Governing equations on differential form: Difference between revisions
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==== Conservation of Momentum ==== | ==== Conservation of Momentum ==== | ||
We start from the momentum equation on differential form derived above | |||
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\frac{\partial}{\partial t}(\rho \mathbf{v}) + \nabla\cdot(\rho \mathbf{v}\mathbf{v}) + \nabla p = \rho \mathbf{f} | |||
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Expanding the first and the second terms gives | |||
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\rho\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v}\frac{\partial \rho}{\partial t} + \rho\mathbf{v}\cdot\nabla\mathbf{v} + \mathbf{v}(\nabla\cdot\rho\mathbf{v}) + \nabla p = \rho \mathbf{f} | |||
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Collecting terms, we can identify the substantial derivative operator applied to the velocity vector and the continuity equation. | |||
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\rho\underbrace{\left[\frac{\partial \mathbf{v}}{\partial t}+\mathbf{v}\cdot\nabla\mathbf{v}\right]}_{=\frac{D\mathbf{v}}{Dt}}+\mathbf{v}\underbrace{\left[\frac{\partial \rho}{\partial t}+\nabla\cdot\rho\mathbf{v}\right]}_{=0}+ \nabla p = \rho \mathbf{f} | |||
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which gives us the non-conservation form of the momentum equation | |||
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\frac{D\mathbf{v}}{Dt}+\frac{1}{\rho}\nabla p = \mathbf{f} | \frac{D\mathbf{v}}{Dt}+\frac{1}{\rho}\nabla p = \mathbf{f} | ||
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==== Conservation of Energy ==== | ==== Conservation of Energy ==== | ||
