A Bingham fluid of viscosity, μ, and yield stress, τ0, flows through a horizontal pipe of radius, R, and length, L. The flow is driven by a pressure drop, ΔP/L along the length of the pipe.
a. Derive an expression for the velocity profile in the pipe. There is a critical radius, rc, within which the fluid moves as a slug, so the velocity profile will be composed of two parts. Within rc, the velocity will be a constant and outside of rc, the profile will depend upon r.
b. Determine the volumetric flow rate for flow through the pipe using the velocity profile from part (a).
c. If τ0 = 1000 N/m2, R = 0.002 m, L = 1 m, and μ = 0.1 Ns/m2, use the expression from part (b) to determine the pressure drop required to deliver a volumetric flow rate of 2 × 10−7m3/s.