23.2.4. Significant Effects near Solids

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. That is what by mg, that weight of the fluid, that is what will give is on this. 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.

In fluid mechanics, we can think about fluid movement in terms of virtual 'fluid balls.' These are just conceptual tools to help us understand how fluids behave. Imagine there are many small balls of fluid, each affected by pressure as they move along different paths. The energy of these fluid balls can be categorized into three types - flow energy, kinetic energy, and potential energy. Each of these types can be expressed per unit weight of the fluid.

Think of an amusement park ride where people in small cars go up and down hills. The cars are like our 'fluid balls' and their movement involves energy transitions. When they go up a hill, they gain potential energy; when they go down, that energy converts to kinetic energy. Just like these cars, fluid balls move and change energy as they navigate through various flow paths.

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. If I consider a balls are moving, a virtual fluid balls are moving it. If I draw the streamlines, I can apply the Bernoulli equations. I should know, the pressure variability, the pressure at the two points or the pressure and velocity.

When using the Bernoulli equation, a key step is to visualize the path of the fluid. This is done by drawing streamlines, which depict the direction of the fluid's motion at various points. By understanding how fluid moves along these streamlines, we can determine changes in pressure and velocity at different locations. Knowledge of these parameters is crucial for solving problems related to fluid dynamics.

Imagine a water slide at a water park. The slide has curves and drops that represent the fluid's streamlines. As a person goes down the slide, they speed up at certain points and slow down at others depending on the steepness. Similarly, when fluid flows through a pipe, its speed and pressure will change based on the shape of the pipe, and we can analyze those changes using Bernoulli's principles.

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. There is two solutions are available for us, I can say it is one mass conservations and energy conservations.

While the Bernoulli equation is a powerful tool in analyzing fluid dynamics, it has limitations. It primarily applies to steady flow situations, where conditions do not change over time. When using this equation, one must be cautious as it is commonly misapplied. Understanding both mass and energy conservation principles is essential to apply Bernoulli's theorem correctly.

Consider a car driving on a smooth highway. If the car moves at a constant speed (steady flow), we can predict its behavior easily. However, if the car starts to speed up and slow down due to traffic (unsteady flow), the simple rules we developed won't apply as straightforwardly, just like how Bernoulli's equation is limited in its application.

Applications of Bernoulli equations is too easy, people do so often it is misused. Another is that the incompressible flow, we discussed a lot of times. It is just we have to most of the fluid flow problems in civil engineering and mechanical engineering and others place where flow Mach number is less than 0.3, we can consider is a incompressible flow because the density variation will be less than 5%, which is we can neglect it.

Bernoulli’s equations apply effectively to incompressible flows, typically where the Mach number is below 0.3. This is because at low speeds, variations in fluid density are minimal and can be ignored. In many engineering applications, this simplification leads to easier calculations and analysis of fluid behavior.

Think of a soft drink in a bottle. If you shake the bottle (create pressure), the liquid remains mostly unchanged; it doesn’t compress much. This is like incompressible flow where density changes are negligible. In civil engineering, when analyzing water flow in pipes, we treat the water as incompressible, making calculations much simpler.

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. So, those reasons we cannot apply it.

The Bernoulli equation assumes a frictionless flow, which is unrealistic near solid boundaries. When fluid flows close to a solid surface, frictional effects, like shear stress, come into play, altering flow patterns and velocities. Thus, applications of Bernoulli's equation in these regions become invalid.

Think about trying to slide down a slide that is covered in thick syrup. You would move slowly or even get stuck because of the friction. Similarly, fluid moving near solid surfaces faces friction that affects its flow, making Bernoulli predictions inaccurate in those conditions.

So, whenever you apply the Bernoulli equations, you have to first look it whether frictional effect is significant or not significant. And it is applied along a streamline. That means, before applying that, we should draw a streamline, then we apply the Bernoulli equation at the two points, or you should justify the flow is irrotational, okay. That means flow is not rotational. There is no vorticity.

Before applying the Bernoulli equation, it is crucial to analyze the flow for friction effects and to determine if the flow is irrotational. Drawing streamlines helps visualize flow patterns and ensures accurate application of Bernoulli's principles. An irrotational flow means there is no swirling or rotational movement of the fluid, which allows simplifications in calculations.

Imagine navigating a river on a kayak. If you move with the current (no rotations), you can predict your path easily. But if you start spinning around (vorticity), it becomes difficult to determine where you'll end up. Streamlines in fluid mechanics serve a similar purpose, helping predict fluid behavior.

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