Bernoulli's equation and applications

Bernoulli's equation is an essential principle in fluid mechanics, and is fundamental to understanding how fluids behave in different situations. It helps explain the relationship between pressure, velocity, and height in a moving fluid.

Understanding Bernoulli's principle

According to Bernoulli's principle, for an incompressible, non-viscous fluid flowing in a streamlined manner, the sum of the pressure energy, kinetic energy and potential energy per unit volume is constant. The equation can be expressed as:

P + 0.5ρv² + ρgh = constant

Where:

• P is the pressure energy per unit volume.
• ρ (rho) is the density (mass per unit volume) of the fluid.
• v is the velocity of the fluid.
• g is the acceleration due to gravity.
• h is the height above the reference level.

Visualise the concept

Imagine water flowing through pipes with different cross-sectional areas. According to Bernoulli's principle, as water flows through a narrow section of the pipe, its speed increases and the pressure decreases.

Let's consider an example using a simple pipe:

Applications of Bernoulli equation

1. Airplane wing lift

The most exciting application of Bernoulli's principle is in airplane wings and how they generate lift. Airplane wings are designed so that air flows faster over the top of the wing than underneath. This difference in speed leads to different pressure levels, with less pressure above and more pressure below, which produces the lift force.

2. Venturi effect

The Venturi effect is another useful application of Bernoulli's equation. It occurs when a fluid flows through a narrowed section of a pipe, causing the velocity to increase while the pressure decreases. This principle is used in devices such as Venturi meters, which measure fluid flow rates.

3. The Coanda effect

The Coanda effect describes the tendency of a liquid jet to remain attached to a convex surface. Bernoulli's principle helps explain this effect. When a fluid flows over a curved surface, the difference in pressure keeps the fluid attached to the surface.

4. Carburetor in the engine

In an internal combustion engine, the carburetor mixes air with fuel vapor. The Venturi effect in the carburetor helps draw in fuel because the air pressure is reduced when the air flows through a narrow section. This principle allows for a balanced fuel and air mixture, improving engine performance.

5. Atomizer

Gold, perfume and fuel atomizers all use Bernoulli's principle, a low-pressure area created when air passes rapidly through a narrow tube. This drop in pressure forces the liquid to rise and mix with the air, creating a spray.

Practical exercises

Consider doing a simple experiment to see Bernoulli's principle. You will need a piece of paper, some straws and water. Try blowing through the straw onto the surface of the paper or water and watch how the paper rises or the water sprays. This demonstrates the lower pressure and resulting lift of fast moving air.

Mathematical example

Example 1: Water flow

Suppose water flows in a pipe from a height of h1 = 10 m to a height of h2 = 5 m. If it is initially flowing at a velocity of v1 = 2 m per second, what is v2 at the lower height?

P1 + 0.5ρv1² + ρgh1 = P2 + 0.5ρv2² + ρgh2

Assume that P1 and P2 are atmospheric and cancel out:

0.5ρv1² + ρgh1 = 0.5ρv2² + ρgh2

Cancel ρ and solve for v2 :

0.5(2²) + 9.81 * 10 = 0.5v2² + 9.81 * 5

Calculation:

0.5 * 4 + 98.1 = 0.5v2² + 49.05

2 + 98.1 = 0.5v2² + 49.05

51.05 = 0.5v2²

v2² = 102.10

v2 ≈ 10.1 m/s

Example 2: Venturi meter

Calculate the flow rate through a venturi meter with cross-sectional areas A1 = 0.1 m² and A2 = 0.05 m², and pressure difference ΔP = 500 Pa.

q = A1*A2 * sqrt((2*ΔP)/(ρ(A1²-A2²)))

Let the fluid density be ρ = 1000 kg/m³:

q = (0.1*0.05) * sqrt((2*500)/(1000*(0.1²-0.05²)))

q = 0.005 * sqrt((1000)/(1000*0.0075))

q = 0.005 * sqrt(133.33)

q ≈ 0.058 m³/s

Conclusion

Bernoulli's equation serves as the cornerstone for understanding fluid dynamics. Its many applications demonstrate the beauty of physics in explaining everyday phenomena. From the lift of airplane wings to the precision of medical instruments, Bernoulli's principle plays a vital role. Understanding this principle not only increases our understanding of natural phenomena but also drives innovation in engineering and technology.

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