Lava River Channel Lab

A viscous river on a slope

Lava is so viscous that the flow is laminar. For a layer of thickness h on a slope α, gravity along the slope is balanced by viscous shear, giving the Jeffreys profile u(z) = ρ g sinα (h z − z²/2) / μ. The mean speed is U = ρ g sinα h² / (3 μ) and the surface moves at 1.5 U. The Reynolds number Re = ρ U h / μ is of order one or less, so inertia plays almost no role.

Cooling stiffens the flow

The lava loses heat to the air by radiation and convection, modelled as Newton cooling with coefficient heff: dT/dx = −heff (T − Ta) / (ρ cp h U). Viscosity grows as it cools, μ(T) = μ0 exp[β (T0 − T)], so the speed U falls, the lava spends longer losing heat per metre and cools even faster: a feedback that makes the flow stall. With β = 0 the speed is constant and the runout has the closed form L = (h U / k) ln[(T0 − Ta)/(Tstop − Ta)]. The crust thickness is the diffusion length δ = √(2 κ t) for the travel time t.

Honest simplifications

The thickness is held constant (a real steady flow thickens as it slows), the channel is wide so side walls are ignored, the lava is Newtonian with no yield strength or crystals, the heat loss is a single linearised coefficient, and the crust is shown but does not insulate the flow. Real flows are far less regular.

Checks in the page

Check 1 compares the marched runout with the closed form for β = 0. Check 2 integrates the velocity profile numerically against the mean-speed formula. Check 3 compares a coarse and a fine step of the cooling march.

• Load Steep slope: the lava is faster, loses less heat per metre and runs much farther.
• Load No stiffening (beta=0) and compare the runout with the default: the feedback of viscosity on cooling shortens the flow by many times.
• Raise the heat loss coefficient with Windy and cold and watch the crust and the stall point move up the slope.
• Read graph 4: thicker flows (red line) keep their heat and travel farther at every slope.