Incompressible Viscous Flows

Two-Dimensional Plane Flows

For two dimensional flows in xy-plane as shown in Figure 1, let

Figure 1. Schematics of plane flows in a Cartesian coordinate system.

That is

and Equation (1) is satisfied.

The nonzero element of

Equation (7) in two-dimensional case reduces to

Using (10), Equation (11) may be restated as

or

Equation (13) is the equivalent to the Navier-Stokes equation and contains a single unknown .