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A Linearized Continuity (Fluids)

  1. May 11, 2017 #1

    joshmccraney

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    Hi PF!

    Continuity for incompressible flow independent of ##\theta## is ##\nabla\cdot u = \partial_ru_r+u_r/r+\partial_z u_z=0##. However, I'm following a problem in cylindrical coordinates, same assumptions as above, and the author states the linearized conservation of mass is ##\partial_ru_r+u_r/r+u_z## (two different authors wrote this so I doubt it's a typo). How are they arriving at this? Also, continuity is already linear in ##u##, so what do they mean by "linearized"?

    Thanks for your help!
     
  2. jcsd
  3. May 11, 2017 #2
    Authors of physics textbooks ramble sometimes. You understand the equation by itself do not you? Then skip this comment and go on.
     
  4. May 11, 2017 #3

    joshmccraney

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    I do, but that still doesn't explain how they get ##u_z## rather than ##\partial_z u_z##.
     
  5. May 11, 2017 #4
    O! now I see what the point is. This is strange indeed. Dimensions must be incomparable
     
  6. May 11, 2017 #5

    joshmccraney

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    Good call on the dimensions. I don't know what they're doing but I'll just move on and forget it (it was the Plateau-Rayleigh instability if you're curious).
     
  7. May 11, 2017 #6
    perhaps in some books ##u_z=\frac{\partial u}{\partial z}##
     
  8. May 11, 2017 #7

    joshmccraney

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    I don't think so because this does not match continuity as they wrote it, since they're using subscripts to denote a particular velocity component.
     
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