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Consider a tank with an outflow a the bottom: e.g here with the unsteady Bernoulli equation:$$\int_S \frac{\partial v}{\partial t} \ dS+\frac{v^2}{2}+\frac{p}{\rho}+gz=0$$where $s$ is the path between $S_1$ and $S_3$. Suppose a inviscid fluid with constant density. Why does the solution of the Bernoulli equation change depending on whether $l$ very small or not? In other words, why is there a term $\frac{\partial v}{\partial t}$ introduced when $l$ is not negligibly small?

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    $\begingroup$ -1. Unclear. Please cite a source for the Bernoulli equation which includes such a term. $\endgroup$ – sammy gerbil Mar 13 '17 at 0:03
  • $\begingroup$ I'm not 100% sure, but I think this term should come from the friction. $\endgroup$ – Alpha001 Mar 13 '17 at 0:04
  • $\begingroup$ @Alpha001 OP says the fluid is inviscid and incompressible. $\endgroup$ – sammy gerbil Mar 13 '17 at 1:25
  • $\begingroup$ Are you referring to the transient version of the Bernoulli equation? $\endgroup$ – Chet Miller Mar 13 '17 at 1:30
  • $\begingroup$ @miller edited, yes i am $\endgroup$ – Ben L Mar 13 '17 at 7:12
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$S_1 \to S_2 = S_2 \to S_3$. The only change here is $v$ and $p$ at $S_1$ and $S_2$. $p_1>p_2$ at $S_1$ and $p_2<p_1$ at $S_2$, $v_1<v_2$ at $S_1$ and $v_2>v_1$. In the Bernoulli E substitute the proper values and that satisfy the same result with L as in question being used in equation on the right. $\rho+\rho g h+\frac{1}{2}\rho v^2 = \rho +\rho g \ell +\frac{1}{2}\rho v^2$. As for the derivative $dv/dt$ it will indicate only the time for the fluid to move in H and L with {H=l} in time; the values of Bernoulli equation do not change because, if the values change then Bernoulli will not be in the books.

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  • $\begingroup$ also is important to consider that the inviscid fluid has an ideal constant. that be temperature and other variables which could increase or decrease the compresion ratio, as having the corresponding constant in the fluid areas meaning same friction of other variables that will keep the dynamics constan in every place of the space the fluid travel in. $\endgroup$ – Isabela fernandez Mar 13 '17 at 2:39

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