3.1. Frictionless and steady flow

Now, 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.

Bernoulli's equation is derived under specific conditions. First, the flow must be frictionless; this means there are no energy losses due to viscosity. Second, it must be steady; the fluid's velocity at any point does not change over time. Also, Bernoulli’s equation assumes constant density, which implies that the fluid is incompressible—this is valid for liquids under typical conditions. Lastly, the analysis is conducted along a streamline, emphasizing that the fluid's behavior is consistent along that path.

Imagine riding a bicycle down a smooth, straight road. If the road is frictionless, you can keep moving without needing to pedal as hard. Similarly, in fluid flow, if there is no friction (like viscosity in the fluid), the flow can continue smoothly and steadily.

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 at two distinct points along the same streamline, the constant remains unchanged. This allows you to analyze the differences in pressure, elevation, and velocity between the two points. By setting both equations equal to each other and considering that 'C' cancels out, you can deduce relationships that govern fluid motion between these points, leading to insights such as how pressure changes with changes in elevation and velocity.

Think of it like comparing two points on a slide. If you slide down from a higher point (where you have more potential energy) to a lower one, the speed at the bottom will increase. Here, the energy at the top is transformed into speed at the bottom, representing the constant energy principle within Bernoulli's framework.

Bernoulli's equation states conservation of mechanical energy: p1/γ + z + V²/2g = C. This includes pressure energy (p/γ), potential energy (z), and kinetic energy (V²/2g), which together represent systemic energy within the fluid.

Bernoulli's equation decomposes the energy of a fluid into three components. The first, pressure energy, indicates how much work the fluid can perform per unit volume due to pressure. The second, potential energy, relates to the height of the fluid relative to a reference point. The last, kinetic energy, relates to the speed of the fluid's motion. The interesting part is that when you sum these energies, they remain constant along a streamline, highlighting energy conservation principles in fluid dynamics.

Consider a water fountain. The water at the bottom has pressure energy from the pump; as it rises, potential energy increases due to height while kinetic energy changes as it flows. In an efficient fountain, all forms of energy contribute to keeping the water flowing, illustrating the balance described by Bernoulli's equation.

Bernoulli’s equation has numerous applications, such as in pitot tubes, orifices, and Venturi meters, which measure flow rates and pressures in pipes and channels.

Bernoulli’s principles help understand how fluids behave under various conditions, allowing the design and analysis of devices used in engineering. Pitot tubes measure the speed of aircraft and fluids in pipes. Orifices and Venturi meters help quantify flow rates based on pressure changes as fluids move through constricted areas. Each device showcases real-world applications of Bernoulli’s equation, using the principles explored in the theoretical framework to achieve practical outcomes.

Think of a garden hose. When you partially cover the nozzle with your thumb, the water shoots out faster due to the change in pressure and area. This is akin to how Venturi meters operate, demonstrating Bernoulli’s principles in action by illustrating how flow rate and pressure interact in everyday experiences.

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