# The applications of Bernoulli’s principle and Bernoulli’s equation

**Torricelli’s theorem**

One of the uses of Bernoulli’s equation is to calculate the speed of a liquid exit from the bottom of a container (see figure).

We apply Bernoulli’s equation at point 1 (surface of the container) and point 2 (surface of the hole). Because the diameter of the hole in the bottom of the container is much smaller than the diameter of the container, the speed of the liquid on the surface of the container is considered zero (v_{1 }= 0). The surface of the container and the surface of the hole are open so that the pressure is equal to atmospheric pressure (P_{1 }= P_{2}). Thus, Bernoulli’s equation for this case is:

If we want to calculate the velocity of the liquid flow in the hole at the bottom of the container, then this equation is changed to:

The density of the fluid liquid is the same so that ρ is eliminated.

Based on this equation, the speed of the flow of water in a hole that ish from the surface of the container is the same as the speed of flow of water that free fall as far as h (compare with the free fall motion). This is known as Torricelli’s theorem. This theorem was discovered by Torricelli, a student of Galileo, a century before Bernoulli found the equation.

**Venturi effect**

In addition to Torricelli’s theorem, Bernoulli’s equation can also be applied to other special cases, namely when the fluid flows in a section of pipe that is almost the same height. To understand this explanation, observe the figure below.

In the figure above it appears that the height of pipe, both the part of the pipe with a large cross-section area and the part of the pipe with a small section area, is almost the same so that the height is equal. If applied in this case, Bernoulli’s equation changes to:

When the fluid passes through a small section of the pipe (A_{2}), the fluid rate increases. According to Bernoulli’s principle, if the speed of the fluid increases, the fluid pressure becomes small. The pressure of the fluid in the narrow section of the pipe is lower, but the fluid flow rate is higher.

This is known as the Venturi effect and shows quantitatively that if the fluid flow rate is high, the fluid pressure becomes small. Likewise, if the fluid flow rate is low, the fluid pressure becomes large.

**Venturi meter**

An interesting application of the venturi effect is Venturi Meter. This tool is used to measure the flow rate of a fluid, for example calculating the flow rate of water or oil flowing through a pipe. There are two types of venturi meters, namely venturi meters without manometers and venturi meters that use a manometer that contains other liquids, such as mercury. The working principle is the same.

**Venturi meter without a manometer**

The figure below shows a venturi meter that is used to measure the flow rate of a liquid in a pipe.

When the liquid passes through a small section of the pipe (A_{2}), the liquid rate increases. According to Bernoulli’s principle, if the rate of fluid increases, the pressure of the liquid becomes small. Thus the pressure of the liquid in a large section is greater than the pressure of the liquid in a small section (P_{1} > P_{2}). Instead v_{2} > v_{1}

Because P_{1}> P_{2 }and v_{2}> v_{1 }then this equation can be changed to:

*Equation 1.*

The equation of continuity :

*Equation 2.*

What is sought is the flow rate of the liquid in a large cross-section (v_{1}). We replace v_{2} in equation 1 with v_{2} in equation 2.

*Equation 3.*

To calculate the fluid pressure at a certain depth, use the equation:

*p = ρ g h → Equation a *

If the difference in fluid density is minimal, then use this equation to determine the difference in pressure at different heights. Thus, the equation a can be changed to:

Δ p = ρ g Δh

For the above case, this equation can be changed to:

p_{1 }− p_{2 }= ρ g h → Equation b

Now we replace p_{1} – p_{2} in equation 3, with p_{1} – p_{2} in the equation b:

Because the liquid is the same, the density is also the same. Eliminate ρ from the equation.

**Pitot Tubes**

If the venturi meter is used to measure the flow rate of a liquid, the pitot tube is used to measure the flow rate of gas or air. The hole at point 1 is parallel to the air flow. The position of these two holes is made far enough from the end of the pitot tube so that the rate and pressure of the air outside the hole are the same as the rate and pressure of free-flowing air. In this case, v_{1 }= the flow rate of free-flowing air (this is what we will measure) and the pressure on the left foot of the manometer (left pipe) = free flowing air pressure (P_{1}).

