What are Navier-stokes equations and why are they difficult to solve?

  • Thread starter Thread starter verdigris
  • Start date Start date
  • Tags Tags
    Navier-stokes
AI Thread Summary
Navier-Stokes equations describe fluid motion in differential form, analogous to Newton's second law (F=ma). Their complexity arises from being non-linear differential equations, where the solution at any point depends on conditions throughout the fluid and its history. These equations couple mass, momentum, and energy, requiring simultaneous solutions under normal circumstances. In specific flow scenarios, such as supersonic flow, they transform into hyperbolic partial differential equations, allowing for reduced complexity based on upstream conditions. The dependence on past instants is due to the finite speed of information transfer within the fluid.
verdigris
Messages
118
Reaction score
0
What are Navier-stokes equations and why are they difficult to solve?
 
Mathematics news on Phys.org
Navier-Stokes is essentially F=ma in differential form for a fluid.

They are difficult to solve mainly because they are non-linear differential equations where the evolution of the solution at anyone place&instant depends upon the solution at (just about) every other place&(past) instant.
 
Last edited:
Just happened to stumble across this thread

To elaborate just a little, they are mass-momentum-and energy written in differential (or sometimes integral) form. They are all coupled meaning that you cannot under normal circumstances solve one, they must be solved simultaneously.

For different types of flows, they are dependent on "solutions" at certain points. For example, for supersonic flow, the equations become hyperbolic partial differential equations. This means that information only travels downstream. Luckily for CFD people, this means that the solution at one point only depends on a "cone of dependence", or points upstream.
 
Indeed, there are many situations in which the physics of the special case allows for a significant reduction of complexity.

For example, if friction can be neglected, and the vorticity of the fluid is zero, then, effectively, we merely have to solve the Laplace equation with, for example, non-linear boundary conditions. :smile:
 
Why does the solution depend on "past instants" and not current instants.How big is a "control volume" used to form the equations and what determines how big it should be?
 
Well, information travel at finite speed. Hence, the information from one part of the fluid cannot affect the behaviour at another, finitely distanced, place until some time has passed.
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...

Similar threads

Back
Top