10.3.5. Momentum Components in Viscous Flow

Good morning all of you. Today we are going to discuss on nebustic locations and its approximation for simple flow problems. We will discuss that part. As we discussed in the last class how we can derive the basic Bernoulli's equations from Navier-Stokes equations that what we did it Navier-Stokes equations from that we derived Euler equations then we have derived Bernoulli's equations.

The introduction sets the stage for today's topic, which focuses on how to apply the Navier-Stokes equations to derive simpler forms, particularly the Bernoulli’s equations. It emphasizes the relevance of these equations in modeling viscous flows, which are critical in engineering applications. Understanding how to move from the complex Navier-Stokes equations to the simpler Bernoulli equations allows for practical calculations in fluid mechanics.

Imagine driving a car in traffic, where you need to navigate smoothly (Bernoulli's equations) versus driving through a construction site where you must deal with barriers and obstacles (Navier-Stokes equations). Just as you simplify your route when possible, engineers simplify equations to solve problems more efficiently.

So basically let us start with Navier-Stokes equations which is long back about 200 years back. So the equations what we derive in last two classes we will go more detail about that. Before going that let me I just write down the basics equations is the Euler equations okay.

This chunk introduces the fundamental equations that govern fluid flow, specifically focusing on the Navier-Stokes equations and their historical evolution. It briefly mentions the Euler equations, which are a simpler form that applies under specific conditions (inviscid flow), setting a foundation for understanding when and how these equations can be applied.

Think of these equations like rules of the road for a driver: while some rules apply in all conditions, others only apply under ideal circumstances (like driving on a clear highway without detours). Just as a driver must know which rules to follow based on the conditions, engineers must know when to use specific flow equations.

The basic idea comes that can you write these equations with a single scalar value okay that is the simple idea comes is that instead of looking at 3 scalar velocity component why we cannot write it with a simple a scalar functions that is what is the velocity potential function.

Here, the concept of velocity potential functions is introduced, suggesting a simplification where instead of dealing with three components of velocity (u, v, w), one can use a single scalar potential function (phi). This is particularly useful in flows that are irrotational, simplifying the analysis significantly.

Imagine a river where water flows in a uniform direction; instead of tracking every droplet, you can just measure the flow level (like measuring a single variable). This simplification helps engineers predict flow behavior without getting overwhelmed by details.

Thus if I define it the velocity is a gradient of phi, phi is a velocity potential function at which condition thus this is what justified it was.

This section emphasizes the conditions under which the velocity potential function can be applied, specifically noting that it is valid for irrotational flows. It discusses how the relationship between velocity and the gradient of the potential function can simplify the analysis of fluid motion.

Consider a calm lake without any disturbances. Here, the surface (potential function) represents stable conditions with no unnecessary complexities. In contrast, if there’s wind (creating rotational flow), the straightforward analysis of the potential function isn’t applicable anymore.

If you look at the very basic components if I velocity potentials. v equal to grade phi is the gradient of a scalar components will be justified okay or the reverse is also true that when you have v is equal to 0 okay.

This chunk explains the relationship between streamlines, velocity potentials, and how they interact in fluid flow. When the flow is irrotational, the velocity can be derived from the potential function, underscoring the elegance of this mathematical relationship.

Imagine a straight river path (streamlines) through a landscape. If the river is undisturbed, you can easily predict its flow (velocity potential). However, if something interrupts the flow (like rocks), the prediction becomes complicated.

That means if I again coming back to the same flow first a tall building if I draw the two streamlines let I draw S1, S2 two streamlines...

This section describes how to visually represent flow fields using streamlines and potential lines, emphasizing their orthogonal relationship in irrotational flow. It underlines the importance of understanding these visual tools to analyze fluid motion effectively.

Think of sketching a winding road on a map (streamlines). The shortcuts you might take (potential lines) need to cross the road at right angles for efficient travel. Both need to be understood to navigate effectively.

Now if you look at very interesting part which I can relate it with very basic equations what you know it is the Newton's first law of motions...

The discussion transitions into how flow fields can become rotational under certain conditions, relating this concept to Newton's laws of motion. When external forces act on the fluid, the initially irrotational flow can transform into a complex rotational flow, affecting the overall behavior of the fluid system.

Consider a spinning top (representing rotational flow) versus a balanced pencil (representing irrotational flow). The laws governing their motion describe how changes in force (like a nudge) can transition between stable and dynamic states.

The outside of this part will show you the viscous is not dominated. That is the reasons we can use Euler equations to solve these problems...

This chunk delineates the boundary between viscous flow and inviscid flow, explaining how and when Euler equations can be applied. It highlights the importance of recognizing flow behavior to choose the appropriate equations for analysis.

Like choosing between two instruments in music; sometimes, you need a soft flute for gentle melodies (viscous flow analysis) or a trumpet for bold sounds (Euler physics for simpler conditions). The right choice dictates the performance.

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