22.3. Applications of Bernoulli's Equations

The Bernoulli’s equation is another conclusion by Bernoulli is used in different fluid flow problems. In short, I can tell you that these equations help us to solve many fluid flow problems by considering energy considerations or a linear momentum equation.

Bernoulli's equation is an important tool in fluid mechanics, allowing us to analyze fluid flow situations by applying the principles of energy conservation. Essentially, it is based on how energy transforms in a moving fluid, and how differences in speed and pressure affect it. This fundamental equation helps predict how fluids behave in various scenarios, simplifying complex flow problems.

Imagine riding a bike down a hill. As you go down, you gain speed (kinetic energy) and your potential energy decreases. Similarly, in fluid flow, when the height of the fluid decreases, its speed increases, demonstrating energy conservation in action.

As you know, we have now the fluid mechanics solvers like computational fluid dynamics, a lot of things have improved, improved mechanics in terms of computational fluid dynamics, in terms of experiments. But whoever the fluid mechanics specialist first looks at two conservation principles: one is mass conservation, and second, he uses the Bernoulli equations to verify the results.

In applying Bernoulli's equations, certain assumptions must be made to validate the results. Primarily, fluid flow is assumed to be steady, incompressible, and frictionless, which means that the fluid density remains constant, and there are no energy losses due to friction between layers of fluid. These assumptions are critical for applying Bernoulli's principle in practical scenarios, such as when using computational fluid dynamics to model flow conditions.

Consider a smooth water slide. When you slide down, the water flows effortlessly and consistently (steady flow), and you do not compress the water (incompressible). If there were friction from the slide, it would slow you down, similar to how friction in a fluid flow impacts energy conservation in Bernoulli’s equation.

If you look at the Euler and Bernoulli both they tried to look at what could be the relationship between the speed at which blood flows and its pressure. Very simple way you know it, at the very beginning of your 12th class that the equations has three components: total energy has three components. One is the flow energy component, another is the kinetic energy component.

The relationship between blood flow speed and pressure is a practical application of Bernoulli's principle. According to Bernoulli’s equation, as the velocity of blood (or any fluid) increases, the pressure within that fluid decreases. This principle can be observed in medical scenarios, especially in diagnosing conditions related to blood circulation. The total energy in flowing blood is composed of dynamic pressure (related to its kinetic energy), static pressure, and gravitational potential energy.

Think of how a garden hose works. When you partially cover the end of the hose with your thumb, the water flows out faster (increased velocity) while simultaneously reducing pressure. This direct relation between flow speed and pressure is much like how blood circulates in our bodies.

Most often, I can go back to the cyclonic disasters. Whenever you have the wind having a velocity of 100 kilometers per hour, as soon as it comes, you will have two streamlines of this. The velocity decreases below this point, the pressure increases as compared to that.

In disaster scenarios like cyclones, Bernoulli's principle explains how pressure changes can lead to significant structural problems, such as roofs being blown off. When wind speed increases, the pressure in its path decreases dramatically. The difference between high pressure inside a structure and lower pressure outside can create forces strong enough to lift roofs off buildings.

Imagine blowing across the top of a piece of paper. The paper rises due to the subtler differences in pressure created by the increased wind speed above the paper. Similarly, in cyclonic winds, structures face a drastic drop in pressure that can lift and even remove roofs.

Now, let us look at that I want to compute the air speed. This is what the streamline hitting this. If I apply this, because it is nearly horizontal.

In engineering applications, Bernoulli’s equation can help calculate various parameters such as air speed over wings of aircraft. By analyzing pressure changes at different points and applying Bernoulli's principle, engineers can derive important details about lift and drag forces affecting a body in motion through a fluid.

Think about how airplanes fly. The wings are designed to create a difference in pressure above and below them. As air moves faster over the curved top of the wing, it creates lower pressure compared to the slower-moving air beneath the wing, producing lift. This application of Bernoulli's equation is crucial for flight.

This is what the verification of Bernoulli equations in fluid mechanics lab in IIT, Guwahati. If you look at these ones, it has two tubes to maintain the contents in supply heads.

In a laboratory setting, Bernoulli's equation can be verified through experiments like the one conducted with venturi meters. These devices measure the difference in fluid velocity and pressure in a controlled setup. By ensuring that total energy remains constant, despite variations in pressure and speed at different points along the pipe, students are able to observe Bernoulli's principles in action.

Think of it like measuring how water flows in a garden stream. By putting various sized rocks in the stream, you can see how water speeds up and slows down depending on the flow path, demonstrating concepts analogous to Bernoulli's equation.

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