Fluid Mechanics seventh edition by Frank M. White · Problem P3.59

Integral Relations for a Control Volume — Frank M. White

Physics — Fluid Mechanics · linear-momentum-equation

Problem

When a pipe flow suddenly expands from

A_1

A_2

, as in Fig. P3.59, low-speed, low-friction eddies appear in the corners and the flow gradually expands to

A_2

downstream. Using the suggested control volume for incompressible steady flow and assuming that

p \approx p_1

on the corner annular ring as shown, show that the downstream pressure is given by

p_2 = p_1 + \rho V_1^2 \frac{A_1}{A_2}\left(1 - \frac{A_1}{A_2}\right)

Neglect wall friction.