Consider a two-dimensional flow $\vec u = \left\langle {u,v} \right\rangle$ of a Newtonian fluid with constant pressure and dynamic viscosity in a static container V with rigid walls S. Define the kinetic energy of the flow as $k = \frac{1}{2}\vec u \cdot \vec u$. Show that $$\vec u \cdot {\nabla ^2}\vec u = {\nabla ^2}k - \left( {\frac{{{\partial ^2}u}}{{\partial {x^2}}} + \frac{{{\partial ^2}u}}{{\partial {y^2}}} + \frac{{{\partial ^2}v}}{{\partial {x^2}}} + \frac{{{\partial ^2}v}}{{\partial {y^2}}}} \right).$$
I tried various identities, but I don't seem to get the desired result. Can anyone help?
