I have been doing some serious review of fluids in order to prep for some CFD. I have been re-deriving the NS Equations in all of their various forms. Something seems to have cropped up that I have worked myself in circles about. Let's take the momentum equation in Conservative Integral form:(adsbygoogle = window.adsbygoogle || []).push({});

$$\sum \vec{F} =\left(\frac{d\vec{M}}{dt}\right)_{system} \qquad (1)$$

$$\sum \vec{F} =

\frac{\partial{}}{\partial{t}}\iiint_{\mathcal{V}}(\rho\vec{V})d\mathcal{V} +

\iint_S(\rho\vec{V})\vec{V}\cdot d\vec{s} \qquad (2)$$

The 3rd term in the equation (2) for a control volume is thenet outflowof momentum from the control volume. What I am a little confused about is the sign convention here. The left-hand side of the equation takes forces acting along the positive coordinate axes as being positive. But the right-hand side takesoutflowto be positive regardless of the coordinate axes.

So it seems that an increase in force acting in the positive coordinate direction is equated to an increase of outflow of momentum from the CV regardless of what direction it flows out from.

I hope that maybe someone can see what I am getting at. I think that my confusion might be stemming from what happens when we go from asystemapproach to acontrol volumeapproach (i.e. application of the Reynold's Transport Theorem (RTT)).

Any thoughts are appreciated.

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# Navier Stokes EQs: Some Insight Please

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