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Is this equation conservative or non-conservative?

  1. Jul 10, 2016 #1
    1. The problem statement, all variables and given/known data
    This is the Navier-Stokes equation for compressible flow. nj is the unit normal vector to the surface 'j', and ni is the unit normal vector in the 'i' direction. Is this equation written for a control volume or a material volume?

    2. Relevant equations
    upload_2016-7-10_19-0-39.png

    3. The attempt at a solution
    I believe it's for a control volume, since it's in integral form and expressing fluxes out of a cube (taking advantage of conservation of momentum). However, I know that integral forms of non-conservative equations also exist, so I'm not sure.
     
  2. jcsd
  3. Jul 11, 2016 #2
    Would the first term on the right hand side be present in the material volume form?
     
  4. Jul 14, 2016 #3
    I'm tempted to say 'no', because no fluid enters or leaves a material volume. So the term would disappear. I'd like to see the integral and differential form for conservative, and the integral and differential form for non-conservative.
     
  5. Jul 14, 2016 #4
    The integral form for material volume is the same as for control volume, except that the first term on the right hand side is absent. The differential forms for both are identical. See this link to see why the integral form of the material volume development reduces to the same differential form as the control volume development: https://en.wikipedia.org/wiki/Reynolds_transport_theorem
     
  6. Jul 15, 2016 #5
    Thank you.
    Great! I seen in another post a reference to Bird's Transport Phenomena book. Thanks.
     
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