Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

I Pipe-Flow Momentum Balance

  1. Oct 19, 2016 #1

    joshmccraney

    User Avatar
    Gold Member

    Hi PF!

    Suppose we have an incompressible UNSTEADY fluid passing through a level pipe. Let station 1 have area, velocity, and pressure ##A_1##, ##V_1(t)## and ##P_1(t)##. Station 2 is defined similarly. I know the unsteady Bernoulli equation could solve this, but if I wanted to make a momentum balance I would have $$\partial_t\iiint_v \vec{V} \rho \, dv + \iint_{\partial v} \rho \vec{V} (\vec{V} \cdot \hat{n}) \, dS = \sum \vec{F}$$ I'm not worried about any specifics here except for one detail, the volumetric time rate of change integral. Since velocity ##\vec{V}## monotonically changes from station 1 to station 2, this integral ##\partial_t\iiint_v \vec{V} \rho \, dv## is definitely not zero; then how do we solve for it? Would we have to look at Navier-Stokes for the fluid to get the fluid velocity profile to solve? I know NS is a momentum balance and takes identical form to the equation I posted, but I'm not sure how to proceed here. Any idea?
     
  2. jcsd
  3. Oct 19, 2016 #2

    boneh3ad

    User Avatar
    Science Advisor
    Gold Member

    Well the usual approach, if the goal is to do this analytically, would be to use the divergence theorem to remove the integrals and solve the integrands as a system of differential equations.
     
  4. Oct 19, 2016 #3
    Josh,

    Have you checked out BSL, Chapter 7 like I suggested. They show how to do what you want for an inviscid fluid. It involves using the rate of change of kinetic energy within the control volume.
     
  5. Oct 19, 2016 #4

    joshmccraney

    User Avatar
    Gold Member

    I don't have the book on me right now. I moved a little while ago and left my book at my old school. I am picking it up this November though, so I was planning on studying it then! I'll be sure to give it a good read. Perhaps I'll check and see if our library has it now though. Then I can read before asking a bunch of questions.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Pipe-Flow Momentum Balance
  1. Flow rate in a pipe (Replies: 7)

Loading...