Fluid Mechanics seventh edition by Frank M. White · Problem 3.4

Integral Relations for a Control Volume — Frank M. White

Physics — Fluid Mechanics · average-velocity-mass-conservation

Problem

For steady viscous flow through a circular tube (Fig. E3.4), the axial velocity profile is given approximately by

u = U_0\left(1 - \frac{r}{R}\right)^m

so that

u

varies from zero at the wall (

r = R

), or no slip, up to a maximum

u = U_0

at the centerline

r = 0

. For highly viscous (laminar) flow

m \approx \frac{1}{2}

, while for less viscous (turbulent) flow

m \approx \frac{1}{7}

. Compute the average velocity if the density is constant.