9.4. BERNOULLI’S PRINCIPLE

Fluid flow is a complex phenomenon. But we can obtain some useful properties for steady or streamline flows using the conservation of energy.

In fluid dynamics, understanding how fluids behave during flow is essential. Bernoulli's Principle provides insights into the energy transformations that occur in flowing fluids. Specifically, it relates pressure, velocity, and elevation in a fluid flowing along a streamline. This principle is fundamental in describing how different parts of the fluid move and interact under various conditions.

Think about water flowing through a garden hose. When you place your thumb over part of the opening, the water jets out faster. This is a practical example of Bernoulli's Principle, where a decrease in pressure (due to the reduced area) leads to an increase in the water's velocity.

Consider a fluid moving in a pipe of varying cross-sectional area. Let the pipe be at varying heights as shown in Fig. 9.9. We now suppose that an incompressible fluid is flowing through the pipe in a steady flow.

When a fluid flows through a pipe with changes in diameter and height, its velocity changes accordingly. According to Bernoulli's Equation, the pressure exerted by the fluid will also vary. If the fluid moves from a wide area to a narrow area, its speed increases, causing its pressure to decrease, and vice versa. This relationship can be mathematically expressed in Bernoulli's Equation, showcasing the conservation of energy in fluid flow.

Imagine running water through a straw that narrows at one point. As the water hits the narrow section, it speeds up. This is similar to how pressure decreases in areas where fluid moves faster, demonstrating Bernoulli's principle precisely.

Bernoulli's equation is a general expression that relates the pressure difference between two points in a pipe to both velocity changes (kinetic energy change) and elevation (height) changes (potential energy change).

Bernoulli's equation summarizes the conservation of mechanical energy for fluid flows in a pipe. It states that as the fluid moves along a streamline, the total energy - which includes pressure energy, kinetic energy due to velocity, and potential energy due to height - remains constant. This equation is crucial for engineers and scientists in designing systems where fluid flow is involved.

Consider a roller coaster. As it rises to the highest point, kinetic energy (speed) is low, but potential energy (height) is high. As it descends, potential energy converts into kinetic energy, and the ride gains speed. Similarly, in a flowing fluid, changes in height and speed correspond to shifts in pressure.

Note that in applying the energy conservation principle, there is an assumption that no energy is lost due to friction. But in fact, when fluids flow, some energy does get lost due to internal friction.

Bernoulli's Principle assumes ideal conditions for its application. In real-life situations, friction and viscosity in fluids can lead to energy losses. These factors can cause discrepancies from the predictions made by Bernoulli’s equation. Therefore, while it offers a solid foundation for understanding fluid dynamics, its ideal application might not always occur in practical scenarios.

Imagine riding a bike on a smooth road versus a bumpy one. On the smooth road, you glide effortlessly, showing minimal energy loss. On the bumpy road, your speed fluctuates due to the bumps, causing energy loss. Similarly, flow in a smooth pipe reflects Bernoulli's principles better than flow in pipes with many bends and rough surfaces, where energy loss is significant.

The speed of efflux from an open tank is given by a formula identical to that of a freely falling body.

Torricelli's Law states that the speed at which a fluid exits an opening is directly related to the height of the fluid above the hole. This principle shows how gravity acts on the fluid, akin to an object in free fall. Hence, the greater the height of the fluid, the faster it flows out.

Think of water pouring from a bucket. The deeper the water in the bucket, the faster water splashes out when you pull the plug. This is similar to how the height of fluid above a hole influences the speed at which it flows out.

Dynamic lift is the force that acts on a body, such as an airplane wing, a hydrofoil or a spinning ball, by virtue of its motion through a fluid.

Dynamic lift results from the variation in fluid speed and pressure around a moving object. This principle collaborates with Bernoulli’s equation, helping explain how wings lift an airplane off the ground by creating lower pressure on top of the wing than underneath it.

Picture a bird flapping its wings. As it swoops and moves forward, the air speeds up over the top of its wings, lowering the pressure and generating lift, which allows it to soar into the skies. This is the same principle that allows airplanes to fly, showcasing dynamic lift in action.

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