Question

A flow field is characterized by the stream function ψ= 3x2y−y3. Demonstrate thatthe flow field represents a two-dimensional incompressible flow. Show that the magnitude ofthe velocity at a point depends only on the distance from the origin of the coordinates.
Plot the strim ψ = 2.

Answer 1

Answer:

δu/δx+δu/δy = 6x-6x =0

9r^2

Explanation:

The flow is obviously two-dimensional, since the stream function depends only on the x and y coordinate. We can find the x and y velocity components by using the following relations:  

u =δψ/δy = 3x^2-3y^2

v =-δψ/δx = -6xy

Now, since:  

δu/δx+δu/δy = 6x-6x =0

we conclude that this flow satisfies the continuity equation for a 2D incompressible flow. Therefore, the flow is indeed a two-dimensional incompressible one.  

The magnitude of velocity is given by:  

|V| = u^2+v^2

    =(3x^2-3y^2)^2+(-6xy)^2

    =9x^4+18x^2y^2+9y^2

    =(3x^2+3y^2)^2

    =9r^2

where r is the distance from the origin of the coordinates, and we have used that r^2 = x^2 + y^2.  

The streamline ψ = 2 is given by the following equation:  

3x^2y — y^3 = 2,

which is most easily plotted by solving it for x:  

x =±√2-y^3/y

Plot of the streamline is given in the graph below.  

Explanation for the plot: the two x(y) functions (with minus and plus signs) given in the equation above were plotted as functions of y, after which the graph was rotated to obtain a standard coordinate diagram. The "+" and "-" parts are given in different colors, but keep in mind that these are actually "parts" of the same streamline.