6. Applications of Bernoulli's equation

So, 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 based on several critical assumptions. Firstly, it assumes that the flow of the fluid is frictionless, which means there are no viscous forces acting on the fluid particles. Secondly, the flow is steady which implies that the fluid properties at a point do not change over time. Thirdly, it assumes a constant density, indicating that the fluid is incompressible, which is typically true for liquids. Finally, Bernoulli’s equation is applied along a streamline, the path followed by a fluid particle in motion.

Think of water flowing through a garden hose. If the water is flowing steadily and without any obstructions, we can assume the same principles of Bernoulli's equation apply, provided the water's density doesn't change significantly while flowing through that hose.

Now, can we eliminate the constant in Bernoulli’s equation? Yes, if we apply the Bernoulli’s equation at 2 points along a streamline, this is important so, the streamline should be the same and if we apply this equation along two streamline, I mean, to 2 points along a streamline the constant would be the same and therefore, it can be eliminated.

When applying Bernoulli's equation to two points along the same streamline, any constant term in the equation can be eliminated. This is because the constant represents the energy level of the fluid system, which remains unchanged between those two points. By knowing the energy expressions at those points, we can find relationships between pressure, height, and velocity.

Imagine a slide at a water park. If you start at the top (with potential energy) and slide down, your energy converts from potential to kinetic. If you take measurements at two points along the slide, you can calculate speeds and heights without knowing the exact total energy.

Now, the Bernoulli equation is a statement of conservation of mechanical energy. So, you see this is potential energy. So, let me take the eraser. And this is let it this is kinetic energy.

Bernoulli's equation essentially reflects the conservation of mechanical energy in flowing fluids. It combines three main types of energy associated with fluid flow: potential energy (due to height), kinetic energy (due to the speed of the fluid), and pressure energy (due to the fluid's pressure). The equation shows that the sum of these energies remains constant along a streamline, indicating that a decrease in one form of energy can lead to an increase in another.

Consider a roller coaster. As the coaster climbs up a hill, it has maximum potential energy and minimum kinetic energy. As it descends, the potential energy converts to kinetic energy, making the coaster speed up. Similarly, as fluid flows through different elevations or constrictions, its energy transforms between pressure, height, and speed based on Bernoulli's principle.

Now, proceeding forward first simple case, very simple case, where V = 0. So, there is point 1 and there is point 2, z direction is shown as above, you know, this is pressure datum and this is elevation datum. Reservoir means V = 0.

In this example, we consider a fluid that is flowing from a reservoir into the atmosphere as a free jet. When the fluid is at rest at the surface of the reservoir, we assume its velocity (V) is zero. The Bernoulli equation can then be simplified to relate the pressures and elevations at different points as it exits the reservoir, allowing us to calculate changes in height and speed as the fluid flows.

Picture filling a container with water. If you poke a hole in the container, the water will shoot out with a certain speed determined by the height of the water above the hole. This is a real-world application of Bernoulli’s equation at play, where the water's potential energy is converted into kinetic energy as it flows out.

So, another application of Bernoulli equation is the stagnation tube. What happens when the water starts flowing in the channel? For example, this and so, the equation along the stream line by Bernoulli equation will be can be given as p / gamma + z + V square / 2g = constant.

The pitot tube is a device used to measure fluid velocity by utilizing Bernoulli's equation. It consists of two tubes, one that measures the total pressure (stagnation pressure) and another that measures static pressure. By calculating the difference between these pressures, we can determine the fluid velocity at that point using Bernoulli’s principle.

Imagine the pitot tube as a specially designed thermometer for measuring how fast the wind is blowing. Just like a thermometer measures temperature based on how much mercury rises, a pitot tube measures airspeed by comparing pressures. In aircraft, this helps pilots understand how quickly they are flying through the air.

One of the examples as I said, we had not seen was venturi meter, how would you find the flow Q given the pressure drop between point 1 and 2 and the diameter of the 2 section?

A Venturi meter is a device shaped like a tube that narrows in the middle. As fluid flows through, it speeds up in the narrow section, causing a pressure drop according to Bernoulli's equation. By measuring this pressure change and knowing the cross-sectional areas of the tube, we can calculate the flow rate of the fluid.

Think of a garden hose with your finger pressed over the nozzle. When you partly block the nozzle, the water speeds up and sprays out further but with less pressure in the hose. Similarly, a Venturi meter uses changes in speed and pressure in fluid dynamics to measure how much fluid is flowing through it, much like using your finger to control water flow.

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