7.1.1. Lec 28: The Navier-Stokes Equation

Good morning all of you. Today we are going to discuss very interesting topics, the topics on Navier-Stokes equations. That is what the basic foundations of computational fluid dynamics and many of the complex fluid flow problems also we can solve it by using Navier-Stokes equations.

The Navier-Stokes equations are critical in fluid mechanics, as they describe how fluids move. They serve as the foundation for computational fluid dynamics (CFD), which is the use of numerical methods to analyze fluid flows. Understanding these equations allows us to model and solve complex fluid flow problems effectively.

Think of the Navier-Stokes equations as the set of rules that govern how water flows in a river. Just as you can use these rules to predict where the water will go, engineers use the equations to predict how fluids will behave in various conditions.

If you look it for vector notation and related to the vectors, operations in fluid mechanics, the best course which is available in MIT OpenCourseWare which on fluid mechanics you can get it and also we are going through this F.M. White book on fluid mechanics. These are 3 books, you can always refer it to understand this Navier-Stokes equations part. Now, if you look go through its who is Navier and who is Stokes okay the way back the French physicist okay Claude-Louis Navier okay so it is almost 200 years back and also this George Stokes from as a mathematicians from England both have derived the Navier-Stokes equations independently.

The Navier-Stokes equations were derived independently by Claude-Louis Navier and George Stokes in the early 19th century. Navier, a French physicist, contributed to the understanding of fluid flow behavior, while Stokes, a British mathematician, worked on the mathematical formulation. Their contributions have become fundamental to studying fluid dynamics today.

Imagine two inventors working on a similar invention in different parts of the world. Even without communication, they both arrive at the same groundbreaking idea, just like Navier and Stokes did with their equations. Their combined legacy impacts fields like weather forecasting, aircraft design, and even ocean currents.

I will talk about the derivations of Navier-Stokes equations with assumptions of incompressible isothermal flow.

Derivations of the Navier-Stokes equations typically start with fundamental principles of mass, momentum, and energy conservation. Key assumptions include incompressibility (constant density) and isothermal flow (constant temperature). These assumptions simplify the equations, making them more manageable and relevant for many practical applications.

Consider a hot air balloon. If the air inside the balloon is heated, it becomes less dense, and the balloon rises. This situation illustrates compressible flow. However, if the balloon was filled with water instead, which is incompressible, the analysis would be simpler, akin to how the Navier-Stokes equations are derived under the assumption of incompressibility.

If you go back very basic things, the the vectors divergence okay. If you look at that del and V the velocity divergence it is talking about the net volume outflow from a control volume.

Velocity divergence refers to how much fluid is expanding or contracting at a point. If it’s positive, more fluid is exiting a volume than entering, indicating a flow source. Conversely, if it’s negative, fluid is accumulating, indicating a sink.

Imagine a sponge absorbing water; as the sponge soaks up water (negative divergence), it expands. On the other hand, if you squeeze a sponge, the water flows out (positive divergence). This analogy helps illustrate how divergence informs us about fluid flow behavior in different scenarios.

If you look at the Cauchy's equations, we are deriving the basic equations for linear momentum equations for control volumes which is having infinitely small control volumes.

Cauchy’s equations describe the motion of fluids through momentum conservation principles. They consider forces acting on an infinitesimal control volume of fluid. Body forces (like gravity) and surface forces (like pressure) are analyzed to understand how fluids behave under different conditions.

Think of a water bottle sitting on a scale. The scale reads the weight of the water (body force) plus the pressure at the bottom of the bottle (surface force). Both forces contribute to the total force the scale measures, similar to how surface and body forces contribute to fluid movement.

Also, I will talk about revisit Newtonian fluid and non-Newtonian fluid part which is quite interesting to look at how we can use for this Navier-Stokes equations.

Newtonian fluids exhibit a linear relationship between shear stress and shear rate, meaning they flow predictably (like water). Non-Newtonian fluids do not have this predictable behavior; their flow characteristics change under different conditions (like ketchup). Understanding this distinction is critical in applying the Navier-Stokes equations correctly.

Consider stirring honey versus water. Honey flows much differently and requires more effort to move; it behaves non-Newtonian where its viscosity can change with the applied force. This analogy serves to understand how fluid properties affect their behavior in different conditions.

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