Introduction

In broad terms, the Navier-Stokes equations represent a mathematical framework that describes the movement of any fluid.

Before delving into these equations, let's clarify some key definitions and concepts that will assist us in our exploration.

First, what constitutes a fluid?

A fluid is defined as a substance without a fixed shape that readily deforms under external pressure; this includes gases and (particularly) liquids. When stress is applied to a fluid, it experiences a continuous change in shape as the positions of its particles shift relative to one another. This irreversible transformation is referred to as flow.

Within this scope, viscosity refers to a fluid's resistance to flow. For example, honey and ketchup exhibit higher viscosity compared to water and oil, which flow more freely. Viscosity is a critical property of fluids that frequently appears in the physics used to model their behavior.

We primarily focus on two types of flow: laminar and turbulent.

Laminar flow is characterized by smooth fluid motion, where the particles remain organized in distinct layers that slide past one another with minimal mixing, similar to playing cards.

In contrast, turbulent flow is marked by chaotic changes, featuring eddies and vortices that affect the pressure and velocity of the fluid.

To simplify our discussion, we will make two primary assumptions: the fluids we examine are Newtonian, meaning their viscosity remains constant regardless of shear applied, and incompressible, indicating their density does not change during flow.

With this foundational knowledge, we can now appreciate the elegance of the Navier-Stokes equations!

The Navier-Stokes Equations

Without further delay, here are the two pivotal equations.

Where:

• u: the fluid velocity vector
• p: the fluid pressure
• ?: the fluid density
• ?: the fluid's viscosity
• ?²: the Laplacian operator (sum of the three second-order spatial derivatives of the vector field it operates on)

Both equations focus on u, the fluid's velocity vector. To clarify, a vector field assigns a vector to each point in space, representing both magnitude and direction. In this case, each vector indicates the fluid's velocity at that specific location.

We will analyze each of the Navier-Stokes equations, starting with the simpler one.

Conservation of Mass

The first equation mathematically expresses the principle of mass conservation.

The divergence of a vector field at a specific point quantifies how much the field tends to "spread out" at that location. Points exhibiting positive divergence can be likened to “sources” where fluid appears to emerge from nowhere, while points with negative divergence act as “sinks” where fluid seems to vanish.

Considering this, the first Navier-Stokes equation indicates that for an incompressible fluid, the velocity vector field must have zero divergence throughout. A non-zero divergence suggests that at certain points, the fluid either compresses or expands, leading to a non-zero time derivative of its density, which contradicts our assumptions.

Now, let's proceed to the second equation.

Conservation of Momentum

At first glance, the second equation may appear daunting, but we can simplify it step by step.

Essentially, this equation represents an application of Newton's second law to fluids.

Left-hand side — Mass times acceleration

The left side of the Navier-Stokes equation can be expressed as follows:

This makes the connection to Newton's second law clearer. Instead of mass, we utilize the fluid's density, which can be mathematically interchanged with mass under constant volume conditions. The acceleration is represented as the derivative of velocity, specifically termed a material derivative, which can be articulated using the chain rule:

We will not delve into the specifics of the material derivative here, as there are numerous resources available online.

Right-hand side — Sum of forces

The right side of the second Navier-Stokes equation sums all the forces acting on the fluid. We won't derive this equation formally, as it doesn't enhance our understanding, but it can be found online.

We can categorize forces into internal and external forces.

• ?p: The first internal force considered is pressure, represented by the pressure gradient. This reflects the pressure differential within the fluid's confines. Fluids naturally flow from areas of higher pressure to lower pressure, a tendency encapsulated in this term.
• ??²u: The second internal force accounts for friction, which is tied to the fluid's viscosity. Fluids with high viscosity exhibit greater friction between molecules than those with low viscosity, and this term succinctly expresses that concept.
• f: Lastly, this term incorporates any external forces acting on the fluid. Often, gravity is the sole external force, allowing us to replace “f” with g, resulting in the term ?g. In other scenarios, one might integrate the Lorentz Force to combine the Navier-Stokes equations with Maxwell’s equations.

Reynolds Number

Often mentioned alongside the Navier-Stokes equations, the Reynolds number, denoted as Re, helps determine whether fluid flow is laminar or turbulent.

The formula for the Reynolds number is as follows:

Note that D represents the characteristic length, which varies based on the type of flow being analyzed. In the above illustration, if the flow occurs within a pipe, D corresponds to the pipe's diameter.

Intuitively, the Reynolds number indicates the ratio of inertial forces to viscous forces. If inertial forces—those propelling fluid movement—are significantly greater than viscous forces—those resisting flow—the flow is turbulent. Conversely, if viscous forces predominate, the flow is laminar. But what constitutes “significantly greater”?

For Reynolds numbers under 2000, flow is typically laminar. For values exceeding 4000, flow is classified as turbulent, while values between 2000 and 4000 indicate a transitional flow, a blend of laminar and turbulent characteristics.

A Million Dollar Challenge

Engineers and physicists leverage the Navier-Stokes equations for various real-world applications. However, they frequently make certain assumptions and simplifications depending on the specific problem.

These equations pose significant challenges in their original form without approximations. The Clay Mathematics Institute has even offered a one million dollar reward for a solution. In essence, the problem to solve is:

For three-dimensional fluids, can you ascertain if a smooth solution exists at every spatial point?

Final Remarks

The Navier-Stokes equations rank among the most vital and widely utilized equations in physics. They facilitate aircraft and automobile design, blood flow studies, power station engineering, pollution analysis, and much more. Due to their inherent mathematical intricacies, solutions often require approximations, while a theoretical proof of a smooth solution promises a reward of one million dollars.

"When I meet God, I am going to ask him two questions: Why relativity? And why turbulence? I really believe he will have an answer for the first."

— Werner Heisenberg on his deathbed

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