Stream function $\left( {{\rm{\psi  = constant}}} \right)$ gives the equation of a streamline and the flow velocity is obtained by differentiating ${\rm{\psi }}$.For a compressible flow $$\rho u = \frac{{\partial \psi }}{{\partial y}}$$ $$\rho v =  – \frac{{\partial \psi }}{{\partial x}}$$ For incompressible flow, $$u = \frac{{\partial \psi }}{{\partial y}}$$ $$v =  – \frac{{\partial \psi }}{{\partial x}}$$ Velocity potential: Velocity potential is defined by $\phi ,\phi  = \phi \left( {x,y,z} \right)$.For an irrotational flow,velocity is given by gradient of $\phi$. $$u = \frac{{\partial \phi }}{{\partial x}},v = \frac{{\partial \phi }}{{\partial y}},w = \frac{{\partial \phi }}{{\partial z}}$$ Stream function is defined for both rotational and irrotational flows,velocity potential is defined for irrotational flows only.However,stream function is defined for two-dimensional flows only,the velocity potential applies to three-dimensional flows also.Since irrotational flow can be described by velocity potential $\phi$,such flow are also called potential flow.