Show that a uniform flow with velocity ${V_\infty }$ is a physically possible incompressible flow and is irrotational.

For a uniform flow $${V_\infty } = u = {\rm{constant}}\,\,{\rm{;}}\,\,\mathop V\limits^ \to   = {V_\infty }\mathop i\limits^ \to$$ Divergence of velocity field $$\nabla  \cdot \mathop V\limits^ \to   = \frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} + \frac{{\partial w}}{{\partial z}}$$ $$= 0 + 0 + 0$$ $$= 0$$ Therefore, it shows that it is a physically possible incompressible flow.

Also, curl of velocity field  $$\nabla  \times \mathop V\limits^ \to   = \left| {\begin{array}{*{20}{c}}{\mathop i\limits^ \to  }&{\mathop j\limits^ \to  }&{\mathop k\limits^ \to  }\\{\frac{\partial }{{\partial x}}}&{\frac{\partial }{{\partial y}}}&{\frac{\partial }{{\partial z}}}\\u&v&w\end{array}} \right| = \mathop i\limits^ \to  \left( {0 – 0} \right) – \mathop j\limits^ \to  \left( {0 – \frac{{\partial u}}{{\partial x}}} \right) + \mathop k\limits^ \to  \left( {0 – \frac{{\partial u}}{{\partial y}}} \right)$$ $$\nabla  \times \mathop V\limits^ \to   = 0$$ This shows that the flow is irrotational.