23.4.2. Applying Continuity and Bernoulli's Equation

Similar way, the potential energy by this. So, if you it, instead of understanding or deriving along the streamlines, the same concept we can visualize it, if a virtual fluid balls is moving from one location to two locations, since it is a virtual fluid balls, again I am to talk about these, where we consider it is not a one fluid flow ball movements, we consider there are n number of fluid balls are there. They are having a pressure exiting one by others. Because of that, there will be a flow energy, which we quantify into pressure into area into delta x.

In fluid dynamics, we can visualize fluids as 'virtual fluid balls' that travel along streamlines. Each 'ball' represents a group of fluid particles interacting based on pressure differences. This interaction results in energy flow, which is defined by the equation: Flow Energy = Pressure × Area × Distance. This means that as pressure changes, the energy associated with the flow also changes, showcasing how fluids behave in various conditions.

Imagine a crowd of people (the fluid balls) moving through a narrow hallway (the streamline). The pressure with which they push against each other and the walls generates movement. If more people (higher pressure) enter the hallway, they move faster and create a flow of individuals much like how fluid movement is dictated by pressure differences.

So, we can say it, any fluid balls if you consider it, the flow energy per weight, the kinetic energy per weight, and the potential energy per weight, that is what is custom. So, this is the difference between a simple ball and the virtual fluid ball. So, what I am telling is that, whenever you apply the Bernoulli equation, you should draw the streamlines. You should visualize how the fluid moves.

Bernoulli's equation relates the flow energy, kinetic energy, and potential energy of fluid to how they change along the streamline. It emphasizes the importance of visualizing the flow by drawing streamlines. Each point along a streamline needs to be analyzed to accurately apply Bernoulli's equation and understand how fluid velocity and pressure interact at different locations.

Think of a water slide. As water flows down, it's similar to a 'virtual fluid ball'. The water at the highest point (potential energy) will eventually gain speed as it moves down (kinetic energy). If you visualize the slide as a streamline, you can predict how fast the water will be at various points along its path, which resembles the application of Bernoulli's equation.

Let us have a very quick, what are the limitations of Bernoulli equations. Bernoulli equations can be applied for unsteady flow, but the simplified derivations what you use it, those are for per steady flow. That means, there is no time component is there. And remember it, this equation is most frequently used, also misused equations, okay.

Bernoulli's equation has limitations. It is primarily derived for steady, incompressible, and frictionless flow. When applying it to unsteady flows, like in real-life conditions, one must be cautious. Misuse often occurs due to misunderstandings about when and where the equation applies, which might lead to incorrect analyses.

Imagine trying to use a detailed math formula for a specific scenario without considering the context. If you were to solve for the time it takes to finish a race at a steady pace, it wouldn’t work if the runner suddenly stopped then sprinted. Similarly, applying Bernoulli's equation without understanding its conditions can lead to errors in predicting how fluids behave.

The most important assumption is that the frictionless flow, that means it cannot be applied near to the solid. Because as you know it, whenever the fluid goes through near the solid, if there is a solid fixed surface, there will be the velocity gradient, there will be the shear stress acting on that. Viscous effect will come to pictures.

Bernoulli’s equation assumes that the flow is frictionless. However, near solid surfaces, friction and shear stress come into play, which complicates how fluids move. This means if a fluid is flowing close to a pipe wall or any surface, Bernoulli's equation cannot accurately describe the flow due to these effects.

Consider sliding a book across a table. The more surface area in contact (like fluid near a wall), the harder it is to slide due to friction. If you tried to calculate the motion of the book using an equation that doesn't account for friction, your results would be inaccurate. Similarly, Bernoulli’s equation needs to account for such friction in real-world fluid situations.

That is the reasons Bernoulli equations has a lot of advantages.

Despite its limitations, Bernoulli's equation has significant advantages in simplifying fluid flow analysis. It can easily account for energy losses or gains in different systems, making it versatile for engineers in various applications like piping systems, aerodynamics, and hydraulics.

Think about how engineers design roller coasters. They use concepts like Bernoulli's equation to predict how fast cars will go down slopes and how high they will climb safely. Understanding energy dynamics helps ensure thrilling but safe rides, showcasing the practical advantages of applying Bernoulli’s equation in transportation engineering.

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