A Bingham fluid flows of viscosity, , and yield stress, 0, through a horizontal pipe of radius, , and length, . The flow is driven by a pressure drop, /, along the length of the pipe.

a. Derive an expression for the velocity profile in the pipe. There is a critical radius, c, within which the fluid moves as a slug, so the velocity profile will be composed of two parts. Within c, the velocity will be a constant, and outside of c, the profile will depend upon .

b. Determine the volumetric flow rate for flow through the pipe using the velocity profile from part (a).

c. If 0= 1000 /^2, = 0.002 , = 1 , and = 0.1 . /^2, use the expression from part (b) to determine the pressure drop required to deliver a volumetric flow rate of 2 (10^-7)^2/.