Bernoulli's Principle — Definition

Definition

Imagine water flowing through a pipe. If the pipe narrows, the water has to speed up to get through the constriction. What happens to the pressure inside the water at that narrower, faster-moving section?

Intuitively, one might think the pressure increases because the water is being 'squeezed.' However, Bernoulli's Principle tells us the opposite: where the fluid speed is higher, the pressure is actually lower.

This might seem counter-intuitive at first, but it's a direct consequence of the conservation of energy.

Let's break it down. Bernoulli's Principle applies to what we call an 'ideal fluid.' An ideal fluid is a theoretical concept that helps us simplify complex fluid behavior. It's assumed to be incompressible (its density doesn't change), non-viscous (it has no internal friction, so it flows smoothly without energy loss due to stickiness), and its flow is steady (the velocity at any point doesn't change over time) and irrotational (it doesn't swirl or rotate about its own axis).

While no real fluid is perfectly ideal, many fluids under certain conditions behave closely enough for Bernoulli's principle to be a very useful approximation.

The principle states that for such an ideal fluid flowing along a streamline (an imaginary line tracing the path of a fluid particle), the sum of three types of energy per unit volume remains constant. These three types are:

1
Pressure Energy: — This is the energy associated with the pressure exerted by the fluid. Higher pressure means more energy.
2
Kinetic Energy: — This is the energy due to the fluid's motion. Faster-moving fluid has more kinetic energy.
3
Potential Energy: — This is the energy due to the fluid's height or elevation in a gravitational field. Higher fluid has more potential energy.

So, Bernoulli's equation mathematically expresses this as $P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$, where $P$ is the static pressure, $\rho$ is the fluid density, $v$ is the fluid velocity, $g$ is the acceleration due to gravity, and $h$ is the height.

If one term increases, at least one of the others must decrease to keep the sum constant. For instance, if the fluid speeds up (increasing kinetic energy term), and its height doesn't change much, then the pressure term must decrease.

This is why a fast-moving fluid exerts less pressure. This principle explains phenomena from why airplanes fly to how a perfume atomizer works, making it a cornerstone of fluid dynamics.