# Two-dimensional flow [closed]

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?

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Comment to OP's question(v1): The dimensions in the equation seems wrong. – Qmechanic Oct 7 '12 at 23:14
What question? What? – glebovg Oct 7 '12 at 23:16
I'd start by applying the product rule to $\nabla^2 u^2$ and rearranging. @Qmechanic is right though, the second term on the right hand side is missing a factor with dimensions of velocity – poorsod Oct 7 '12 at 23:19
@poorsod Yes, I tried that, it did not seem to get me anywhere. – glebovg Oct 7 '12 at 23:23
You should mention in the question what you've tried already. My advice in this case is to do the same calculation again, extra carefully. I haven't done the entire derivation, but I'm pretty sure that's what they want. – poorsod Oct 7 '12 at 23:33
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## closed as too localized by David Zaslavsky♦Oct 8 '12 at 3:50

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