Navier-stokes and 1 million dollars

[kurious]

Help me find solutions for the Navier-stokes equations and you could get rich.
The problem is this:
A fluid enters a pipe and flows through it smoothly at the outset.
Will it keep flowing smoothly?
Sounds easy to solve but it isn't because nobody has
won the million dollars yet.

Here is one idea for how to deal with the problem:

Assume the equations which describe smooth flow are really the sums of two types of other equation describing vortices:
one set of equations about a vortex spinning clockwise and moving along the pipe, and the other set about a vortex spinning anticlockwise and also moving along the pipe.
The Navier-Stokes question then becomes:
Do the vortices cancel each other out permanently as time goes on?

 

[terrabyte]

Make a pipe with an inner frictionless surface

problem solved

 

[Chronos]

Safe bet by NS. The vortices will never cancel out. Even in a frictionless tube, there will always be swirls due to quantum effects.

 

[Antonio Lao]

Tensor equations in need of scalar solutions

The Navier-Stokes equation is really a tensor equation in need of scalar solutions.

\frac{\vec{F}}{V} = - \vec{\nabla} P

The major difficulty of its solution is the fact that the pressure term, P, is believed to be a scalar. If pressure is defined as Force per unit area, how could it be a scalar? Unless pressure is the scalar product of two vectors. It's obvious that force is always a vector or tensor but is 1/area a vector?

Macroscopically, pressure is a scalar. It is the average force that a bounding closed volume received on a point of its control surface. Once there is a hole on this closed surface, the definition of pressure becomes invalid.

But by taking the negative gradient of pressure, one inadvertently created a preferred direction for the pressure force and change the macroscopic model into a microscopic one. Hence, it is changed from a continuum dynamic to a discrete quantum problem.

Nevertheless, Navier-Stokes equations remain a fundamental model of classical mechanics not quantum mechanics. It remains just a continuum model of Newton's laws of motion. And the gravity law is proven correct only to a tenth of a millimeter.

 

[Antonio Lao]

Do the vortices cancel each other out permanently as time goes on?

At the quantum level, these vortices do not cancel. I think, these are the spin component of each atom and even molecules have spin components.

 

[Lama]

What if we have a low air pressure in the outset's side?

 

[Antonio Lao]

In fiber optics, the problem of losses, I think, was solved by total internal reflection of light as waves not as particles.

 

[Antonio Lao]

The theory of specific heat is based on the idealization of defining what is pressure as force per unit area. And the ideal triangular area is found to be the equilateral triangle whose area is 1/2 \sqrt{3} giving the ratio of hypotenuse to altitude as \frac{\sqrt{3}}{2} whose twice square gives 3/2 instead of 5/2, an important factor in the kinetic theory of heat. But the 5/2-triangle is based on 1/2 unit area of an isosceles triangle whose base and altitude are equal to 1 but making the other side as inverse of its altitude whose magnitude is \frac{2}{\sqrt{5}}.

 

[kuengb]

At the quantum level, these vortices do not cancel. I think, these are the spin component of each atom and even molecules have spin components.

Safe bet by NS. The vortices will never cancel out. Even in a frictionless tube, there will always be swirls due to quantum effects.

The "Million-Dollar-Problem" is purely mathematical and does not deal with physical reality at such a low scale. Of course there is no such thing as a non-smooth velocity field in a real liquid.

 

[Antonio Lao]

Of course there is no such thing as a non-smooth velocity field in a real liquid.

The velocity field of an ideal fluid is composed of massless field points which is a true continuum but real fluid is not so that a real fluid field point is associated with the electronic energy, the vibrational energy, the rotational energy, the translational energy.

Furthermore, to make NS equation soluble, the assumption that the pressure and temperature are constants is made and also that the divergence of the velocity is zero as in an incompressible fluid.

 

[kurious]

What is the equation for a vortex?
What makes a vortex grow, what makes it shrink?

 

[Antonio Lao]

What is the equation for a vortex?

I think it is the curl of a vector field, curl A or \nabla \times A. It grows, assuming the vector is constant, to a maximum when the angle between the curl and the vector is an odd integer multiple of 90 degrees or pi/2.

 

[Chronos]

The answer is a bit complicated. It starts with the Helmholtz vortex theorems. It gets ugly after that. Fluid dynamics give me a headache.

 

[Gza]

Anyone know of a good set of notes on fluid dynamics, I think i'll set to work on the problem right now.