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Navier–Stokes equations
Definition
$ n\in\mathbb N $ |
$ \rho: \mathbb R^n\times\mathbb R\to\mathbb R^n $ |
$ p \in C(\mathbb R^n\times\mathbb R,\mathbb R) $ |
$ \boldsymbol{\mathsf{T}} \in C(\mathbb R^n\times\mathbb R\times\mathbb R^n,\mathbb R^{n^2}) $ |
$ \mathbf{f} \in C(\mathbb R^n\times\mathbb R,\mathbb R^n) $ |
$ ::\rho(\mathbf{x},t) $ |
$ ::p(\mathbf{x},t) $ |
$ ::\boldsymbol{\mathsf{T}}(\mathbf{x},t,\mathbf{v}) $ |
$ ::\mathbf{f}(\mathbf{x},t) $ |
$ \mathbf{v} \in \mathrm{it} $ |
$ \mathbf{v} \in C^2(\mathbb R^n\times\mathbb R,\mathbb R^n) $ |
$ ::\mathbf{v}(\mathbf{x},t) $ |
$ \rho\ \left(\frac{\partial}{\partial t} + \mathbf{v} \cdot \nabla \right)\ \mathbf{v} = -\mathrm{grad}(p) + \nabla \cdot \boldsymbol{\mathsf{T}} + \mathbf{f} $ |
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