For an incompressible flow \(\rho  = {\rm{constant}}\). \(\nabla  \cdot V\) is physically the time rate of change of volume of a moving fluid element per unit volume. For an incompressible flow, volume of a fluid element is constant, therefore \[\nabla  \cdot V = 0\]This can be also shown from continuity equation. Continuity equation is given as\[\frac{{\partial \rho }}{{\partial t}} + \nabla  \cdot \rho V = 0\]Since, for an incompressible flow, \(\rho  = {\rm{constant}}\) we have \[\frac{{\partial \rho }}{{\partial t}} = 0\]therefore the continuity equation becomes \[0 + \nabla  \cdot \rho V = 0\]\[\nabla  \cdot V = \frac{0}{\rho } = 0\]For an irrotational flow, velocity potential \(\phi \) is defined as \[V = \nabla \phi \]Therefore, a flow which is both incompressible and irrotational the equation can be written as \[\nabla .\left( {\nabla \phi } \right) = 0\]\[{\nabla ^2}\phi  = 0\]This equation \({\nabla ^2}\phi  = 0\) is called the Laplace’s equation.In Cartesian coordinates \[\phi  = \phi \left( {x,y,z} \right)\] therefore\[{\nabla ^2}\phi  = \frac{{{\partial ^2}\phi }}{{\partial {x^2}}} + \frac{{{\partial ^2}\phi }}{{\partial {y^2}}} + \frac{{{\partial ^2}\phi }}{{\partial {z^2}}} = 0\]Also, \[u = \frac{{\partial \psi }}{{\partial y}},v =  – \frac{{\partial \psi }}{{\partial x}}\]For an incompressible and irrotational flow \[\frac{{\partial v}}{{\partial x}} – \frac{{\partial u}}{{\partial y}} = 0\]\[\frac{\partial }{{\partial x}}\left( { – \frac{{\partial \psi }}{{\partial x}}} \right) – \frac{\partial }{{\partial y}}\left( {\frac{{\partial \psi }}{{\partial y}}} \right) = 0\]\[ \Rightarrow \frac{{{\partial ^2}\psi }}{{\partial {x^2}}} + \frac{{{\partial ^2}\psi }}{{\partial {y^2}}} = 0\]Therefore, velocity potential as well as stream functions both satisfies Laplace’s equation, for an incompressible and irrotational flow.