A Bingham fluid of viscosity, \(\mu\), and yield stress, \(\tau_{0}\), flows through a horizontal pipe of radius, \(R\), and length, \(L\). The flow is driven by a pressure drop, \(\Delta P / L\) along the length of the pipe.

a. Derive an expression for the velocity profile in the pipe. There is a critical radius, \(r_{c}\), within which the fluid moves as a slug, so the velocity profile will be composed of two parts. Within \(r_{c}\), the velocity will be a constant and outside of \(r_{c}\), 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 \(\tau_{0}=1000 \mathrm{~N} / \mathrm{m}^{2}, R=0.002 \mathrm{~m}, L=1 \mathrm{~m}\), and \(\mu=0.1 \mathrm{Ns} / \mathrm{m}^{2}\), use the expression from part

(b) to determine the pressure drop required to deliver a volumetric flow rate of \(2 \times 10^{-7} \mathrm{~m}^{3} / \mathrm{s}\).