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I Solving Navier Stokes & energy equations with different coordinates

  1. Mar 30, 2016 #1
    Hi all I am conducting a fluid analysis on water flowing through a subsea pipe.

    Having used navier stokes equation, i derived the equation for velocity in the r-direction (using cylindrical coordinates.

    But when initially solving the energy equation to determine temperature distribution I have used the cartesian coordinates, x & y

    From the picture I have attached am I correct in proceeding to solve the energy equation, if I were to just differentiate my velocity equation result and substitute back into the energy equation?
    From my knowledge the Y direction in cartesian corresponds with the R direction in the cyclindrical hence my reasoning for proceeding this way.
     

    Attached Files:

  2. jcsd
  3. Mar 31, 2016 #2
    These are the actual relationships between cylindrical and cartesian coordinates.
    [tex]x = r \cos \theta[/tex]
    [tex]y = r \sin \theta[/tex]
    [tex]r^2 = x^2 + y^2[/tex]
    Why don't you just use the energy equation in cylindrical coordinates? That's the easiest way to proceed.
    [tex]\frac{1}{r} \frac{d}{dr} \left( r \frac{dT}{dr} \right) = - \frac{\mu}{k} \left( \frac{dv_z}{dr} \right)^2[/tex]
    However, this model is only valid if the temperature is a function of radius only. If it also depends on z, your energy equation becomes a PDE.
     
  4. Mar 31, 2016 #3
    yes this is what i assumed.
    of course it is better to work in just one coordinate system.

    thanks
     
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