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verdigris
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What are Navier-stokes equations and why are they difficult to solve?
What are Navier-stokes equations and why are they difficult to solve?
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Discussion Overview
The discussion centers on the Navier-Stokes equations, exploring their definition, complexity, and the challenges associated with solving them. It encompasses theoretical aspects, technical explanations, and conceptual clarifications related to fluid dynamics.
Discussion Character
- Technical explanation
- Conceptual clarification
- Debate/contested
Main Points Raised
- Some participants describe the Navier-Stokes equations as the differential form of F=ma for fluids, highlighting their non-linear nature and the interdependence of solutions across space and time.
- Others elaborate that the equations represent mass, momentum, and energy in coupled forms, necessitating simultaneous solutions under typical conditions.
- A participant notes that for specific flow types, such as supersonic flow, the equations transform into hyperbolic partial differential equations, which restrict information propagation to downstream points.
- It is mentioned that in certain scenarios, simplifying assumptions like neglecting friction and zero vorticity can reduce the problem to solving the Laplace equation with non-linear boundary conditions.
- Questions are raised about the dependence of solutions on past instants and the concept of control volumes, including inquiries about their size and determining factors.
- A response indicates that information travels at finite speed, meaning that changes in one part of the fluid cannot instantaneously affect another part until a certain time has elapsed.
Areas of Agreement / Disagreement
Participants express various viewpoints regarding the complexity and characteristics of the Navier-Stokes equations, with no consensus reached on specific aspects such as the implications of past instants or the definition of control volumes.
Contextual Notes
Some discussions involve assumptions about flow conditions and the applicability of simplifications, which may not hold in all scenarios. The dependence on definitions and the implications of finite information travel are also noted as areas of complexity.
- #2
arildno
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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.
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.
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minger
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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.
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.
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arildno
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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.
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.
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verdigris
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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?
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arildno
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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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