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.
|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.