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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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In summary: Thus, the solution at a current instant is influenced by the solution at a past instant, hence the dependence on "past instants".In summary, Navier-Stokes equations are a set of non-linear differential equations that describe the behavior of fluids in terms of mass, momentum, and energy. They are difficult to solve because they are coupled and require simultaneous solutions. However, for certain types of flows, the equations can be simplified. The solution also depends on past instants due to the finite speed of information travel. The size of the control volume used to form the equations is determined by the physics of the particular situation.
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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.
1. What are Navier-Stokes equations?
Navier-Stokes equations are a set of partial differential equations that describe the motion of fluids. They relate the velocity, pressure, temperature, and density of a fluid to its surrounding environment and any external forces acting upon it.
2. Why are Navier-Stokes equations important?
Navier-Stokes equations are used in fields such as fluid dynamics, aerodynamics, and weather forecasting to model and predict the behavior of fluids. They are essential for understanding and analyzing fluid flow phenomena in various natural and engineered systems.
3. What makes Navier-Stokes equations difficult to solve?
Navier-Stokes equations are highly non-linear and involve complex interactions between different variables. They also contain terms that are difficult to express mathematically, making it challenging to find exact solutions. Additionally, the equations are coupled, meaning that changes in one variable can affect the entire system, making it difficult to isolate and solve for individual variables.
4. What are some methods used to solve Navier-Stokes equations?
There are several methods used to solve Navier-Stokes equations, including numerical methods, analytical methods, and experimental techniques. Numerical methods involve dividing the equations into smaller, simpler equations that can be solved iteratively. Analytical methods involve using mathematical techniques to find exact solutions to simplified versions of the equations. Experimental techniques involve conducting physical experiments to gather data and validate the solutions.
5. What are some real-world applications of Navier-Stokes equations?
Navier-Stokes equations have a wide range of real-world applications, including predicting weather patterns, analyzing air and water flow in airplanes and ships, designing turbines and other machinery, and studying ocean currents and atmospheric circulation. They are also used in the development of new technologies, such as wind turbines and aerodynamic designs for cars and airplanes.
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