Time Derivative of Kinetic Energy in Fluid Dynamics I'm trying to work out the value of $dT/dt$ where $T$ is the total kinetic energy of the flow and $t$ is time, for a fluid motion with velocity $\underline{u}$, pressure $p$ and constant density $\rho$ inside a closed region with a fixed solid boundary, where $\underline{u}$ satisfies the Euler Equations;
$$ u_t + (u \cdot \nabla )u = \frac{-1}{\rho} \nabla p $$ $$\nabla \cdot u = 0 $$
I've started the problem but am now stuck, I'm told I need to use two vector identities and the divergence theorem. Here are my partial workings, 
$$T = \frac{1}{2}\rho\iiint_Vu^2dV$$
$$\implies \frac{dT}{dt} = \frac{d}{dt} (\frac{1}{2}\rho\iiint_Vu^2dV) $$
$$  =  \frac{1}{2}\rho\iiint_V \frac{d}{dt}(u^2)dV $$
$$  =  \frac{1}{2}\rho\iiint_V 2u \cdot u_t dV $$
$$  =  \rho\iiint_V u \cdot u_t dV $$
Any pointers would be appreciated, thank you.
 A: You did alright, but you also need the other terms. Let me present the most usual derivation: multiplying the momentum conservation equation by $\mathbf{u}$,
$$\mathbf{u}_t\cdot\mathbf{u} + \mathbf{u} \cdot \nabla \cdot (\mathbf{u} \mathbf{u}) = -\frac{1}{\rho} \mathbf{u} \cdot \nabla p.$$
Using $\nabla \cdot \mathbf{u}=0$ (from the continuity equation) and $T=\rho \mathbf{u}\cdot\mathbf{u}/2=\rho \| \mathbf{u} \|^2/2$,
$$T_t +  \nabla \cdot (\mathbf{u} T) = -\frac{1}{2} \nabla \cdot (\mathbf{u} p).$$
The identity $\nabla \cdot(\mathbf{u} p) = \mathbf{u} \cdot \nabla p + p \nabla \cdot \mathbf{u}$ is probably the one you mentioned.
Integrating this equation along the volume $V$,
$$\int_V T_t\ dV =-  \int_V \nabla \cdot (\mathbf{u} T) dV  -\frac{1}{2} \int_V\nabla \cdot (\mathbf{u} p) dV,$$
and using the divergence theorem,
$$\int_V T_t\ dV = - \oint_{\partial V} \mathbf{u} T \cdot \mathbf{n}\ dS  -\frac{1}{2} \oint_{\partial V}\mathbf{u} p \cdot \mathbf{n} \ dS.$$
Since $V$ is a fixed volume,
$$\frac{d}{dt} \int_V T\ dV = - \oint_{\partial V} \mathbf{u} T \cdot \mathbf{n}\ dS -\frac{1}{2} \oint_{\partial V}\mathbf{u} p \cdot \mathbf{n} \ dS,$$
which is the kinectic energy conservation equation in integral form.
