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navier-stokes_equations [2013/11/10 19:57] nikolaj |
navier-stokes_equations [2014/03/21 11:11] (current) |
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===== Navier–Stokes equations ===== | ===== Navier–Stokes equations ===== | ||
- | ==== Definition ==== | + | ==== Set ==== |
- | | @#88DDEE: $ n\in\mathbb N $ | | + | | @#55CCEE: context | @#55CCEE: $ n\in\mathbb N $ | |
- | | @#88DDEE: $ \rho: \mathbb R^n\times\mathbb R\to\mathbb R^n $ | | + | | @#55CCEE: context | @#55CCEE: $ \rho: \mathbb R^n\times\mathbb R\to\mathbb R^n $ | |
- | | @#88DDEE: $ p \in C(\mathbb R^n\times\mathbb R,\mathbb R) $ | | + | | @#55CCEE: context | @#55CCEE: $ p \in C(\mathbb R^n\times\mathbb R,\mathbb R) $ | |
- | | @#88DDEE: $ \boldsymbol{\mathsf{T}} \in C(\mathbb R^n\times\mathbb R\times\mathbb R^n,\mathbb R^{n^2}) $ | | + | | @#55CCEE: context | @#55CCEE: $ \boldsymbol{\mathsf{T}} \in C(\mathbb R^n\times\mathbb R\times\mathbb R^n,\mathbb R^{n^2}) $ | |
- | | @#88DDEE: $ \mathbf{f} \in C(\mathbb R^n\times\mathbb R,\mathbb R^n) $ | | + | | @#55CCEE: context | @#55CCEE: $ \mathbf{f} \in C(\mathbb R^n\times\mathbb R,\mathbb R^n) $ | |
- | | @#DDDDDD: $ ::\rho(\mathbf{x},t) $ | | + | | @#DDDDDD: range | @#DDDDDD: $ ::\rho(\mathbf{x},t) $ | |
- | | @#DDDDDD: $ ::p(\mathbf{x},t) $ | | + | | @#DDDDDD: range | @#DDDDDD: $ ::p(\mathbf{x},t) $ | |
- | | @#DDDDDD: $ ::\boldsymbol{\mathsf{T}}(\mathbf{x},t,\mathbf{v}) $ | | + | | @#DDDDDD: range | @#DDDDDD: $ ::\boldsymbol{\mathsf{T}}(\mathbf{x},t,\mathbf{v}) $ | |
- | | @#DDDDDD: $ ::\mathbf{K}(\mathbf{x},t) $ | | + | | @#DDDDDD: range | @#DDDDDD: $ ::\mathbf{K}(\mathbf{x},t) $ | |
- | | @#FFBB00: $ \mathbf{v} \in \mathrm{it} $ | | + | | @#FFBB00: definiendum | @#FFBB00: $ \mathbf{v} \in \mathrm{it} $ | |
- | | @#55EE55: $ \mathbf{v} \in C^2(\mathbb R^n\times\mathbb R,\mathbb R^n) $ | | + | | @#55EE55: postulate | @#55EE55: $ \mathbf{v} \in C^2(\mathbb R^n\times\mathbb R,\mathbb R^n) $ | |
- | | @#DDDDDD: $ ::\mathbf{v}(\mathbf{x},t) $ | | + | | @#DDDDDD: range | @#DDDDDD: $ ::\mathbf{v}(\mathbf{x},t) $ | |
- | | @#55EE55: $ \rho\ \left(\frac{\partial}{\partial t} + \mathbf{v} \cdot \nabla \right)\ \mathbf{v} = -\mathrm{grad}(p) + \nabla \cdot \boldsymbol{\mathsf{T}} + \mathbf{K} $ | | + | | @#55EE55: postulate | @#55EE55: $ \rho\ \left(\frac{\partial}{\partial t} + \mathbf{v} \cdot \nabla \right)\ \mathbf{v} = -\mathrm{grad}(p) + \nabla \cdot \boldsymbol{\mathsf{T}} + \mathbf{K} $ | |
==== Discussion ==== | ==== Discussion ==== | ||
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=== Subset of === | === Subset of === | ||
[[PDE system]] | [[PDE system]] | ||
- | === Context === | + | === Related === |
[[Macroscopic observables from kinetic theory]] | [[Macroscopic observables from kinetic theory]] |