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.
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