Fluid Dynamics: Static pressure in compressible Liquids

[FreezingFire]

Homework Statement

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"Derive a relation between the static pressure P at a point and its depth y from the free surface of the liquid. Given the surface density of liquid is ρ_ο, and compressibility of the liquid is k."

Homework Equations

$ρ(P) = ρ_οe^{kP}$
$dP = ρg dy$

The Attempt at a Solution

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I have already found that:
$ρ(P) = ρ_οe^{kP}$

Then, from the relation:
$dP = ρg dy$

and using $ρ(P) = ρ_οe^{kP}$, we get:

$e^{-kP} dP = ρ_ο g dy$

Integrating L.H.S. from 0 to P and R.H.S. from 0 to y, we get:

$$P = \frac {-1}{k} \ln(1 - kρ_ο gy)$$

Are my limits correct? Is the answer correct? I ask this because the answer looks like it could easily become undefined (if $kρ_ο gy$ became greater than 1). If it is wrong, then why? If it is correct, then how does it not become undefined?

 

[Chestermiller]

The answer is correct. In practice, you would never encounter values of y huge enough for the expression to become undefined. Imagine how much pressure it would take to compress liquid water to half its volume (say). Also, in reality, k varies gradually with P, but the starting equation is a good approximation over a substantial range of pressures.

Chet

 

[FreezingFire]

The answer is correct. In practice, you would never encounter values of y huge enough for the expression to become undefined. Imagine how much pressure it would take to compress liquid water to half its volume (say). Also, in reality, k varies gradually with P, but the starting equation is a good approximation over a substantial range of pressures.

Chet

Thank you very much! :)