Understanding Navier-Stokes Equation

[Sabbatical]

In layman's terms, can someone explain what the Navier-Stokes Equations express?

And also, can someone explain to me, what the real problem is in trying to understand the Navier-Stokes problem/Turbulence?

 

[arildno]

1. Navier-Stokes is, simply F=ma per unit mass, as expressed in terms of how the velocity field must be in the fluid, rather than an expression for the particle paths as such (those are derivable from the N-S equations, so no loss of generality has occurred9
The Navier-Stokes equations are based on a specific modelling of the relevant forces in the fluid, where in the most common formulation, a) the isotropic pressure has been extracted as an explicity term b) gravity is included and c) A viscous stress-strain rate tensor model has been adopted, with a constant viscosity parameter.
generalizations can be made for hydro-magnetic flows, the equations as they would appear in a non-inertial frame of reference, and with a variable viscosity, typically linking N-S with the thermodynamical equations, due to viscosity dependence on temperature.
2. the Navier-Stokes problem poses:
Are these equations well-posed, in the sense that they cannot "blow up" at a finite time?
That is, can they, under all conditions be regarded as realistic physical equations? (In REALITY, we "know" that fluids can't get infinite accelerations, if our equations said they would get that, our equations are WRONG, not reality!)

3. The problem of turbulence is, essentially, that macroflow is significantly RANDOMIZED in turbulent flow. This means it is basically impossible to find a sufficiently accurate full (numerical) solution to N-S, and earlier, one sought to improvize and simplify the equations for the behaviour of the mean flow, averaged over time, rather than solve for the full flow. the trouble was, however, that one thereby introduces a pseudo-force term which shows the effect the random flow directly has on the evolution on the mean flow. This pseudo-force term, usually called Reynolds' stress tensor is notoriously hard to model in a good manner.
In newer times, with vastly improved computers, many have chosen to eschew the mean-flow approach altogether, and attack N-S head-on instead.
Here, the resolution problem will occur, because, for example, energy dissipation mainly takes place on microscopic scales, and having a full resolution down to THAT level would overpower the computers we have today and in the future..

 

[Sabbatical]

So this is what I am understanding:

The N-S equations explain the forces produced by a fluid (or by another force that acts on a fluid which produces another force per unit mass, etc.) but it is expressed by understanding the velocity caused within that disturbance? So... because there are collisions, there are different accelerations, which can be understood at microscopic levels.

This is difficult to understand and properly map out because different forces from different particles are continuously colliding in any amount of different directions, so on and so forth.

There has been a suggestion of a pseudo-force or Reynold's stress tensor so as to denote for us when something has changed during the averaging of the forces at a particular time frame.

So, if I have designed a way to pour out a glass of water, there will always be a different way that the water, on a microscopic level, changes its flow of particles? Am I thinking about this as an appropriate example?

Also could you elaborate what you mean by saying that fluids do not have an infinite acceleration?

And what did you mean when you said that the representation is in velocity, if and accelerated mass is force?

I apologize in advance if my questions are silly. I clearly don't have any background in physics. Please help me come to an appropriate understanding, as I know I'm missing many things here.

 

[Alpha Floor]

I'll try to explain as best as I can in plain words.

First of all, the Navier-Stokes equations are valid as long as we can apply the continuum hypothesis, which is the most basic hypothesis in continuum mechanics. It assumes that the characteristical distances of the problem are much larger than the distances between molecules. This way you can model the material (in this case fluid) as a continuum.

The concept of fluid particle (fp) is also important. A fluid particle is a control mass (a constant amount of molecules) which is macroscopically very small but microscopically very large. Let me explain this better, microscopically very large means that a fluid particle contains a lot of molecules, and its characteristical length is much larger than the distance between molecules. Macroscopically very small means that a fp is very small compared to the characteristical length of the problem. Any fp has a density, a pressure, a temperature and a velocity vector assigned to it.

Now we've got to talk about Eulerian and Lagrangian descriptions of fluid motion. The lagrangian description is the one that anyone who has studied classical mechanics would intuitively use at first. To study the motion of a fluid one might consider focusing on one fluid particle alone and following it along its motion. We would have the problem solved if we knew the position of the fp over time (trayectory) and its pressure, density and temperature. In practice, this is completely nuts since any fluid field consists of millions of fluid particles and trying to solve the motion of each one of them is impossible using this viewpoint.

Instead, in fluid mechanics we use the eulerian description of motion. Euler instead of focusing on a given fp focuses on a given position of the fluid field and studies the velocity, pressure, density and temperature of the fp that at one given time happens to occupy that position. A very short time (infinitely short) later, there is another fp occupying that same position, but now this fp has a diffent velocity, pressure...

This way, the problem would be solved if we knew the velocity, pressure, density and temperature of the fluid as a function of position (x,y,z) and time t. How do we achieve this? With the Navier - Stokes equations:

The NS equations are a set of 6 equations for 6 unknowns and 4 independent variables. These unknowns are the 3 components of velocity (u,v,w), density, pressure and temperature of the fluid. The equations are basically:

1) Conservation of mass or continuity equation. This equation states that mass can either be created out of thin air nor destroyed. It is a scalar equation and can be expressed both in differential (applied to a fp) or in integral form (applied to a control volume)

2) Conservation of momentum. This is Newton's second law F=m·a. It is a vector equation so it is actually like three equations in one. This equation can also be expressed in differential or in integral form.

3) Conservation of energy, or first principle of thermodynamics. It is an energy balance to a control volume. This equation has multiple variants like the enthalpy equation, the entropy equation, the inner energy equation... but they are all equivalent.

4) Equation of state, for ideal gases this would be pV=nRT. For a liquid this would be ρ = constant (constant density or incompressibility)

These 6 equations are a set of second order, non-linear partial differential equations, thus the huge complexity around them. They can be solved (numerically or analitically) provided with the necessary boundary and/or initial conditions. It is important to notice that the boundary or initial conditions are the only thing that change from one problem to the other, because the NS equations don't change.

I think that's enough as for a brief introduction to fluid mechanics :D