Show the relation when W is constant

[mathmari]

Hey!

If the space $W$ is constant (doesn't move with the flow), show that $$\frac{d}{dt}\int_{W}\left (\frac{1}{2}\rho |\overrightarrow{u}|^2+\rho \epsilon\right )dV=-\int_{\partial{W}}\rho \left (\frac{1}{2}|\overrightarrow{u}|^2+i\right )\overrightarrow{u} \cdot \overrightarrow{n}dA$$

where $p$ the pressure, $\rho$ the density, $\epsilon=i-\frac{p}{\rho}$ the internal energy per unit mass, $i$ the enthalpy.

How can we show that?? Do we have to use the divergence theorem?? (Wondering)

 

[jvicens]

Hello! Yes, we can use the divergence theorem to show this relationship. Let's start by writing out the left side of the equation:

$$\frac{d}{dt}\int_{W}\left (\frac{1}{2}\rho |\overrightarrow{u}|^2+\rho \epsilon\right )dV$$

Using the product rule for differentiation, we can split this into two separate integrals:

$$\frac{1}{2}\frac{d}{dt}\int_{W}\rho |\overrightarrow{u}|^2 dV + \frac{d}{dt}\int_{W}\rho \epsilon dV$$

Now, we know that the density, $\rho$, is constant in this case since the space $W$ is not moving with the flow. Therefore, we can take it out of the integrals:

$$\frac{1}{2}\rho\frac{d}{dt}\int_{W}|\overrightarrow{u}|^2 dV + \rho\frac{d}{dt}\int_{W}\epsilon dV$$

Next, we can use the continuity equation, which states that $\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \overrightarrow{u}) = 0$, to simplify the first integral:

$$\frac{1}{2}\rho\frac{d}{dt}\int_{W}|\overrightarrow{u}|^2 dV = -\frac{1}{2}\rho\int_{W}\frac{\partial \rho}{\partial t} |\overrightarrow{u}|^2 dV$$

Using the divergence theorem, we can rewrite this as a surface integral over the boundary of $W$:

$$-\frac{1}{2}\rho\int_{\partial{W}}\rho |\overrightarrow{u}|^2 \overrightarrow{u} \cdot \overrightarrow{n} dA$$

Similarly, we can use the continuity equation to simplify the second integral:

$$\rho\frac{d}{dt}\int_{W}\epsilon dV = -\rho\int_{W}\frac{\partial \rho}{\partial t} \epsilon dV$$

Again, using the divergence theorem, this becomes:

$$-\rho\int_{\partial{W}}\rho \epsilon \overrightarrow{u} \cdot \overrightarrow{n} dA