I am trying to follow along with the text's derivation of the Bernoulli equation and I am a little confused with a particular line. The images show an elemental streamtube control volume (CV) of variable area A(s) and length ds.(adsbygoogle = window.adsbygoogle || []).push({});

We assume that though the properties may change with time and position 's' they remain uniform over the cross section 'A.'

So treating as a 1-dimensional inlet/outlet CV we can write conservation of mass as:

[tex]\frac{d}{d\,t}(\int_{CV}\rho d\,V) +\dot{m}_{out} - \dot{m}_{in} = 0 \approx \frac{\partial{\rho}}{\partial{t}}d\,V + d\,\dot{m}[/tex]

Okay. I am missing somethings here:

(1) I am assuming that they set [itex]\dot{m}_{in} =\dot{m}[/itex]

and

[itex]\dot{m}_{out} =\dot{m} + d\,\dot{m}[/itex]

so

[itex]\dot{m}_{out} - \dot{m}_{in} =d\,\dot{m} [/itex].

Does that sound right?

(2) So the thing that is really bothering me is this. Where did the integral go? AND why did [itex]d/d\,t[/itex] turn into [itex]\partial/\partial{t}[/itex]

I keep getting jammed up on these total derivatives turning into partials. What does it all mean?

thanks

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# Deriving the Bernoulli EquationFluids

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