Coupon Accepted Successfully!


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 h1 above a standard height, and point 2 is at height h2. (See Figure 10-11.)

..\art 10 jpg\figure 10-11.jpg

Figure 10-11

For incompressible, laminar, inviscid (no viscosity) flow, if points 1 and 2 are on the same streamline, we have


Let's see if we can make sense of this equation, which can also be written
P + ρgh + 1/2 ρv2 = 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 mv2 = const

Wait! This looks like an energy conservation equation, the second and third terms being potential and kinetic energy. But what is the first term? This is a bit more difficult, but we can make this look like an expression for work. For a moving fluid, ΔV can be replaced with AΔx so that PΔV = PAx = FΔx. Thus, PV is the work that one portion of the fluid does on another portion of the fluid as it moves along.
Bernoulli's principle is an expression of energy conservation, and that is why there are so many caveats in the statement of the principle: we are trying to make sure energy does not leak out into heat and ruin the equation.

A large barrel of water has a hole near the bottom. The barrel is filled to a height of 4.5 meters above the bottom of the barrel, and the hole is a circle of radius 1 centimeter in the side of the barrel at a height 0.5 meters above the bottom.
  1. What is the flow velocity v just outside the hole?
  2. What is the flow rate f out of the hole?

a. First, we DRAW A DIAGRAM with a streamline. (See Figure 10-12).


..\art 10 jpg\figure 10-12.jpg
Figure 10-12

Bernoulli's principle applies, so we have

P1 + ρgh1 + 1/2 ρv12 = P3 + ρgh3 + 1/2 ρv32

We are looking for v3. At point 1, the pressure is atmospheric pressure Patm, and at point 3, we have P3 = Patm as well. Also, h1 = 4.5 m and h3 = 0.5 m.

The tricky part is realizing that v1 is very, very small. This is because continuity guarantees that A1v1 = A3v3, where A1 is the cross-sectional area of the barrel and A3 is the area of the hole. Thus we can set v1 to zero in the above equation:


Patm + ρgh1 = Patm + ρgh3 + 1/2 ρv32


ρgh1 = ρgh3 + 1/2 ρv32


gh1 = gh3 + 1/2 v32


(10 m/s2)(4.5 m) = (10 m/s2)(0.5 m) + 1/2 v32


v32 = 80 m2/s2


v3 = 9 m/s


Another way to get the same result is to realize that the pressure at point 2 must be (from Section D)

P2 = P1 + ρgh = Patm + (103 kg/m3)(10 m/s2)(4 m)


Patm + 4 x 104 Pa



Then we can use Bernoulli's principle between points 2 and 3 and use h2 = h3 to obtain
P2 + ρgh2 + 1/2 ρv22 = P3 + ρgh3 + 1/2 ρv32
(Patm + 4 x 104 Pa) + 1/2 ρv22 = Patm + 1/2 ρv32


Again, we set the very small velocity v2 to zero to obtain
4 x 104 Pa = 1/2 (103 kg/m3) v32
v3 = 9 m/s


b. The answer to part b we get through the definition of flow rate, so we have
f3 = A3v3
π (0.01 m)2(9 m/s)
3 x 10–3 m3/s
In this chapter we studied fluids in static equilibrium and fluids in motion. Pressure is a unifying concept for fluids in equilibrium. Pressure is related to force by the equation P = F/A (where the units of pressure are often Pa = N/m2). Pressure at one point in a body of fluid can be related to pressure at another point using P2 = P1 + ρgh (for vertical separation) and Pascal's law (for horizontal separation). If we know the pressure everywhere in a situation, we can often understand the physics and answer questions about it.
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 + 1/2ρv2 + 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.

Test Your Skills Now!
Take a Quiz now
Reviewer Name