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Homework Help: Fluid Dynamics: Static pressure in compressible Liquids

  1. Oct 9, 2015 #1
    1. The problem statement, all variables and given/known data

    "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 ρο, and compressibility of the liquid is k."

    2. Relevant equations

    ##ρ(P) = ρ_οe^{kP}##
    ##dP = ρg dy##

    3. The attempt at a solution

    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?
    Last edited: Oct 9, 2015
  2. jcsd
  3. Oct 9, 2015 #2
    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.

  4. Nov 24, 2015 #3
    Thank you very much! :)
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