3. Assumptions of Bernoulli's equation

Now, the Bernoulli equation, the assumptions that are needed for Bernoulli’s equation, what have we assumed, the flow was frictionless, the flow was steady, the final equation that we have derived we have assumed, constant density that means the flow was incompressible and we have done it for along a streamline.

The Bernoulli equation is built on several key assumptions. The first assumption is that the flow of the fluid is frictionless, meaning there are no energy losses due to viscosity within the fluid. This condition simplifies the calculations as it enables us to apply the conservation of energy principles directly without accounting for losses. Additionally, the flow must be steady, indicating that the fluid properties at any given point do not change over time. Finally, we assume constant density, which implies that the fluid is incompressible, meaning changes in pressure do not significantly affect the fluid's density, which is particularly valid for liquids under normal conditions. All of this is applied along the same streamline where the effects can be uniformly measured.

Think of a perfect water slide where the water flows smoothly without any friction. In reality, water experiences some resistance due to friction, but if you imagine that there is no friction, you can understand why the equations simplify. Just as a frictionless slide allows the rider to travel without slowing down, a frictionless flow allows us to predict how fluid moves without losing energy.

Bernoulli's equation along a streamline can be applied when the fluid is flowing along a single streamline, ensuring the assumptions hold for that specific path.

When using Bernoulli's equation, it is important to apply it along a single streamline rather than across different streamlines. This is because each streamline may have different fluid properties. If two points are taken on the same streamline, we can share constants between them, allowing us to derive useful equations from the assumptions of frictionless flow, steady state, and incompressible density. However, if we stray from one streamline to another, the outcomes may change due to varying pressures and velocities, which would invalidate the assumptions.

Imagine a roller coaster where the ride strictly follows one track. If the coaster goes from one point on the track to another, we can predict its speed and height along that path. However, if it suddenly left the track and tried to roll onto a different path, the physics would change, making it much harder to predict what would happen next. In this analogy, staying on the same streamline ensures reliable predictions using Bernoulli’s equation.

Bernoulli's equation is essentially a statement of conservation of mechanical energy which incorporates pressure, kinetic energy as velocity, and potential energy as elevation.

Bernoulli’s equation relates the pressure, kinetic energy, and potential energy along a streamline to the concept of energy conservation. It states that the total energy per unit volume in a moving fluid remains constant, which can be articulated mathematically as: p/γ + z + V²/2g = constant. Here, p/γ represents the pressure head, z is the elevation head, and V²/2g is the velocity head. This relationship indicates that at points where the fluid velocity is high, the pressure will be lower, and vice versa, effectively describing how energy is distributed in a moving fluid.

Consider a garden hose with a nozzle at the end. When you squeeze the nozzle, the water shoots out much faster, indicating that the pressure inside the hose falls. This is a direct application of Bernoulli's principle as it beautifully illustrates how energy is redistributed: at high speeds (kinetic), the pressure drops (potential) while the elevation remains relatively constant.

In order to use Bernoulli's equation, we assume that the fluid has constant density, which means that it is incompressible.

The assumption of constant density is crucial when applying Bernoulli’s equation. Incompressible flow supposes that the fluid density remains constant despite changes in pressure or velocity. This makes calculations simpler and holds true for liquids under most conditions, as they are generally not sensitive to pressure changes. However, for gases, particularly under high-pressure or high-temperature conditions, this assumption may not apply, necessitating more complex equations to describe their behavior.

Imagine a basketball filled with air. If you squeeze it, not much happens to its volume, so you can think of it as incompressible under normal conditions. However, if you were to take that basketball to significantly high altitudes or depths under water, its volume would change, reflecting compressibility. This example highlights that while liquids are often treated as incompressible in fluid dynamics, gases require careful consideration of density changes.

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