What are Navier-stokes equations and why are they difficult to solve?
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 any one place&instant depends upon the solution at (just about) every other place&(past) instant.
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.
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.
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