Example 2

Consider the the other potential we saw:
u=x^2 y-1/3 y^3.
Check for yourself if it satisfies Laplace's equation.

In[46]:=

  Clear[u,v]
  u[x_,y_]:=x^2 y-1/3 y^3

The flow has velocity v=grad u.

In[47]:=

  v[x_,y_]:=grad2[u][x,y]
  v[x,y]

Out[47]=

           2    2
  {2 x y, x  - y }

The flow is irrotational because it comes from a potential, that is, curl v=curl(grad u)=0. This is a freebie.

In[48]:=

  curl2[v][x,y]

Out[48]=

We had the streamlines from above.

In[49]:=

  Clear[x,y]
  stream=ContourPlot[1/3 x^3-y^2,{x,-5,5},{y,-5,5},ContourShading->False,
  AspectRatio->Automatic];

Are there sources or sinks?

In[50]:=

  div2[v][x,y]

Out[50]=