# How to use Navier-Stokes equation to model incompressible flow when pressure moves towards infinity?

I have a container of inviscid, incompressible fluid with a piston at one end. It is completely closed, except the fact that I can move the piston(say $$x_p$$). From definition of Bulk Modulus-

$$\mathcal{B}=-V\frac{\partial P}{\partial V}$$

Since $$\mathcal{B}=\infty$$ for any incompressible fluid, so any displacement change to the piston will give rise to infinite pressure.

I am thinking how to model this rise in pressure using Navier-Stokes equation. Let me write the simplified equations-

$$\frac{\partial{u}}{\partial{x}}=0 \\ \frac{\partial{u}}{\partial{t}} + u \frac{\partial{u}}{\partial{x}}=-\frac{1}{\rho} \frac{\partial p}{\partial x}$$

If I give some finite value of displacement to piston, how to show that $$p \rightarrow\infty$$?

## 1 Answer

For an incompressible fluid in such a situation, the pressure is indeterminate. Any pressure can be applied to the fluid (with no motion), not just an infinite pressure. The Euler equation just gives p = constant.

• Thanks for your help. What if I apply some displacement, u to the piston, which will lead to some change in volume and hence infinite pressure should come. – user283587 Feb 15 at 5:22
• Don't worry. I found my doubt. Velocity cannot be applied out of nowhere, because I have the continuity equation also. Thanks for the answer. – user283587 Feb 15 at 7:20