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.