23.3.2. Fluid Flow Analysis Techniques

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 this chunk, we are introduced to the concept of visualizing fluid flow using imaginary 'virtual balls' of fluid. As these virtual balls move through the fluid, we can observe how pressure acts on them. By considering many fluid balls instead of just one, we can analyze the flow energy, kinetic energy, and potential energy in a more comprehensive way. The formula for flow energy is given by pressure multiplied by area and a small displacement, which connects the physical properties of the fluid to its motion.

Imagine a crowded subway train where each passenger represents a 'virtual ball' of fluid. The pressure of the crowd pushes each passenger toward the exits during an emergency. Just like in fluid dynamics, where many interconnected fluid balls (passengers) affect the overall flow, understanding how each individual (ball) moves helps us comprehend the flow of the entire train (fluid) through the urban landscape.

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. 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. If I know any of them, then you know I can solve the problems. That is the idea.

This chunk emphasizes the importance of visualizing fluid movement through streamlines when using Bernoulli's equation. Streamlines help in understanding how pressure and velocity change within the fluid. Knowing the properties at different points enables us to effectively apply Bernoulli’s principle to solve flow problems. It suggests that drawing a streamline diagram allows for a better comprehension of the flow and the dynamics involved.

Consider the example of wind blowing through trees. The paths that leaves take as they flutter is akin to streamlines in fluid dynamics. By observing how leaves (representing fluid particles) move and change direction, we can predict airflow around obstacles, just as we visualize fluid flowing around objects using streamlines.

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. Applications of Bernoulli equations is too easy, people do so often it is misused.

In this chunk, we explore the limitations associated with using Bernoulli's equation. It can be used for unsteady flows, but most derivations are based on steady flows where time does not affect the situation. As a commonly used tool, it is often misapplied. The chunk highlights the importance of understanding when it is appropriate to apply this equation, ensuring that the principles of mass and energy conservation are respected.

Think of navigating a bike down a steep hill. The bike can move under steady conditions, but if you suddenly start pedaling harder (unsteady flow), the equations governing your speed change. Misapplying a steady-state equation while pedaling faster could lead to incorrect predictions about how fast you'll reach the bottom. Just like you should consider the changing conditions when biking, we must remember Bernoulli's limitations when analyzing fluid flows.

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. 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.

This chunk discusses key assumptions critical for the application of Bernoulli’s equation, such as incompressible flow and frictionless conditions. It indicates that for most engineering problems, flows with low Mach numbers are treated as incompressible because the density changes are negligible. Additionally, it observes that Bernoulli's equation cannot be applied near solid surfaces due to shear stresses and velocity gradients that introduce frictional forces.

Imagine trying to slide a hand across a smooth glass surface versus a rough one. The smooth surface represents a frictionless flow where Bernoulli's equations can easily apply. However, if you attempt to slide your hand across a rough surface (like a conveyor belt), the friction makes it complicated. Here the rough surface illustrates conditions where Bernoulli’s assumptions break down and lead to inaccuracies.

So, that is the reasons Bernoulli equations has a lot of advantages. 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.

Finally, this chunk summarizes the practical advantages of using Bernoulli's equation, stressing the need to assess friction effects before application. It reaffirms the necessity of drawing streamlines to ensure accurate application and the justification of irrotational flow conditions. Understanding these steps can lead to successfully applying Bernoulli’s principles to real-life problems.

Think of planning a route for a road trip. Before you set off, you’d analyze the map for smooth routes (streamlines) to avoid heavy traffic (friction). Just like selecting the best path for driving, drawing streamlines and assessing flow conditions ensure the best outcomes in fluid dynamics with Bernoulli's equation.

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