Unsteady Pipe-Flow Momentum Balance: Solving for Volumetric Time Rate of Change

[member 428835]

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?

 

[boneh3ad]

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.

 

[Chestermiller]

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.

 

[member 428835]

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.