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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 $p \rightarrow\infty$?

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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.

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  • $\begingroup$ 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. $\endgroup$
    – user283587
    Commented Feb 15, 2021 at 5:22
  • $\begingroup$ 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. $\endgroup$
    – user283587
    Commented Feb 15, 2021 at 7:20

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