Basaltic melt viscosity follows an Arrhenius temperature law, then gets a crystal-loading correction (Einstein–Roscoe form) because a crystal-bearing melt resists shear far more than a crystal-free one:
η(T) = A · exp( Ea / (R·T) )
η_eff = η(T) · (1 − φ/φmax)^(−2.5) [φmax = 0.6]
γ̇ = γ̇0 · sin(slope) [kinematic shear rate from grade]
De = η_eff · γ̇ / G [G = apparent crust modulus]
De is a Deborah number: it compares how fast the crust is being strained to how fast it can relax the stress viscously. De < 1 — the skin has time to deform plastically as it stretches, so it folds into smooth, ropy pahoehoe. De > 1 — the strain outruns viscous relaxation, so the cooling skin fractures into rubbly a'a clinker. All three sliders push the same lever: hotter/less crystalline lava lowers η_eff (favours pahoehoe); a steeper grade raises γ̇ (favours a'a); more crystals raise η_eff sharply (favours a'a) — exactly the documented field observation that a single flow can turn from pahoehoe to a'a purely by speeding up or cooling further downslope.
- Cross-section — the flow front advancing left→right; skin texture is computed live from De, not painted on.
- De gauge — the current Deborah number against the De=1 pahoehoe/a'a threshold.
- Constants — Ea≈200 kJ/mol and φmax=0.6 follow documented basalt-melt rheology (Marsh 1981; Pinkerton & Stevenson 1992); G is an illustrative apparent modulus for a thin, still-hot crust, not a measured field constant.