Bernoulli’s Equation

1. Bernoulli's Equation

1.1. Bernoulli's equation states that for an incompressible, frictionless fluid, the following sum is constant.

1.1.1. P+12ρv2+ρgh=constant,

1.2. P is the absolute pressure, ρ is the fluid density, v is the velocity of the fluid, h is the height above some reference point, and g is the acceleration due to gravity.

1.2.1. P1+12ρv21+ρgh1=P2+12ρv22+ρgh2.

1.3. Bernoulli's equation is a form of the conservation of energy principle.

1.3.1. 12ρv2=12mv2V=KEV.

1.3.2. ρgh=mghV=PEgV,

2. Bernoulli's Equation for Static Fluids

2.1. Let us first consider the very simple situation where the fluid is static—that is, v1=v2=0. Bernoulli's equation in that case is

2.1.1. P1+ρgh1=P2+ρgh2.

2.2. This equation tells us that, in static fluids, pressure increases with depth. As we go from point 1 to point 2 in the fluid, the depth increases by h1, and consequently, P2 is greater than P1 by an amount ρgh1. In the very simplest case, P1 is zero at the top of the fluid, and we get the familiar relationship P=ρgh

2.2.1. P2=P1+ρgh1.

3. Bernoulli's Equation at Constant Depth

3.1. Another important situation is one in which the fluid moves but its depth is constant—that is, h1=h2. Under that condition, Bernoulli's equation becomes

3.1.1. P1+12ρv21=P2+12ρv22.

3.2. Situations in which fluid flows at a constant depth are so important that this equation is often called Bernoulli's principle. It is Bernoulli's equation for fluids at constant depth.

4. Wnet=12mv2−12mv20.

5. When a fluid flows into a narrower channel, its speed increases. That means its kinetic energy also increases. The increased kinetic energy comes from the net work done on the fluid to push it into the channel and the work done on the fluid by the gravitational force,

6. Pressure difference results in a net force on the fluid: recall that pressure times area equals force. The net work done increases the fluid's kinetic energy. As a result, the pressure will drop in a rapidly-moving fluid, whether or not the fluid is confined to a tube.