Debris Flow Rheology: Bingham Plug Flow & Levee Formation

Rheology Controls

Slope angle θ 18°

Yield stress τ_y 1200 Pa

Plastic viscosity μ 120 Pa·s

Flow depth h 1.4 m

Flow State

Flowing — plug core over a shearing basal layer.

Plug (surface) velocity

— m/s

Plug thickness δ = τ_y/(ρg sinθ)

— m (—%)

Bingham number Bi = δ/h

—

How it works

Debris flows are not water — they are dense mixtures of mud, sand, gravel and boulders that behave as a Bingham plastic: the material stays rigid until the local shear stress exceeds a yield stress τ_y, and only then does it shear like a viscous fluid.

τ = τ_y + μ (du/dz)      if τ > τ_y  (shearing)
du/dz = 0                if τ ≤ τ_y  (rigid)

Steady flow, depth h, slope θ, bed at z=0:
τ(z)  = ρ g sinθ (h − z)
δ     = τ_y / (ρ g sinθ)        ← plug thickness
z_p   = max(0, h − δ)           ← plug starts here

Shear zone (0 ≤ z ≤ z_p):
u(z) = (ρg sinθ/μ)(hz − z²/2) − (τ_y/μ) z

Plug (z_p ≤ z ≤ h):  u(z) = u(z_p)  — moves as one rigid block

The dimensionless Bingham number Bi = δ/h = τ_y / (ρg sinθ h) compares the yield stress to the gravitational driving stress at the bed. Bi → 0 means the flow behaves almost like a viscous fluid (thin plug, strong shear everywhere); Bi ≥ 1 means the driving stress never exceeds the yield stress and the whole mass locks up and stops.

Slope θ and flow depth h set the driving stress ρg sinθ·h at the bed.
Yield stress τ_y controls how thick the rigid plug is — real debris flows have τ_y from a few hundred to several thousand Pa depending on clay/water content.
Plastic viscosity μ controls how fast the basal shear layer accelerates once it yields.
Across the channel width the drag from the side walls lowers the local driving stress, so the margins fall below τ_y and stall — this is exactly why real debris flows build self-formed levees of coarse, stationary material flanking a fast central channel (Johnson, 1970; Iverson, 1997).

Rigid plug core Fast shear layer Stalled margin / levee

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