# 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 $$p \rightarrow\infty$$?