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Fluid Mechanics: Compressible Fluids, density function

  1. Feb 18, 2013 #1
    1. The problem statement, all variables and given/known data
    "Suppose that a liquid has an appreciable compressibility. Its density therefore varies with depth and pressure. The density at the surface is ρ0
    (a) Show that the density varies with pressure according to ρ=ρ0ekp where P is gauge pressure at any depth and k is the compressibility, a constant.
    (b) Find P as a function of depth y.

    2. Relevant equations
    ρ = m/V (Density definition)
    k = (1/V)(ΔV/ΔP) (Compressiblity definition)
    dP/dy = ρg (Pascal's Law)
    My professor also hinted that I should replace any deltas with differentials, that I need to eliminate V, and to find dρ/dV (which is -m/V2
    3. The attempt at a solution
    From the second equation, I multiplied both sides by dP (after changing ΔV to dV and ΔP to dP) to obtain the following:
    k dP = (1/V) dV
    I then integrated both sides:
    ∫k dP (from P0 to P) = ∫(1/V) dV (from 0 to V)
    kP(gauge)=ln(V)
    ekP(gauge) = V
    I get the feeling that I'm going in the correct direction, but haven't found a viable solution from this point. I fooled around with the density definition and Pascal's Law to no avail. Any assistance would be much appreciated.
     
  2. jcsd
  3. Feb 18, 2013 #2
    Good start!!! Your integration limits on V are incorrect. They should be from V0 to V. Also, for a given amount of mass the product of density and volume is constant.
     
  4. Feb 18, 2013 #3
    When I intergrate on those limits I get kP(gauge)=ln(V)-ln(V0), which seems to be worse than before.
    Also, if m = ρV is constant, how does that work with ρ and V both being varying functions?
     
  5. Feb 18, 2013 #4
    The compressibility condition should have a minus sign (the volume decreases with increasing pressure):

    k dP = -(1/V) dV

    [tex]\frac{V}{V_0}=\frac{\rho_0}{\rho}=e^{-kP}[/tex]
     
  6. Feb 18, 2013 #5
    I understand where the v/v0 came from (the ln) but how did you derive p0/p? Did it come from another integral somewhere?
    EDIT: nvm, I understand now. Thanks a bunch!
     
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