Fully developed flow of a Bingham plastic fluid moving through a 12-in diameter pipe has the given velocity profile. The flow of a Bingham fluid does not shear the center core, producing plug flow in the region around the centerline

Find the total volume flow rate Q using the relationship Q = ∫^r2 _r1 2π r vdr + vc A_c, where r is the radial axis of the pipe. R is the radius of the pipe, υ is the velocity, υ_c is the velocity at the core, and A_c is the cross-sectional area of the plug. Solve the problem using two different approaches.

(a) Fit a polynomial curve to the noncore data and integrate.

(b) Use multiple-application Simpson’s 1/3 rule to integrate.

(c) Find the percent error using the integral of the polynomial fit as the more correct value.

Radius, r, in 1 3. Velocity, v, ft/s 5.00 5.00 4.62 4.01 3.42 1.69 0.00