7.1.4. Derivations of Navier-Stokes Equations

Fluid Mechanics Prof. Subashisa Dutta Department of Civil Engineering Indian Institute of Technology-Guwahati Lec 28: The Navier-Stokes Equation Good morning all of you. Today we are going to discuss about very interesting topics, the topics on Navier-Stokes equations.

In this introduction, the professor emphasizes the significance of Navier-Stokes equations in fluid mechanics, highlighting their application in computational fluid dynamics and complex fluid flow problems. The historical context of Claude-Louis Navier and George Stokes, who derived these equations independently over 200 years ago, is also presented. Their contributions form the foundation of fluid mechanics as we study it today.

Consider the way we use equations in everyday life: similar to how the laws of physics help us understand motion and forces, the Navier-Stokes equations enable engineers and scientists to predict how fluids behave in various situations—from the gentle flow of water in a stream to the turbulent gusts of wind around a skyscraper.

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.

This chunk introduces two classifications of fluids: Newtonian and non-Newtonian. Newtonian fluids, like water and air, follow a linear relationship between shear stress and shear strain rate, which allows their behavior to be described using simple equations. Non-Newtonian fluids, such as blood and certain slurries, do not adhere to this linearity, requiring more complex formulations to describe their viscous behavior.

Imagine honey versus water. When you try to pour both, honey flows more slowly and doesn’t easily follow the same rules as water. This difference in behavior exemplifies how non-Newtonian fluids (like honey) require specialized treatment compared to Newtonian fluids (like water) when using the Navier-Stokes equations.

We can derive where sigma ij is a stress tensor. Now if I look at the component wise before going to these derivations, I just want to look at what is the component wise of this part.

In fluid mechanics, the concept of a 'control volume' is essential. It involves analyzing the fluid flow within a defined volume to understand the forces acting on it. The stress tensor, denoted as sigma ij, represents the internal forces within the fluid that result from stresses, often derived from the momentum equations that form part of the Navier-Stokes equations.

Consider a soccer ball being kicked. The area where the foot meets the ball is like a control volume. The stress (force) applied at that point causes the internal structure of the ball to respond, transmitting forces throughout it. Similarly, analyzing how fluid behaves utilizes control volumes to understand the forces acting throughout the fluid.

If you look at that what we are deriving this that we are not bringing a new physics to that. The same concept we are talking about that we know from Newton s second laws the force is equal to mass into acceleration.

This chunk focuses on deriving the Navier-Stokes equations from established principles, specifically Newton's second law of motion: force equals mass times acceleration. The text explains that the fluid dynamics are analyzed through control volumes, and the resulting equations articulate the balance of momentum due to pressures and stresses within the fluid.

Think of a car accelerating on a highway. The push of the engine (force) acts on the mass of the car resulting in acceleration. Similarly, when analyzing fluids, we consider how forces (pressure, gravity, viscous forces) affect the fluid's motion (acceleration) to arrive at the Navier-Stokes equations.

Now if you consider it that I have this big control volumes which is movable control volume okay it is moving it.

In this section, the derivation of the Navier-Stokes equations is simplified under specific assumptions, such as considering incompressible and isothermal flow conditions. These assumptions help reduce the complexity of the governing equations, making them easier to solve by focusing on how density and viscosity behave under steady conditions.

Think of a kitchen sponge soaked with water. If you squeeze it gently, the water (fluid) flows out without changing density significantly. This scenario mimics incompressible flow conditions, allowing us to employ simplified versions of fluid equations like the Navier-Stokes under certain assumptions.

So we have a four equations and four unknowns. So, we can solve it but still it has the problems is comes it here is it is a unsteady time component is there nonlinear partial differential equations.

The final expressions of the Navier-Stokes equations yield four equations with four unknowns—pressure and the three velocity components. However, due to the inherent nonlinearities and the time dependence of these equations, obtaining exact solutions can be problematic and often requires numerical methods or approximations.

Imagine trying to predict the exact motion of a crowded bus stop where people are constantly moving in and out. Just like analyzing complex flow behaviors requires sophisticated simulations, understanding fluid dynamics through the Navier-Stokes equations often involves approximations and computational models to handle the complexities.

If you look at this the equations what we have in Kaji's equations we have a stress tensors okay which is having 9 components.

In practical terms, the equations derived from Cauchy's principles incorporate stress tensors with components that must be modeled to account for real-world fluid behaviors. When dealing with multiple forces acting on a fluid, it's essential to know how these stress components interrelate to accurately predict the fluid's motion and time evolution.

Think about the water flowing through a pipe with different bends and fittings. Each bend can introduce changes in the pressure and velocity of the fluid, akin to how stress components influence the flow in real-world situations. Engineers must consider these variables when designing systems that manage fluid movement effectively.

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