About Torricelli's law derivation: $p_1=p_2$?

When deriving Torricelli's law, pressure at points 'a' and 'b' are said be equal to atmospheric (Figure 1 below). Torricelli's law became a specific case of the Bernoulli's principle, whose derivation uses the diagram below (Figure 2), and requires the pressure at point 2 be tangential to fluid velocity. As a waterjet is formed in Torricelli's law experiment, atmospheric pressure is tangential only at the start (t=0), and then, at the very end of the waterjet when it's falling (describing a parabola), until it hits the "ground" (a table, the floor, etc.).

Hence, I would say that pressure at point b of Figure 1 cannot be the atmospheric, also because this point is in the start of a streamline (at t>0). But as this law has been discovered in 1643, many years have passed since then and there must be something I don't understand well enough about Pa=Pb. Thanks in advance.

• The water surface at point "a" is obviously at atmospheric pressure, because it is exposed to the atmosphere. The water surface at point "b" must be at atmospheric pressure, because it is outside the tank (independent of the pressure of the water column) and it is in free fall (there is no pressure on any "piece" of water in the jet from the water that is above it). – David White Aug 10 '17 at 1:44
• David, it would mean Torricelli's law is valid only at t=0. But undergraduate students solve math problems about tanks draining, using this law, for t greater than 0, when I'm not sure enough about pressure at point "b" be the atmospheric. – ilich qynn Aug 10 '17 at 2:01
• While i think it's good that you are thinking critically about these type of problems, don't overthink it. Bernoulli's (and by extension Torricelli's) law are tools which help you simplify a problem and get a solution to engineering accuracy. There are simply to many assumptions made using these laws for use to a high degree of mathematical accuracy which is usually not even required in engineering analyses. Another assumption not taken into account are the effects of frictional losses due to viscosity; my guess is these are much less negligible than the assumption of atmospheric pressure. – nluigi Aug 10 '17 at 9:05
• @ilichqynn, Torricelli's law is a function of liquid height above the hole. You would have to go through some amount of mathematical development, and arrive at a (undoubtedly) logarithmic equation, to solve such an equation as a function of time. – David White Aug 10 '17 at 20:21