8.2. Approximations of Navier-Stokes equations

Today I will talk about the Navier-Stokes equations, how we do the approximations of Navier-Stokes equations for a simplified fluid flow with a series of illustrations that how we can simplify these Navier-Stokes equations.

The Navier-Stokes equations are fundamental in fluid mechanics, representing the motion of fluid substances. They consist of four equations that account for mass conservation and linear momentum. Because of their complexity, especially in real-world applications, approximations are needed. This section discusses how we can simplify these equations through various assumptions, facilitating easier calculations and analysis in fluid dynamics.

Imagine a complex recipe with many ingredients and steps. Simplifying the recipe by focusing only on the main ingredients and basic steps (e.g., substituting ingredients or skipping non-essential steps) allows you to create a quick version of the dish while retaining its essence. Similarly, approximating the Navier-Stokes equations helps us solve fluid mechanics problems more easily without losing sight of key principles.

If you look at this Navier-Stokes equations which is in Cartesian coordinates if I can write in vector forms which is easy to remember it is del cross dot v divergence of velocity vectors is 0 for incompressible fluid flow and that is the equations of continuity equations.

The continuity equation expresses the principle of mass conservation within a fluid flow. For incompressible flows, the density remains constant, which simplifies our calculations. The equation can be represented in vector form where the divergence of the velocity vector is equal to zero, indicating that the fluid's volume remains constant as it flows. This concept is fundamental in fluid dynamics, illustrating that the amount of fluid entering a region must equal the amount exiting, assuming no accumulation within that region.

Think of water flowing through a garden hose. If you squeeze the hose (creating a constriction), the water flows faster through that narrowed section, but the total volume of water entering and exiting remains the same, illustrating the continuity principle. Just like how the hose maintains a constant volume flow rate, incompressible fluid flow obeys the continuity equation.

So, if you look at the next the linear momentum equations as we have derived this is linear momentum equations in vector forms. So, in vector forms it comes like rho So, it is the accelerations part is equal to is rho g grade p pressures into mu Laplace operators of velocity field.

The linear momentum equations describe how the motion of fluid particles changes under the influence of various forces such as pressure, gravitational pull, and viscous shear forces. The term 'rho' refers to fluid density, while the Laplace operator indicates how the velocity field varies within the fluid. Therefore, these equations help predict how fluids will behave under exerted forces, enhancing our understanding of fluid motion dynamics.

Consider a car accelerating on a road. The car's acceleration (change in velocity) is influenced by forces like the engine's power (analogous to pressure), gravity (acting downhill), and friction (similar to viscosity). Likewise, the linear momentum equations help us quantify how fluids respond to similar forces at play within their environment.

So, if I write that equations in x directions I will get a local accelerations, the convective accelerations which will have non-linear terms, I will have a pressure gradient, the gravity force components.

In fluid dynamics, non-linear terms arise due to the convective acceleration—movement of the fluid affects its velocity. This non-linearity complicates the solutions to the Navier-Stokes equations, as these terms involve products of the velocity components and their derivatives. Understanding these non-linear dynamics is crucial as they lead to complex flow behaviors such as turbulence, which are challenging to predict and control.

Imagine a crowded subway train. The movement of passengers pushing and swaying creates complex interactions that are hard to predict. Similarly, in fluids, the non-linear interactions lead to unpredictability in flow patterns, especially when factors like turbulence come into play.

So, if these terms are 0 in a particular fluid flow, then these equations becomes a very linear form of very simplified equations that is what we called the Euler equations.

The Euler equations simplify the Navier-Stokes equations by disregarding the effects of viscosity (when viscosity is much smaller or negligible) and focusing on inviscid flow. These equations allow for an analytical approach to fluid dynamics in idealized conditions where fluid does not experience resistance. Understanding when it is appropriate to apply these equations is key when analyzing flows where viscous effects are minimal.

Think about swimming in a pool (viscous) versus swimming in a large lake (less viscous). When swimming in the lake, you feel less resistance from the water, and your strokes are more straightforward, similar to how the Euler equations simplify the analysis of fluid flow by ignoring viscosity.

So, we will try to simplify the equations to try to get analytical solutions for that. That is what the strategy.

In fluid mechanics, simplifying equations such as the Navier-Stokes and approximating them through methods like the Euler equations enables engineers and scientists to derive analytical solutions for complex fluid behaviors. These simplifications facilitate understanding under ideal or constrained conditions where exact solutions would be cumbersome or not feasible.

When planning a road trip, you can choose the most straightforward route using maps or GPS rather than tracking every possible detour considering every road condition. Similarly, in fluid dynamics, using simplified equations allows engineers to focus on the main aspects of fluid flow without getting lost in complexities.

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