Exact Solutions to the Navier-Stokes Equation

Elliptical Pipes

Consider an elliptical cross-section pipe shown in Figure 7 with its boundary given as

(62)

We assume that the velocity field is given by

(63)

Figure 7. Elliptical cross-section pipe subject to a constant pressure gradient.

(64)

Hence

(65)

The flow rate is given as

(66)

After integration, it follows that

(67)