Bernoulli's Principle
For laminar flow there is an equation which relates conditions at one point in the flow with points downstream. Consider two points, 1 and 2, along a streamline. Point 1 is at some height h_{1} above a standard height, and point 2 is at height h_{2}. (See Figure 1011.)
Figure 1011
For incompressible, laminar, inviscid (no viscosity) flow, if points 1 and 2 are on the same streamline, we have

P + ρgh + 1/2 ρv^{2} = const  ...(17) 
The expression ρgh reminds us of mgh, the difference being only a factor of ΔV, that is, volume. If we multiply the above expression by ΔV, then we obtain
PΔV + mgh + 1/2 mv^{2} = const
 What is the flow velocity v just outside the hole?
 What is the flow rate f out of the hole?
a. First, we DRAW A DIAGRAM with a streamline. (See Figure 1012).
Figure 1012
Bernoulli's principle applies, so we have
P_{1} + ρgh_{1} + 1/2 ρv_{1}^{2} = P_{3} + ρgh_{3} + 1/2 ρv_{3}^{2}
We are looking for v_{3}. At point 1, the pressure is atmospheric pressure P_{atm}, and at point 3, we have P_{3} = P_{atm} as well. Also, h_{1} = 4.5 m and h_{3} = 0.5 m.
The tricky part is realizing that v_{1} is very, very small. This is because continuity guarantees that A_{1}v_{1} = A_{3}v_{3}, where A_{1} is the crosssectional area of the barrel and A_{3} is the area of the hole. Thus we can set v_{1} to zero in the above equation:
P_{atm} + ρgh_{1} = P_{atm} + ρgh_{3} + 1/2 ρv_{3}^{2}
ρgh_{1} = ρgh_{3} + 1/2 ρv_{3}^{2}
gh_{1} = gh_{3} + 1/2 v_{3}^{2}
(10 m/s^{2})(4.5 m) = (10 m/s^{2})(0.5 m) + 1/2 v_{3}^{2}
v_{3}^{2} = 80 m^{2}/s^{2}
v_{3} = 9 m/s
Another way to get the same result is to realize that the pressure at point 2 must be (from Section D)
P_{2} = P_{1} + ρgh = P_{atm} + (10^{3} kg/m^{3})(10 m/s^{2})(4 m)
= P_{atm} + 4 x 10^{4} Pa
Then we can use Bernoulli's principle between points 2 and 3 and use h_{2} = h_{3} to obtain
Again, we set the very small velocity v_{2} to zero to obtain
b. The answer to part b we get through the definition of flow rate, so we have
The important concepts for fluids in motion are continuity and Bernoulli's principle. Continuity expresses the conservation of mass as the fluid flows, so we have the product Av being a constant along a streamline. Bernoulli's principle expresses the conservation of energy along the fluid flow, so we have the sum P + 1/2ρv^{2} + ρgh being a constant along a streamline, as long as energy is not lost to heat or other energy sinks. These two principles allow you to solve most simple problems involving flowing fluids.