While describing speed of efflux, using Bernoulli equation, we use value of $P$ at both openings as atmospheric pressure. Why don't we use $P_\text{atmosphere}+\rho gh$ for $P$ at bottom opening?
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3 Answers
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The pressure inside the tank is not $P_{atm}+\rho g h$ over the entire bottom of the tank. Within the tank, there is a disturbance in the pressure distribution in close proximity to the exit hole. The fluid flow is converging toward the exit hole, and, as it does, its velocity is building up; and its pressure is correspondingly decreasing. This all takes place within a distance of a few hole diameters approaching the exit hole (in all directions). At the very exit hole, the velocity reaches the exit velocity and the pressure reaches the atmospheric pressure.
answered Dec 3, 2018 at 18:09
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At the bottom opening the jet is exposed to atmosphere, and therefore the jet is not constrained by boundaries. Therefore its pressure must adjust to that of the atmosphere; if there were any pressure difference the jet would bulge out or shrink (while respecting continuity) until pressures were equalized. We are assuming that pressure change due to surface tension is negligible.
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While writing Bernoulli's equation you write the equation for fluid just inside and just outside the container this leads to the fact that just outside pressure is equal to $P_{atm}$ and just inside it is $P_{atm}+ \rho gh$ . Note that the $P_{atm}$ term cuts off from both sides so you don't feel it's presence.
If you wish to apply Bernoulli's equation for top and bottom then you should realise that at the bottom you are dealing with the fluid which is just outside the container and hence it's pressure is equal to atmospheric pressure and not the one given by the formulae of hydrostatic pressure as fluid is flowing in exposure to atmosphere
