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Navier-Stokes Eqs.


stingray78
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Chapman and Cowling's The Mathematical Theory of Non-Uniform Gases has a derivation based on kinetic theory of gases

 

Batchelor's An Introduction to Fluid Dynamics has a good derivation, as does Bird, Stewart and Lightfoot's Transport Phenomena.

 

In fact, any fluid mechanics book that covers the more mathematical side of the subject -- as opposed to the "rules of thumb" side of the subject like pump sizing and estimating pressure drop in a pipe based on a chart of friction factors, etc -- all of which is very important, don't get me wrong, it just isn't related to the mathematical/differential equation side of fluid mechanics -- will have a decent derivation. The wikipedia derivation isn't all the great, unless you already know it. Definitely try the Bird, Stewart, and Lightfoot reference.

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  • 7 years later...

When I was in college, I derived the Navier-Stokes equations beginning with Pascal's Principles. While I'm certainly not up for delivering my full 10 page derivation tonight, let's see If I can give ya the basics.

 

First, understand that we need to derive an equation of motion for a fluid, that is F=ma. Beginning with the definition for pressure we have, P=F/A thus F=PA. Let's take an outward unit normal vector to a circle, and call it df. df has the unit of area, so we can now integrate P dot df. Applying the divergence theorem tells us that we now have an integral with respect to Volume of the pressure gradient. (this should be a negative quantitity so that pressure decreases as area increases). Now that we have a force, the negative pressure gradient, let's see what we can do with the RHS of F=ma.

 

If I let my Volume be a unit volume, then because density equals mass over volume, I can say that our mass is equal to density or rho=m/V. Really, all that's left is to find the acceleration(s). This is where it gets tricky for most people. What I'll need are two accelerations, one to explain the motion of the fluid as a whole, and one to explain the motion of fluid particles. The acceleration of the fluid as a whole will simply be a temporal derivative of velocity, delta v over delta times dt, where delta indicates a partial derivative.

 

Finding the acceleration of fluid particles, requires that I define a displacement between those particles, call it dr. Because we are dealing with particles in space, I want something that is capable of explaining that situation. Divergence will suit our needs perfectly. This divergence will act parallel to our velocity field (or velocity profile if you prefer). So you should have (grad dot dr)v. Applying the chain rule from multivariable calculus here gives the "material derivative" (our change in velocity associated with the particles in the fluid), I like to think of it as the spatial derivative. You should find that it is equal to (grad dot v)v. Thus the acceleration, dv/dt of the whole mess is: dv/dt=delta v over delta t + (grad dot v)v. Finally we can allow an external body force to act on our system (such as gravity or a centrifugal force). We'll denote this with a lower case f, and put the whole equation together.

 

f - (grad P) = rho * (delta v over delta t + (grad dot v)*v) This is F = ma

 

The above equation is Euler's Equation for a fluid. You may notice that it is almost identical to the Navier-Stokes equation for incompressible flow, yet it is missing what is arguably the most important term, viscocity. Deriving viscocity can be a rather long and technical process, so right now I'm tempted just to tell you to add it to the LHS. The gist for viscosity is this however. All of those little infinitessimal fluid layers that build up our velocity vector field are sliding across each other. Stokes realized that in doing so they must be obeying some sort of concept of friction. In short the fluid itself is resisting the shear force of the pressure gradient that is acting perpendicular to it and causing the fluid's motion.

 

If your interested in how to derive the viscocity, let me know. I'll be happy to type up something here, it is afterall the explanation to solving the problems that winemakers were having during the time. And what could be more important than using physics to solve problems associated with wine.... or water I suppose. LOL :)

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... let's see If I can give ya the basics.

Please note that stingray78 was last active here on May 13, 2008.

 

Bringing up old threads risks the original poster not being active any more.

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But we'd all be interested in knowing about the problems they were having with wine that prompted Navier and Stokes to come up with the viscous flow equations.

 

I like science history, what can I say ?

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The wine problems are related to filling a vat with wine at some height, and then using the pressure generated from the potential energy to fill wine jugs at distances along a pipe. The jugs further down the line would of course fill more slowly, regardless of how much pressure could be generated at the beginning of the line. A logical conclusion was to increase the height of the tank, but they encountered even more problems when they did so. The concepts associated with (dare I say) energy loss due to friction, were not well understood until Stokes solved viscocity, (and for that matter, still aren't all that well understood. LOL) The texts I have read on the subject treat the problems associated with winemakers as mostly anecdotal. Nonetheless, it is reasonable to conclude that there were strong motivations from the wine sector to have a better explanation for dynamic flows.

 

I saw that the question had been posted some time ago, yet the topic is the Navier Stokes Equations. Those equations are so important to physics, mathematics, engineering, life etc., that I was compelled to keep the post active. Thanks for your reply.

Edited by Casey Wood
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