2. Bernoulli's equation

Welcome back in this lecture we are going to start elementary fluid dynamics or more commonly Bernoulli's equation. So, we are going to cover some basic derivation in this regard and then we proceed further to some of the solved examples.

In this portion, the lecture is introducing Bernoulli's equation, a fundamental principle in fluid dynamics. It indicates that the focus will be on deriving the equation and working through practical examples to help students understand its applications.

Think of Bernoulli's equation as the 'balance of flow' in a system. Just as a tightrope walker must balance to stay upright, fluids have to balance pressure, velocity, and height to maintain flow in pipes.

So, we are going to see the Bernoulli along a streamline... There is no shear forces there the flow must be frictionless. This is also a steady state so, there is no change in p with respect to time, p is pressure.

In this part, the equation is derived based on the conditions of flow along a streamline. For Bernoulli's equation to apply, the flow must be steady, without friction or shear forces, meaning it needs to be idealized. Here, pressure is represented and comes into play as constant for the derivation.

Imagine water flowing smoothly through a straight pipe without any bumps or turns. This scenario represents ideal flow—where water moves without distractions, allowing us to predict its behavior using Bernoulli's equation.

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...

This section summarizes the key assumptions underlying Bernoulli's equation: the flow must be frictionless, steady, and incompressible—meaning the density of the fluid does not change. These assumptions allow the equation to simplify complex fluid dynamics into manageable calculations.

Think of riding a bicycle on a smooth and flat road. The ride is easy and predictable (steady flow) compared to riding over a bumpy or steep surface where things get unpredictable and chaotic.

Bernoulli equation... p1/γ + gz + V²/2g = C, where p/γ is the pressure head, z is the elevation head and V²/2g is the velocity head...

In this portion, the components of the equation are explained: pressure head, elevation head, and velocity head. Each component represents a form of energy within the fluid system. The sum of these energies remains constant along a streamline, embodying the principle of conservation of energy.

Consider a seesaw: when one end goes up (elevation increases), the other must go down (pressure or velocity decreases), showcasing how energy is conserved in the system.

So, proceeding forward, so, we take another simple case, where pressure is 0 or constant... Thus, we can write z1 + V1²/2g = z2 + V2²/2g...

Practical applications of Bernoulli's equation are illustrated with examples, helping solidify understanding of how the theoretical equation applies to real-world scenarios. In this instance, it depicts situations where pressure changes or remains constant while tracking fluid elevation and velocity.

Imagine an amusement park ride: as you go higher up in the ride (elevation increases), the speed (velocity) might decrease, mimicking how energy exchange occurs through the Bernoulli principle.

One of the other applications of Bernoulli equation is the stagnation tube... this is used to measure flow velocities in an airplane...

This section discusses how Bernoulli's equation aids in measuring flow velocities, specifically through instruments like the stagnation tube and Pitot tube. These devices help translate fluid velocity into measurable physical quantities, important in fields like aerodynamics.

Think of the Pitot tube as a water slide: it helps us measure how fast the water flows at a given point, akin to how a speedometer shows how fast your car is moving at any moment.

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