Will Navier-Stokes ever be solved?

Will Navier-Stokes ever be solved?

In particular, solutions of the Navier–Stokes equations often include turbulence, which remains one of the greatest unsolved problems in physics, despite its immense importance in science and engineering. Even more basic (and seemingly intuitive) properties of the solutions to Navier–Stokes have never been proven.

What does the Navier-Stokes equation calculate?

Navier-Stokes equation, in fluid mechanics, a partial differential equation that describes the flow of incompressible fluids. The equation is a generalization of the equation devised by Swiss mathematician Leonhard Euler in the 18th century to describe the flow of incompressible and frictionless fluids.

What are the assumptions of the Navier-Stokes equations?

The Navier–Stokes equations are based on the assumption that the fluid, at the scale of interest, is a continuum, in other words is not made up of discrete particles but rather a continuous substance.

What is the Stokes hypothesis?

2.2. With Stokes’ hypothesis, 2μ+3ζ=0, which expresses that the changes of volume do not involve viscosity, the viscous stress tensor may be written as σ = μ [ ∇ ⊗ u + ( ∇ ⊗ u ) T − ( 2 / 3 ) ( ∇ · u ) I ] . The Navier–Stokes equations also assume that the fluid follows the Fourier law of diffusion.

Why is Navier Stokes unsolvable?

The Navier-Stokes equation is difficult to solve because it is nonlinear. This word is thrown around quite a bit, but here it means something specific. You can build up a complicated solution to a linear equation by adding up many simple solutions.

Is Diane Adler a real person?

If you’re asking whether she solved the problem in real life, then no; Diane Adler is a fictional character, and if anyone solved a Millennium Prize problem, then it would have been publicized in media outlets worldwide.

What is Navier Stokes equation in CFD?

The Navier-Stokes equations are the basic governing equations for a viscous, heat conducting fluid. It is a vector equation obtained by applying Newton’s Law of Motion to a fluid element and is also called the momentum equation. Usually, the term Navier-Stokes equations is used to refer to all of these equations.

What is Navier-Stokes derived from?

The Navier-Stokes equations can be derived from the basic conservation and continuity equations applied to properties of fluids. The basic continuity equation is an equation which describes the change of an intensive property L. An intensive property is something which is independent of the amount of material you have.

What is rank of strain tensor?

The strain tensor, εkl, is second-rank just like the stress tensor. Sijkl is called the compliance tensor and is also fourth-rank. The strain tensor is a field tensor – it depends on external factors. The compliance tensor is a matter tensor – it is a property of the material and does not change with external factors.

What is the stress tensor in fluid mechanics?

A key step in formulating the equations of motion for a fluid requires specifying the stress tensor in terms of the properties of the flow, in particular the velocity field, so that the theory becomes “closed”, that is, that the number of variables is reduced to the number of governing equations.

What are the eigenvectors of the new stress tensor?

There will be three eigenvalues corresponding to the three diagonal elements of the new stress tensor. For each eigenvalue there will be an eigenvector ajm,m=1,2,3. Since the stress tensor is symmetric the eigenvectors corresponding to different eigenvalues are orthogonal.

Why is the stress tensor isotropic?

The fact that the stress tensor is isotropic implies that the normal stress in any orientation is always P, and the tangential stress is always zero (This isotropy of pressure is often called Pascal’s Law). When the fluid is moving, the pressure (often defined as the average normal force on a fluid element) need not be the thermodynamic pressure.

Can a stress tensor have off diagonal components in another frame?

In the example of the last chapter we saw that a stress tensor that had only a diagonal component in one coordinate frame would have, in general, off diagonal components in another frame.