In the last section, when discussing the equation of continuity, we saw that when a fluid travels through a smaller cross-sectional area it's velocity increases. How does this affect the pressure of the fluid at that location? The relationship between the velocity of a fluid and its pressure was discovered by Daniel Bernoulli (1700-1782). In words, Bernoulli's Principle basically says that in fast moving fluids, the pressure is lowered. This concept explains why planes fly and how a pitcher can throw a curveball.
In the above picture the wing has a curved shape over the top. This makes air travel over the top of a wing faster than it does underneath. According to Bernoulli's Law this means that the pressure above the wing will be less than the pressure below the wing causing the plane to have lift.
When a pitcher puts a spin on a baseball, the air going by the ball in the same direction as the spin will be moving faster than the air going against the spin on the other side of the ball. This reduces the pressure on one side of the ball causing it to curve into this lower pressure on its way from the pitcher's mound to home plate.
Bernoulli's Equation
Bernoulli's equation can be easily derived from the conservation of energy work theorem but we will ignore that derivation at this time and work on applications of the equation. Bernoulli's equation relates the Pressure, Velocity and Height at any two points.
Bernoulli Equation
P1 + ½ρv12 + ρgh1 = P2 + ½ρv22 + ρgh2
The equation uses h1 and h2 instead of y1 and y2 as shown in the graphic. Bernoulli's equation combines two concepts we have covered already: Pressure due to depth and pressure due to velocity. Let's examine how Bernoulli's equation can be simplified for the two different situations shown below.
In the first picture, the moving fluid is contained in a horizontal pipe so all parts of it have the same elevation (h1 = h2). This simplifies Bernoulli's equation to:
P1 + ½ρv12 = P2 + ½ρv22
From this we see what Bernoulli's Principle stated: If the velocity of the fluid goes up the pressure must go down.
In the second picture, the moving fluid is at different elevations but maintaining a constant velocity (v1 = v2). This simplifies Bernoulli's equation to:
P1 + ρgh1 = P2 + ρgh2
From this we see that if the elevation of the fluid is low (deep) then the pressure must be higher.
Efflux Speed
Before moving on, we want to examine one more application of Bernoulli's equation. The large tank on the right has water emerging from a small hole near the bottom. Bernoulli's equation can be used to determine the efflux speed at which the water leaves the pipe simply by knowing the height of the fluid above the pipe.
At points 1 and 2 the water is exposed to the atmosphere so the pressure would be the same at both locations. This simplifies Bernoulli's equation to:
Before moving on, we want to examine one more application of Bernoulli's equation. The large tank on the right has water emerging from a small hole near the bottom. Bernoulli's equation can be used to determine the efflux speed at which the water leaves the pipe simply by knowing the height of the fluid above the pipe.