Governing equations on differential form: Difference between revisions
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The governing equations for compressible inviscid flow on partial differential form: | The governing equations for compressible inviscid flow on partial differential form: | ||
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{{NumEqn|<math> | {{NumEqn|<math> | ||
\frac{\partial \rho}{\partial t} + \nabla\cdot(\rho\mathbf{v})=0 | \frac{\partial \rho}{\partial t} + \nabla\cdot(\rho\mathbf{v})=0 | ||
</math>}} | </math>|background-color=whitesmoke|color=steelblue|description=Continuity:|nonumber=1|padding=2em}} | ||
{{NumEqn|<math> | {{NumEqn|<math> | ||
\frac{\partial}{\partial t}(\rho \mathbf{v}) + \nabla\cdot(\rho \mathbf{v}\mathbf{v}) + \nabla p = \rho \mathbf{f} | \frac{\partial}{\partial t}(\rho \mathbf{v}) + \nabla\cdot(\rho \mathbf{v}\mathbf{v}) + \nabla p = \rho \mathbf{f} | ||
</math>}} | </math>|background-color=whitesmoke|color=steelblue|description=Momentum:|nonumber=1|padding=2em}} | ||
{{NumEqn|<math> | {{NumEqn|<math> | ||
\frac{\partial}{\partial t}(\rho e_o) + \nabla\cdot(\rho h_o\mathbf{v}) = \rho\mathbf{f}\cdot\mathbf{v} + \dot{q}\rho | \frac{\partial}{\partial t}(\rho e_o) + \nabla\cdot(\rho h_o\mathbf{v}) = \rho\mathbf{f}\cdot\mathbf{v} + \dot{q}\rho | ||
</math>}} | </math>|background-color=whitesmoke|color=steelblue|description=Energy:|nonumber=1|padding=2em}} | ||
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=== The Differential Equations on Non-Conservation Form === | === The Differential Equations on Non-Conservation Form === | ||
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==== Summary ==== | ==== Summary ==== | ||
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{{NumEqn|<math> | {{NumEqn|<math> | ||
\frac{D\rho}{Dt}+\rho(\nabla\cdot\mathbf{v})=0 | \frac{D\rho}{Dt}+\rho(\nabla\cdot\mathbf{v})=0 | ||
</math>}} | </math>|background-color=whitesmoke|color=steelblue|description=Continuity:|nonumber=1|padding=2em}} | ||
{{NumEqn|<math> | {{NumEqn|<math> | ||
\frac{D\mathbf{v}}{Dt}+\frac{1}{\rho}\nabla p = \mathbf{f} | \frac{D\mathbf{v}}{Dt}+\frac{1}{\rho}\nabla p = \mathbf{f} | ||
</math>}} | </math>|background-color=whitesmoke|color=steelblue|description=Momentum:|nonumber=1|padding=2em}} | ||
{{NumEqn|<math> | {{NumEqn|<math> | ||
\rho\frac{De_o}{Dt} + \nabla\cdot(p\mathbf{v}) = \rho\mathbf{f}\cdot\mathbf{v} + \dot{q}\rho | \rho\frac{De_o}{Dt} + \nabla\cdot(p\mathbf{v}) = \rho\mathbf{f}\cdot\mathbf{v} + \dot{q}\rho | ||
</math>}} | </math>|background-color=whitesmoke|color=steelblue|description=Energy:|nonumber=1|padding=2em}} | ||
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=== Alternative Forms of the Energy Equation === | === Alternative Forms of the Energy Equation === | ||
