# 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.

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.