The famous Navier-Stokes

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It is time now to add together advection, pressure, and diffusion. For this we will use the Newton second law: acceleration of a body multiplied by the mass of the body is equal to the force acting on the body. In our case the forces acting on the body are pressure and viscous friction, and their effects should be added together. It is easier to remember the result if it is written in the form of a simple formula:

ρdu/dt = -p + μΔu.

Here u stands for fluid velocity, p for pressure, ρ for density, t for time and μ for dynamic viscosity. The values of μ and ρ depend on the fluid and can be found in handbooks. The formula means:

<density> times <acceleration> = <pressure force> + <friction force>.

The next important step is to rewrite this in the non-dimensional form. This might come as a surprise, but the earlier we do this the better is the final outcome for you.  For this, we select two characteristic scales, U for velocity and L for length. We then understand u in our formula as the actual velocity divided by our scale U. Our coordinate vector x will now be the actual coordinate vector divided by L. For time, t will be the actual time divided by the time scale, which we take to be equal to L/U. For pressure the scale is ρU2. As a result, out new quantities are now non-dimensional. For example, the velocity could initially be measured in m/sec, but after dividing it by the velocity scale, also measured in m/sec, the dimension, m/sec, is cancelled out. In these non-dimensional variables the law of fluid motion takes the form

du/dt = -∇p + (1/Re)Δu

where Re=ρUL/μ is the famous Reynolds number. Its verbal equivalent is still

<acceleration> = <pressure force> + <friction force>,

but now it can also be put in a different way, namely,

<the rate of velocity variation> = <rate due to pressure> + (1/Re)<diffusion>.

This expresses the law of fluid motion from the Lagrangian viewpoint. That is, du/dt is the rate of the velocity variation as observed from a particle moving with the fluid. The Euler viewpoint is often more convenient. Switching to Euler viewpoint will result in a different way of expressing the rate of velocity variation. It leads to

u/∂t = -u·∇u – ∇p + (1/Re)Δu.

This is the famous Navier-Stokes equation. In words, this can be described in the following way. At a fixed point in space, the velocity changes with time with the rate ∂u/∂t for three reasons. First, fluid particles come to this point and bring their own velocity there. If it is different from the velocity at the particle that occupied the same point in space shortly before that, the velocity at that point is changing with the rate -u·∇u. Then there is pressure. If pressure on one side of the fluid volume is greater than the pressure on the other side of the fluid volume, the fluid will accelerate, and the corresponding rate of the velocity change is denoted -∇p. Finally, friction results in the diffusion of velocity, which is expressed by  (1/Re)Δu.

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