Hydraulic Conduit Laminar Volumetric Flow Rate (Civil & Environmental Engineering)

A civil and environmental water resources engineer at the UNLV Howard R. Hughes College of Engineering is designing a pressurized subsurface drainage culvert to convey stormwater and treated municipal effluent under Tropicana Avenue in Las Vegas.

The culvert is constructed as a circular reinforced concrete conduit of internal radius $R = 0.5\text{ meters}$. Due to viscous fluid friction and boundary layer adhesion along the conduit wall, the stormwater exhibits a fully developed axisymmetric parabolic velocity profile, modeled in Cartesian coordinates as:

To verify hydraulic discharge capacity and prevent upstream street flooding during heavy desert flash-flood events, the engineer must determine the total volumetric flow rate $Q = \iint_D v(x, y) \, dA$.

1. Identify the circular boundary of the pipe cross-section and convert the domain $D$ into polar coordinate limits for radial distance $r$ and azimuth angle $\theta$.
2. Transform the Cartesian velocity profile $v(x, y)$ into polar form $v(r, \theta)$, and express the volumetric flow rate integral using the polar area element $dA = r \, dr \, d\theta$.
3. Evaluate the double integral to determine the exact volumetric discharge $Q$ in cubic meters per second ($\text{m}^3/\text{s}$).
4. Explain why transforming the double integral into polar coordinates avoids complicated radical integration and provides the governing discharge equation used in civil hydraulic infrastructure design.