Solving Navier-Stokes for Pressure in 1D

[stanley.st]

Hello, I have Navier stokes in 1D

$$\rho\left(\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}\right)=-\frac{\partial p}{\partial x}+\mu\frac{\partial^2u}{\partial x^2}$$

Condition of imcompressibility gives

$$\frac{\partial u}{\partial x}=0$$

So I have Navier stokes

$$\rho\frac{\partial u}{\partial t}=-\frac{\partial p}{\partial x}+\mu\frac{\partial^2u}{\partial x^2}$$

How to find pressure p(x,t)?

 

[GunnaSix]

To start, if $\frac{\partial u}{\partial x} = 0$, then what is $\frac{\partial^2 u}{\partial x^2} = \frac{\partial}{\partial x} \left(\frac{\partial u}{\partial x}\right)$?

You still need some information about $\frac{\partial u}{\partial t}$ to solve for $p(x, t)$, however. What is the problem you're trying to solve?

 

[stanley.st]

Thank you so much. I have no specific problem to solve. I wanted to find general solution of NS in 1D. What is an example of such information?

 

[bigfooted]

if you also neglect the time-dependency, the problem reduces to that of Couette flow. The equation can be integrated twice to get a solution in terms of the unknown pressure gradient.

Also note, as hinted before, that you cannot use the incompressibility assumption like that because you now imply that the velocity is a constant (but time-varying). First use the incompressibility equation on the 2D or 3D equation, and then reduce to 1D.

If you keep the nonlinear term but neglect the pressure gradient, you get the Burgers equation, which is a much more interesting problem to study.

 

[stanley.st]

Thank you ! Navier-Stokes without pressure? It is strange, because in order to get particles move, we have to include pressure considerations. How would you interpretate that ?

 

[bigfooted]

Thank you ! Navier-Stokes without pressure? It is strange, because in order to get particles move, we have to include pressure considerations. How would you interpretate that ?

When the pressure is (nearly) constant, you can neglect the pressure gradients.
This is the case for (Couette) flow between two flat plates where one of the plates is moving and causing the flow motion.

The Burgers equation is mostly used to study shock waves.

 

[menelaos]

The inclusion of pressure in the equation causes additional mathematical difficulties because pressure is non-local. Also, the derivation of the time dependent NS through variation assumes that time is stationary and in effect it is questionable whether it is present as a dynamical variable or a parameter. Moreover, since it describes fields it should display some form of gauge invariance which is still under investigation and its relationship with any form of Noether's theorem is at best weak.