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 navier-stokes_equations [2013/11/10 19:56]nikolaj navier-stokes_equations [2014/03/21 11:11] (current) Both sides previous revision Previous revision 2013/11/10 19:57 nikolaj 2013/11/10 19:56 nikolaj 2013/09/17 14:33 nikolaj 2013/09/17 14:33 nikolaj 2013/09/17 14:32 nikolaj 2013/09/17 14:32 nikolaj 2013/09/17 14:31 nikolaj 2013/09/17 14:30 nikolaj 2013/09/17 14:29 nikolaj 2013/09/17 14:29 nikolaj 2013/09/17 14:28 nikolaj 2013/09/17 14:22 nikolaj 2013/09/17 14:15 nikolaj created Next revision Previous revision 2013/11/10 19:57 nikolaj 2013/11/10 19:56 nikolaj 2013/09/17 14:33 nikolaj 2013/09/17 14:33 nikolaj 2013/09/17 14:32 nikolaj 2013/09/17 14:32 nikolaj 2013/09/17 14:31 nikolaj 2013/09/17 14:30 nikolaj 2013/09/17 14:29 nikolaj 2013/09/17 14:29 nikolaj 2013/09/17 14:28 nikolaj 2013/09/17 14:22 nikolaj 2013/09/17 14:15 nikolaj created Line 1: Line 1: ===== 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: $::\mathbf{f}(\mathbf{x},​t)$ | + | @#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: $\rho\ \left(\frac{\partial}{\partial t} + \mathbf{v} \cdot \nabla ​ \right)\ \mathbf{v} = -\mathrm{grad}(p) + \nabla \cdot \boldsymbol{\mathsf{T}} + \mathbf{f}$ | + | @#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 ==== Line 25: Line 25: === Subset of === === Subset of === [[PDE system]] [[PDE system]] - === Context ​=== + === Related ​=== [[Macroscopic observables from kinetic theory]] [[Macroscopic observables from kinetic theory]]