Incompressible Flow: Questions and Answers

[mathmari]

Hey!

In my notes there is the following example:

One dimensional incompressible flow in a channel of constant section.

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$$\overline{u}=(u(x, t), 0) \ \ \ \ \ p(x, y, t)=p(x)$$

incompressible flow: $\partial_x u=0$

Euler equations: $\rho_0 \partial_t u=-\partial_x p$

$$\Rightarrow \partial_x^2 p=0 \Rightarrow p(x)=p_1-\left (\frac{p_1-p_2}{L}\right )x$$

$$\partial_t u=-\frac{1}{\rho_0}\partial_x p \Rightarrow u=u(t)=\frac{p_1-p_2}{\rho_0 L}t+\text{ constant }$$

That the flow is incompressible is equivalent to the following:

1. $div \overrightarrow{u}=0$ everywhere
   
2. $\frac{D\rho }{Dt}=0$

Is this the same as $\partial_x u=0$ ?? (Wondering)

The Euler equations for imcompressible fluids are the following:

1. $\rho \frac{\partial}{\partial{t}}\overrightarrow{u}+\rho \overrightarrow{u} \cdot \nabla \overrightarrow{u}+\nabla p=0$
   
2. $\frac{\partial}{\partial{t}}\rho+\overrightarrow{u}\cdot \nabla \rho=0$
   
3. $div \overrightarrow{u}=0$

Are these equations the same as $\rho_0 \partial_t u=-\partial_x p$ ?? (Wondering)

 

[jvicens]

Hello! Yes, the equations you have mentioned are all related to incompressible flow in a channel of constant section. The equation $\partial_x u=0$ represents the fact that the flow is incompressible, meaning the velocity does not change in the x direction. This is also equivalent to the condition $div \overrightarrow{u}=0$, which states that the divergence of the velocity field is equal to zero everywhere.

The Euler equations for incompressible fluids are indeed the same as the equation $\rho_0 \partial_t u=-\partial_x p$. This equation is known as the momentum equation and it represents the balance between pressure and acceleration. The other two equations you have mentioned are the continuity equation, which states that the density does not change over time, and the condition $div \overrightarrow{u}=0$ which we have already discussed.

I hope this helps clarify your doubts. Let me know if you have any other questions. Keep up the good work!