Viscoplastic Flow — Bingham Fluid

Bingham fluids (mud, toothpaste, lava) require a yield stress τ₀ before flowing. The velocity profile shows a plug flow core (rigid zone) surrounded by shear regions.

Pressure gradient (dP/dx)8.0
Yield stress τ₀3.0
Viscosity η (plastic)1.5
Bingham model:
τ > τ₀: γ̇ = (τ−τ₀)/η
τ ≤ τ₀: γ̇ = 0 (plug)

Velocity profile:
r > R(1−τ₀/τ_w):
u = (G/4η)(R²−r²) − (τ₀/η)(R−r)
r ≤ plug: u = u_plug
Plug radius: —
Max velocity: —
Flow rate: —