1. Concept Overview
Bernoulli’s principle explains how the speed of a fluid is connected to its pressure. When a fluid speeds up, its pressure drops, and when it slows down, its pressure rises. This relationship comes from the idea that the total energy of a flowing fluid remains constant along a streamline.
This principle is behind many everyday phenomena such as aircraft lift, swinging cricket balls, and faster flow through narrow pipes.
2. Definition
3. Bernoulli’s Equation
3.1. Mathematical Form
Bernoulli’s equation is written as:
\( P + \dfrac{1}{2}\rho v^2 + \rho g h = \text{constant} \)
Where:
- \(P\): pressure energy per unit volume
- \(\dfrac{1}{2}\rho v^2\): kinetic energy per unit volume
- \(\rho g h\): potential energy per unit volume
3.2. Meaning of Each Term
Pressure Energy: Energy stored in the fluid due to its pressure.
Kinetic Energy: Energy due to the motion of fluid particles.
Potential Energy: Energy due to height in a gravitational field.
Bernoulli’s equation says that if one type of energy increases, at least one of the others must decrease to keep the total constant.
4. Pressure–Speed Relationship
4.1. High Speed → Low Pressure
When fluid flows faster, kinetic energy increases. To keep total energy constant, pressure energy must decrease. So high-speed regions have low pressure.
4.2. Low Speed → High Pressure
When fluid slows down, its kinetic energy decreases. The pressure rises to conserve total energy.
5. Applications of Bernoulli’s Principle
5.1. Lift on an Airplane Wing
Air moves faster over the curved top surface of the wing than under it. Faster air has lower pressure, creating an upward lift force.
5.2. Swing of a Cricket Ball
The ball moves through air at different speeds on its two sides due to seam position. The pressure difference causes the ball to swing.
5.3. Venturi Effect
When a pipe narrows, the fluid speeds up and pressure drops. This is used in devices like carburetors and flow meters.
5.4. Chimney Draft
Fast-moving wind over the top of a chimney lowers pressure, helping smoke rise upward.
5.5. Perfume Atomizer
Fast-moving air at the nozzle reduces pressure and draws liquid perfume upward, creating a fine spray.
6. Situations Where Bernoulli’s Principle Works Best
6.1. Steady Flow
The flow must be smooth (not turbulent) so that energy remains consistent along streamlines.
6.2. Incompressible Fluid
Bernoulli’s equation assumes the density of the fluid does not change, which is true for liquids and low-speed gases.
6.3. Negligible Viscosity
The fluid should not lose much energy due to internal friction. Real fluids have some viscosity, but the principle still works well under many practical conditions.
7. Examples to Build Intuition
7.1. Fast Flowing River at Narrow Sections
Water flows faster in narrow parts of a river, meaning pressure there is slightly lower compared to wider parts.
7.2. Holding a Sheet of Paper
If you blow air across the top of a loose sheet, the sheet lifts because the pressure above it becomes lower due to the high-speed air.
7.3. Two Ships Passing Each Other
As two ships pass closely, water speeds up between them, reducing pressure and pulling them slightly toward each other.