I was wondering if anyone could help me show that $\nabla \phi . \nabla \psi=0$ Where we let $\phi$ be the velocity potential and let $\psi$ be the stream function for a 2-dimensional,irrational flow of in compressible fluid.
my attempt was to express $\nabla \phi$ and $\nabla \psi$ in terms of partial derivatives of $\psi$ and $\phi$, i.e $$\nabla \psi=\frac{\partial \psi_{x}}{\partial x}i+\frac{\partial \psi_{y}}{\partial y}j$$ and $$\nabla \phi=\frac{\partial \phi_{x}}{\partial x}i+\frac{\partial \phi_{y}}{\partial y}$$
then Calculating $\nabla \psi\cdot\nabla \phi$,
$\nabla \psi\cdot\nabla \phi =$ is where i get stuck.
And could anyone give me a physical argument that explains this result as i cannot seem to visualise it
For velocity field $\mathbf{u} = (u,v,0)$, we have that \begin{alignat*}{2} u & = \frac{\partial\phi}{\partial x} && = \frac{\partial\psi}{\partial y} \\ v & = \frac{\partial\phi}{\partial y} && = -\frac{\partial\psi}{\partial x} \end{alignat*} Hence, $$ \nabla\phi\cdot\nabla\psi = \frac{\partial\phi}{\partial x}\frac{\partial\psi}{\partial x} + \frac{\partial\phi}{\partial y}\frac{\partial\psi}{\partial y} = -uv + uv = 0. $$