Bernoulli’s principle and Bernoulli’s equation

When we ride a motorcycle, the clothes we use are swollen backward. Sometimes if the wind blows hard, the door can close itself. Though the wind blows outside the house, while the door is inside the house.

This can be explained using the Bernoulli’s principle. Daniel Bernoulli (1700-1782) discovered a principle that could be used to explain the above phenomenon.

Bernoulli’s principle

Bernoulli’s principle states that where the speed of the fluid flow is high, the pressure of the fluid is low. Conversely, if the speed of the fluid flow is low, the pressure of the fluid is high. When a motorcycle moves quickly, the airspeed on the front and side of your body is high. Thus, the air pressure becomes low. The back of your body is obstructed by the front of your body, so the air velocity on the back of your body does not turn high. As a result, the air pressure in the back of your body becomes higher. Because there is a difference in air pressure, wherein the back of the body the air pressure is higher then the air pushes your shirt back so that your clothes look bloated backward.

What about the house door that closes itself when the wind blows outside the house? The air outside the house moves faster than the air inside the house. As a result, air pressure outside the home is smaller than the air pressure inside the house. Because there is a pressure difference, where the air pressure inside the house is bigger, the door is pushed out. In other words, the door leaf moves from a place where the air pressure is large towards a place where the air pressure is small.

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Bernoulli’s equation

Previously we have learned about Bernoulli’s principle. Bernoulli also developed the principle quantitatively. To derive Bernoulli’s equation, we assume that fluid flow is steady & laminar, uncompressed, the viscosity is very small so that it can be ignored.

In the discussion of the equation of continuity, we have learned that the fluid flow rate can also vary depending on the flow area of the flow tube. Based on the Bernoulli’s principle described above, fluid pressure can also vary depending on the flow rate of the fluid. Fluid pressure can also vary depending on the height of the fluid. The relationship between pressure, flow rate, and flow height can be obtained in the Bernoulli equation.

Bernoulli’s equation is very important because it can be used to analyze aircraft flights, hydroelectric power plants, piping systems, etc. For the Bernoulli’s equation to be derived in general, we assume that the fluid flows through a flow tube with a cross-sectional area that is not the same and the height is also different. To derive the Bernoulli equation, we apply the theorem of the work and energy to the fluid in the flow tube.

The opaque color in the flow tube in the figure below shows fluid flow while the white color shows no fluid.

Bernoulli’s principle and Bernoulli’s equation 1

Fluid in the cross-sectional area 1 (left side) flows as far as L1 and forces the fluid in section 2 (right side) to move as far as L2. Because the cross-sectional area 2 on the right is smaller, the speed of the fluid flow on the right side of the flow tube is larger (Remember the equation of continuity). This causes a pressure difference between section 2 (the right side of the flow tube) and section 1 (the left side of the flow tube) – Remember Bernoulli’s principle. The fluid that is to the left of section 1 gives pressure (P1) on the fluid to the right and does work:

Bernoulli’s principle and Bernoulli’s equation 2

Then the W1 equation can be written:

W1 = p1 A1 L1

In section 2 (the right side of the flow tube), the work done on the fluid is:

W2 = − p2 A2 L2

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A negative sign indicates that the force applied is opposite to the direction of motion. So the fluid does work to the right of the cross-section 2. Also, the gravitational force also does work on the fluid. In the case above, some fluid masses are transferred from section 1 as far as L1 to section 2 as far as L2, where is the volume of fluid in section 1 (A1 L1) = volume of fluid in section 2 (A2 L2). The work done by gravity is:

W3 = − m g (h2 − h1)

W3 = − m g h2 + m g h1)

W3 = m g h1 − m g h2

A negative sign is caused by the fluid flowing upwards, in contrast to the direction of gravity. Thus, the net work done on the fluid is:

W = W1 + W2 + W3

W = P1 A1 L1 ‐ P2 A2 L2 + m g h1 ‐ m g h2

The theorem of work-energy states that the net work done on a system is the same as the change in the kinetic energy. Thus, we can replace work (W) with the changes in the kinetic energy (EK2 – EK1).

The equation above can be written again :

W = P1 A1 L1 ‐ P2 A2 L2 + m g h1 ‐ m g h2

EK2 ‐ EK1 = P1 A1 L1 ‐ P2 A2 L2 + m g h1 ‐ m g h2

1⁄2 m v22 ‐ 1⁄2 m v12 = P1 A1 L1 ‐ P2 A2 L2 + m g h1 ‐ m g h2

The fluid mass that flows as far as L1 in cross section A1 = the mass of fluid that flows as far as L2 (cross section A2). The mass of the fluid, say m, has a volume of A1 L1 and A2 L2 where A1 L1 = A2 L2 (L2 is longer than L1).

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Now we substitute m in the above equation with m = ρ A L:

Bernoulli’s principle and Bernoulli’s equation 3

Bernoulli’s principle and Bernoulli’s equation 4

This is the Bernoulli’s equation. Bernoulli’s equation is derived based on the principle of the work-energy, so that it is a form of the of the Conservation of energy.

P = pressure, ρ =density, v =speed of fluid, g = acceleration of gravity, h = height of pipe above the ground.

The left and right segments of the Bernoulli’s equation above can refer to two points anywhere along the flow tube so that we can rewrite the above equation into:

Bernoulli’s principle and Bernoulli’s equation 5

Now let’s review Bernoulli’s equation for some cases.

Bernoulli’s equation in static fluids

The special case of Bernoulli’s equation is for the static fluid where the fluid has no speed. Thus v1 = v2 = 0. In the case of a static fluid, we can formulate Bernoulli’s equation to be:

Bernoulli’s principle and Bernoulli’s equation 6

If h2 – h1 = h, this equation can be written as:

p1 − p2 = ρ g (h2 − h1)

p1 − p2 = ρ g h

Bernoulli’s equation on the same height pipe

If the height of the pipe is the same, Bernoulli’s equation is changed to:

Bernoulli’s principle and Bernoulli’s equation 7

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