# Pipe-Flow Momentum Balance

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?

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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.

joshmccraney
Chestermiller
Mentor
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.

Gold Member
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.
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.