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.