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Fluid Mechanics -continuity equation

  1. Dec 4, 2005 #1
    Fluid Mechanics --continuity equation

    This is for a 300 level fluid mechanics class.

    "Consider the y-component of the velocity (v) at any cross-section in the entrance region. For simplicity, take the lower half of the pipe; the content of the flow field is going to be symmetric with respect to the center-line. Using the continuity equation and the viscous phenomena taking place in the entrance region, explain the behavior of the v (y-component of the velocity)."

    The pipe diameter is 0.1 m and is 20 m long. We used Flowlab to find the friction factor for different reynolds numbers 200-1000 (by changing the velocity to get a different reynolds #).

    I am trying to use (rho)1A1V1 = (rho)2A2V2
    to explain this. I am not really sure what the "viscous phenomena" means.

    Thanks for any help.
  2. jcsd
  3. Dec 4, 2005 #2


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    Staff: Mentor

    With a Reynolds number below 2000, the flow will be laminar and have a parabolic velocity distribution profile across the tube.

    The continuity equation simply implies that the fluid in-flow = fluid out-flow.
    Last edited: Dec 4, 2005
  4. Dec 4, 2005 #3


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    The viscous phenomena for the entrance region is what is responsible for the entrance losses a fluid sees when entering a region like a tank, valve, pipe, etc...You might want to read up on the idea of fully developed flow in a pipe entrance region. The location of fully developed flow defines the entrance region.

    Question, in what direction is the y component in this assignment?
  5. Dec 4, 2005 #4
    Thanks for your help so far.

    The lenght of the pipe is the x-component. So i am assuming the y-component is the height of the pipe (if the pipe is laying horizontally). i was not given a picture or any additional info.

    On flowlab, i assumed the fully developed region to be the point at which the friction function stopped changing (where it became almost a straight line).

    At the entrance of the pipe (a lower x-value), the friction function was higher. So does this mean the parabola is longer (has more of a point to it) at the entrance of the pipe, and then the parabola gradually becomes more rounded as it goes further through the pipe?
  6. Dec 4, 2005 #5


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    When you say parabola, are you refering to the boundary layer or the velocity profile across the field?
  7. Dec 4, 2005 #6
    hmm good question. i was thinking of them as the same thing. But since you mentioned it, i think the prof. is asking about the velocity profile. Although wouldn't the boundary layer be the same thing when it first enters the pipe?
  8. Dec 4, 2005 #7


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    Not really. If memory serves me correctly, the boundary layer grows as it goes down the length of pipe. I don't have any references in front of me right now. I'll have to see if I can hunt something down on-line.
  9. Dec 5, 2005 #8


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    I've got tomorrow two finals, man, nevermind I am going to explain this to you. Let x be the axial coordinate of the pipe adimensionalized with the pipe diameter D, being x=0 the point of inlet.

    At x=O(1), the fluid is in the hydrodynamic entrance lenght. The boundary layer begins to grow, but at this point its thickness d is too small. The profile of axial velocity is almost plane, except in a zone near walls such that d/D<<<1. In this zone, [tex]\partial u/\partial x<0[/tex] and by means of continuity [tex]\partial v/\partial y>0[/tex]. This means in this zone there is an Entraintment of fluid due to the sucking process. At distances of the order x=Re^(-1), the boundary layer has grown too much as to produce a parabolic profile. In this situation the flow is fully developed and so [tex]\partial u/\partial x=0[/tex] and v=0.
  10. Dec 6, 2005 #9
    Yes, the flow is developing in the entrance region. The developed flow should have parabolic velocity distribution due to viscous effect, which means lower velocity close to the wall.
    Close to wall, in entrance, the u velocity should be reducing, (u)in+(v)out=(u)out (continuity). Therefore, v velocity doesn't equal to zero.
    I think this is what question asked for.
    Assumption, flow is incompressible.
    Last edited: Dec 6, 2005
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