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I need a clarification about Bernoulli's principle.

The standard way in which this principle is taught is related to a picture like this one enter image description here

where usually the two sections have different areas and are located at different values of height. Then the relation $p_1+\rho g h_1 + 1/2\rho v_1^2 = p_2+\rho g h_2 + 1/2\rho v_2^2$ is derived.

However, I'd like to know if - for some applications - Bernoulli's principle can be applied to two different points at the same height but one inside the tank, and the other one outside the tank but really near the opening. Like in this picture enter image description here

Thus: $p_0+\rho g h_1 + 1/2\rho v_1^2 = p_0+\rho g h_2 + 1/2\rho v_2^2$ yields to $\rho g h_1 = 1/2\rho v_2^2$ (by neglecting $v_1^2$, the opening is small)

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  • $\begingroup$ Short answer: yes. Longer answer: The Bernoulli equation is a balance of energy for an incompressible fluid. It says that work energy + flow energy + potential energy is constant in the flow. $\endgroup$ – Jeffrey J Weimer May 22 at 22:40
  • $\begingroup$ @JeffreyJWeimer there is no such thing as "work energy" or "flow energy". Bernoulli equation is not about energy of a fluid element being constant in time, it is about how work of pressure forces changes kinetic and potential energy of the fluid. $\endgroup$ – Ján Lalinský May 22 at 23:16
  • $\begingroup$ @JeffreyJWeimer the terms have units energy per unit mass or unit volume, but this alone does not imply those terms give some contribution to energy of the liquid. Incompressible liquid cannot change its internal energy. The pressure term does not give some kind of energy of the fluid, it is just pressure, present due to fact that work on a liquid element is given by gradient of pressure. When element of liquid moves along streamline in accord with the Bernoulli equation, its energy is given solely by kinetic and potential terms, so it actually changes. $\endgroup$ – Ján Lalinský May 23 at 10:58
  • $\begingroup$ Energy per volume or energy per mass are the units I would consider. The enthalpy of the fluid will change as pressure changes. I should express this term perhaps better as the capacity of the fluid to do mechanical work. So, the expression is conservation of energy as capacity to do work (energy) + kinetic energy + potential energy. $\endgroup$ – Jeffrey J Weimer May 23 at 15:22
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Bernoulli equation is valid for any two points only if some neccessary conditions are satisfied: the flow has to be incompressible, the two points can be connected by a line in the liquid that has zero curl of velocity on all its points. In practice, this is most often a steady or slowly changing non-turbulent incompressible flow.

If the flow of liquid out of the tank is steady and non-turbulent, and if both points have the same pressure (which requires that the second point is far from the tank so the liquid has pressure equal to the atmospheric pressure), the use of the Bernoulli principle the way you described is valid. What you get is the Torricelli formula for velocity of liquid jet discharging from a tank containing the liquid of given height.

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