The application of the continuity equation (Equation 3.47) assumes one-dimensional flow, where the velocity profile is a rectangular distribution. Therefore, the velocity, v is the mean or average...

The application of the continuity equation (Equation 3.47) assumes one-dimensional flow, where the velocity profile is a rectangular distribution. Therefore, the velocity, v is the mean or average velocity, vave = Q/A   = vmax/2   (mean over the pipe diameter, D or channel depth, y, and mean over the pipe or channel differential distance, ds at a given cross section in the flow). For the case of real laminar pipe flow for which the velocity profile is a parabolic distribution, prove that vave = Q/A   = vmax/2  .

Equation 3.47