4.1. Mechanical energy conservation

The Bernoulli equation is a statement of conservation of mechanical energy:

$\frac{p_1}{\gamma} + gz + \frac{V^2}{2} = C$

The Bernoulli equation expresses that the sum of pressure energy, potential energy, and kinetic energy per unit weight of fluid remains constant along a streamline. Here, $p$ represents pressure, $gz$ represents potential energy due to elevation, and $V^2/2$ is the kinetic energy term. When we consider a fluid moving through a pipe or around an obstacle, these three forms of energy can transform into one another without any loss, provided the flow is steady and frictionless.

Think of the Bernoulli equation like a roller coaster. As the coaster climbs to a peak, its speed decreases, and its potential energy rises. As it zooms down, that potential energy converts to kinetic energy, making the coaster speed up again. The total energy (which includes both forms) remains constant, just like the total energy in the Bernoulli equation.

Each term in the Bernoulli equation represents a type of energy:

• $\frac{p}{\gamma}$ is called pressure head,
• $z$ is the elevation head,
• $\frac{V^2}{2g}$ is the velocity head.

In fluid mechanics, the terms in the Bernoulli equation have specific meanings. The pressure head $\frac{p}{\gamma}$ tells us how high a column of the fluid would rise based on its pressure. The elevation head $z$ indicates the height in gravitational field relative to a reference point. The velocity head $\frac{V^2}{2g}$ indicates how much energy the fluid possesses due to its motion at that point. Together, these terms encompass the mechanical energy present in the fluid system.

Imagine a lake. The water has pressure due to its weight (pressure head), it sits at a certain height above sea level (elevation head), and it flows to a river where you can see how fast it travels (velocity head). All parts of this energetic system work together to determine behaviors like how fast water will flow out of a dam.

Bernoulli’s equation comes with certain assumptions: flow is frictionless, steady, and incompressible.

When using the Bernoulli equation, we assume that the fluid is incompressible (its density doesn't change regardless of pressure), the flow is steady (conditions don't change over time), and there are no friction losses due to viscosity. These assumptions help simplify calculations but mean that Bernoulli's equation may not hold under all conditions, particularly in real-world scenarios where factors like turbulence or compressibility affect flow.

Think of smooth, clear water flowing smoothly through a straight, wide straw - that’s the ideal, frictionless flow we think of when applying Bernoulli's equation. Now, imagine trying to sip a thick milkshake through a narrow straw; that's a scenario where friction and changing properties come into play, and the ideal conditions of Bernoulli's assumptions don't apply.

Bernoulli's equation can be applied between two points along a streamline to predict changes in pressure, velocity, and elevation.

In practice, we can use Bernoulli's equation to analyze how the speed and pressure of a fluid change as it moves along a streamline from point 1 to point 2. For example, if the flow enters a narrower section of a pipe, according to Bernoulli's principle, the speed must increase and pressure must decrease in that section, since energy must be conserved.

Consider a garden hose. If you partially cover the end with your thumb, the water speed increases dramatically, indicating it is moving faster but with lower pressure than when it was fully open. This simple action demonstrates how Bernoulli's equation shows the relationship between speed and pressure in the flow of fluid.

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