Question

If we define a stream function

\[\Psi(x,y) = \ln\sqrt{x^{2} + y^2}\]

What are the corresponding velocity components of

V {vector} = u * i + v * j

Answer 1

If our stream function is defined in terms of psi, then the u will equal the partial derivative of psi with respect to y and v will equal the partial derivative of psi with respect to x and it's negative.

\[\Psi _{y}=(1/\sqrt{x^2+y^2})(1/(2\sqrt{x^2+y^2}))(2y)\]
\[\Psi _{y}=y/\sqrt{x^2+y^2}\]

The partial of x is identical except the numerator is an x.
\[-\Psi _{x}=-x/\sqrt{x^2+y^2}\]

v (vector) = \[iy/\sqrt{x^2+y^2}-jx \sqrt{x^2+y^2}\]