- #1

joshmccraney

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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?