2.1. Derivation along a streamline

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

In this introduction, the focus is on the transition from basic fluid mechanics to more complex topics, specifically Bernoulli’s equation. This equation is central to understanding fluid dynamics and is derived based on assumptions about fluid flow.

Think of flowing water in a river. Just like studying the smooth flow of currents will help understand water flow dynamics, learning about Bernoulli's equation provides insights into how fluids behave in different scenarios.

So, if you remember this equation we had got from in the last week's lecture that is on fluid statics. ... is normal to the flow that is our standard assumption.

The text introduces the concept of a streamline, which is an imaginary line that represents the flow of fluid. The coordinate system and the orientation of axes (i, j, k, s) are fundamental in determining how fluids behave under gravity and flow conditions.

Consider a skateboarder going down a ramp. The skateboarder’s path down the ramp follows a streamlined route, similar to how fluid follows a streamline in a current.

So, that is the component of g in s direction. ... is pressure.

This section discusses how acceleration due to gravity plays a role in fluid dynamics when analyzing flow along a streamline, while considering that the flow is steady (i.e., it doesn’t change over time) and frictionless.

Imagine a water slide; as you go down, the only forces acting are gravity pushing you down and the water slide shaping your path, much like gravity influences fluid along a streamline.

So, this equation that we have obtained... V del v del s.

The derivation progresses with a focus on changes in pressure (dp) and velocity (dv) along the streamline. Using calculus, these derivatives help define the relationship between pressure and velocity under the influence of gravity.

Think of a garden hose: as you pinch the end (simulating a change in area), the water (pressure) speed increases. This relationship is critical in understanding how fluids behave under different pressure conditions.

Now, we have to use force is equal to mass into acceleration along a streamline and integrate it.

By integrating the equation from the previous chunks, we derive Bernoulli's equation. This process involves understanding how pressure, density, and velocity interrelate over a streamline, illustrating the conservation of energy in a flowing fluid.

Imagine filling a balloon with water. The pressure inside at one point affects how the water flows to another point. Integrating these concepts shows how energy is conserved in fluid motion.

Now, the Bernoulli equation, the assumptions that are needed for Bernoulli’s equation... mechanical energy to thermal energy conversation, heat transfer or shaft work.

This chunk highlights critical assumptions for applying Bernoulli's equation: frictionless flow, steady state, and incompressible fluid. Understanding these assumptions is vital for correctly using Bernoulli's in real-world applications.

When driving a car, speeding on a frictionless road would give different results than on a bumpy, friction-filled highway. Likewise, Bernoulli's principles apply under ideal conditions for accuracy.

So we apply the Bernoulli’s equation between these 2 points. ... point 2 will be V = 2g.

This application demonstrates how Bernoulli's equation relates pressure to velocity in a fluid. For a situation involving a reservoir and an outlet, it shows how to calculate exit velocities and other important values.

Imagine measuring the speed of a water jet shooting out of a hose; understanding Bernoulli's helps predict how fast and far that water will travel based on pressure differences.

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