10.5.2. Continuity and Navier-Stokes Equations

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.

The Navier-Stokes equations describe the motion of fluid substances. These equations have been developed over the past 200 years and summarize the principles of fluid dynamics, including how fluids flow and behave under various conditions. In fluid mechanics, these equations are crucial for predicting how fluid will move in different scenarios, such as air flowing over an aircraft or water flowing through a pipe.

Imagine a person swimming through water. The movements of the person's body create waves and currents in the water. The Navier-Stokes equations help us understand how those waves and currents change based on the person's movements and the water's properties.

When I talk about Euler equations you can understand it. It is for incompressibles and non-viscous or fixed or less flow. So if you consider flow past a tall building.

Euler equations are a form of the Navier-Stokes equations but are simplified by assuming the fluid is incompressible and non-viscous. These equations can be used to model flows where viscosity has a negligible effect, such as in large-scale phenomena. For example, when analyzing the airflow around tall buildings, the flow outside the immediate vicinity of the building can be approximated using Euler's equations, as viscosity's influence diminishes at larger distances from the building.

Think of a smooth river flowing around a rock. The water flows around the rock without much turbulence in the broader flow, making it possible to use simplified equations, like Euler’s, to describe the water movement without considering every tiny eddy and whirlpool.

Instead of looking at 3 scalar velocity components why we cannot write it with a simple a scalar function that is what is the velocity potential function.

In fluid dynamics, using velocity potential functions allows for a simplified approach to dealing with flow fields. Rather than handling three different velocity components (u, v, w), we can describe the flow using a single scalar function, known as the velocity potential function (φ). This function's gradients in different directions give us the velocity components, thus reducing the complexity of our calculations significantly.

Imagine trying to navigate a large, crowded room. If you focus on every person's movement (u, v, and w), it can be overwhelming. Instead, if you have a general idea of how people will flow based on the entrances and exits (the velocity potential function), you can easily predict traffic patterns without tracking every individual.

If I define the velocity as the gradient of phi, phi is the velocity potential functions, the condition is that it should be irrotational flow.

The use of velocity potential functions requires specific conditions. One key condition is that the flow must be irrotational, which means there is no rotation or vorticity in the fluid. When the flow is irrotational, it simplifies calculations, allowing the velocity fields to be expressed as the gradient of a single scalar function (the velocity potential). This means that the flow behaves in a predictable manner without swirling eddies or chaotic turbulence.

Think of a calm pond where you drop a rock. The ripples that form move outward in circular patterns without causing any spinning water. This is similar to irrotational flow in a fluid where the potential function can accurately predict the flow’s behavior.

The streamlines and the potential lines they will have a orthogonality to each other.

In fluid dynamics, streamlines represent the paths that fluid particles follow, while potential lines are connected to the velocity potential function. One important characteristic of these two types of lines is that they are orthogonal or perpendicular to each other at the points where they intersect. This orthogonality indicates that the direction of fluid flow (streamlines) is at a right angle to the direction of potential gradient, which portrays how velocity is changing in space.

Consider a playground slide and a path leading up to it. The path (potential line) intersects the slide (streamline) at a right angle, indicating how one leads to the other. As you walk up, you can glance down the slide; however, your direction of travel is always perpendicular to the slide’s surface.

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