The hole leading to the right foot of the manometer, perpendicular to the air flow. Therefore, the rate of air flowing through this hole (the middle) decreases and the air stops when it arrives at point 2. In this case, v_{2} = 0. The pressure on the right foot of the manometer is equal to the air pressure at point 2 (P_{2}). The height of point 1 and point 2 is almost the same (the difference is not too large) so that it can be ignored.

The difference in pressure (P_{2} – P_{1}) = the hydrostatic pressure of the liquid in the manometer (the black color in the manometer is liquid, mercury). Mathematically can be written:

p_{2} – p_{1} = ρ ‘g h → Equation 2

ρ ‘= density of the liquid in a manometer

Equation 1 and equation 2. The left segment is the same (P_{2 }– P_{1}). Therefore equations 1 and two can be changed to:

**Drink with a pipette**

Have you ever drank using a pipette? When we suck water using a pipette, we make the air in the pipette move faster. In this case, the air in the pipette at our mouth has a higher speed. As a result, the air pressure in the pipette section becomes smaller. The air in the pipette section close to the drink has a smaller speed. Because the speed is small, the pressure is greater. This difference in air pressure makes the water we drink flow into our mouths. In this case, the liquid moves from the part of the pipette whose air pressure is high towards the pipette section where the air pressure is low.

**Chimney**

Have you seen a chimney? Why can smoke move up through the chimney? First, the smoke produced by combustion has a high temperature. Because of the high temperature, the air density is small. Air with a small density is easy to float or moves upward. The reason is not only this. Bernoulli’s principle is also involved in this problem.

Second, Bernoulli’s principle states that if the air flow rate is high then the pressure becomes small, otherwise if the air flow rate is low, the pressure is large. The top of the chimney is outdoors. There is a wind blowing at the top of the chimney so that the air pressure around it is smaller. In a closed room there is no wind blowing, so the air pressure is higher. Therefore the smoke was out through the chimney. The air moves from a place where the air pressure is high to where the air pressure is low.

**Rat hole in the ground**

Pay attention to the figure below. This is a figure of a mouse hole in the ground. Mice also know Bernoulli’s principle. Mice do not want to die because of shortness of breath, so mice make two holes at different heights. As a result of the height difference in the surface of the ground, the air is jostling with other air. It’s like water flowing from a pipe with a large diameter crossing to a pipe with a small diameter. So the air speed increases and the air pressure decreases.

Because there is a difference in air pressure, the air is forced to flow through the mouse hole. Air flows from a place where the air pressure is high to a place where the air pressure is low.

**Airplane wings and dynamic lift**

One factor that causes aircraft to fly is wings. The wing shape is curved, and the front is thicker than the rear. This wing shape is called aerofoil. This idea was imitated from the bird’s wings. The shape of the bird’s wing is also like that (the bird’s wings are curved, and the front is thicker). The difference is, the wings of birds can be flapped, while the wings of the aircraft do not. Birds can fly because they flap their wings, so there is air flow through both sides of the wing. For air to flow on both sides of the aircraft wing, the plane must be moved forward. Humans use machines to move planes.

The front of the wing is designed to bend upwards. The air flowing from below jostled with air on the top. It’s like water flowing from a pipe with a large cross-section to a narrow-section pipe. As a result, the rate of air above the wing increases. Because of the air rate increases, the air pressure becomes small. Conversely, the air flow rate below the wing is lower, because the air is not jostling (the air pressure is greater). The existence of this pressure difference, makes the aircraft wing lift up.

Bernoulli’s principle is only one of the factors. Another cause is momentum. Usually, the aircraft wing is tilted slightly upwards. The air that hits the lower surface of the wing is deflected down. Because the plane has two wings, on the left and right, the air that is deflected down collides with each other. Changes in the molecular momentum of air that collide produce additional lift.

Because the shape of the wing curves down to the tail so that the air is forced by the wing to flow down again. According to Newton’s third law, because there is an action force, so there is a reaction force. Because the wings force the air down, the air must force the wings to move upward. In this case, the air provides lift force on the wing. So it’s not just Bernoulli’s principle that causes the plane to lifted up.