Q r − L ρw r dr/dt) / ka.
• Diffusion-limited growth with the Kelvin effect: r dr/dt = A [ (RH − 1) + b r − κ/r ] / (1 + Φ), A = D ρvs/ρw, b = β Q / ka, β = L / (Rv T2), κ = 2γ/(ρw Rv T). Drops smaller than rc ≈ √(κ/b) evaporate, bigger ones grow roughly linearly in time.
• Beads on silk: a film of outer radius R on a thread of radius a is unstable and breaks into beads every λ = 9.02 R (inviscid Rayleigh, maximising kR I1(kR)/I0(kR) (1 − (kR)2)). Film volume per length V' = π(R2 − a2), bead volume V' λ, so a bead has r ≈ 1.9 R and condensation feeds dV'/dt = 4π r (r dr/dt) / λ.

Checks run in the page

• Dew point from Magnus against an independent Antoine-equation bisection.
• Drop growth law integrated numerically against its closed form.
• Plateau-Rayleigh fastest wavelength found numerically (about 9.02 R).
• Kelvin supersaturation of a 1 µm drop and the critical radius against its closed form.
• Water budget on the thread (RK4 against a fine Euler integration) and the clear-sky radiative loss range.

Honest simplifications

• Every thread is assumed to carry wetted nuclei of about 3 µm (hygroscopic dust, spindle knots of the silk); the model starts there and does not simulate nucleation.
• The drops are treated as isolated spheres in still air: no wind, no fog droplets landing on the web, and no vapour competition between neighbouring drops. Silk wetting and the real contact angle are not modelled; beads are drawn as spheres.
• The Plateau-Rayleigh wavelength is the inviscid value; viscosity and the silk's hygroscopic knots change it in real webs. Coalescence is drawn as pairs merging when the spacing doubles.
• The web is drawn with a thread about 200 times thicker than a real 2.5 µm silk strand and the beads are magnified (slider) so they are visible; the true-scale drawing in the third graph shows the real proportions. At most a few thousand beads are drawn; the readout gives the true count.
• The sky emissivity is an empirical clear-sky formula, the sun heats the drop through an assumed 15 % absorptivity, and latent heat is included only through the quasi-steady drop energy balance.

• Run the default night: nothing happens until the air reaches its dew point, then the bead radius grows about linearly with time.
• Choose Cloudy night: clouds return heat to the web, the drops stay close to air temperature and growth slows.
• Choose Sunrise: warming and sunlight push the drops below the Kelvin critical radius and the beads shrink again.
• Tick the close-up box to see the individual beads along the capture spiral.