Problem Diagram

The given steady two-dimensional velocity profile is

where u = 3(x² - y²) and v = -6xy.

Since and, thus. The continuity equation is satisfied, and it can be concluded that the flow is incompressible.

To this end, the flow field is known to be two-dimensional and incompressible, hence a stream function can be introduced.

According to the definition of the stream function,

The two equations above can be integrated to yield,

     ψ = 3x²y - y³ + f₁(x)
     ψ = 3x²y + f₂(y)

In order for the stream function, ψ, to satisfy the two equations simultaneously, it must have the form

     ψ = 3x²y - y³ + C

where C is an arbitrary constant. For simplicity, set C = 0 (recall, the value of an individual stream function is irrelevant) to yield,

     ψ = 3x²y - y³

Streamlines can now be drawn by setting the stream function to a constant. For instance, the plots of ψ = 0, 1 and 2 are shown in the figure. Note that ψ = 0 in this case corresponds to the wall.