equations:continuity_equation
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equations:continuity_equation [2018/04/19 10:06] jakobadmin [Abstract] |
equations:continuity_equation [2020/03/03 10:38] (current) 128.179.254.165 |
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| - | <WRAP lag> $\color{blue}{\frac{\partial \rho}{\partial t}} + \color{magenta}{\rho \vec \nabla \vec v} = \color{red}{\sigma} $</WRAP> | + | <WRAP lag> $\color{blue}{\frac{\partial \rho}{\partial t}} = \color{red}{\sigma} - \color{magenta}{\rho \vec{\nabla} \cdot \vec{v}} $</WRAP> |
| ====== Continuity Equation ====== | ====== Continuity Equation ====== | ||
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| <tabbox Intuitive> | <tabbox Intuitive> | ||
| - | The continuity equation states that the $\color{red}{\text{total amount of a quantity (like water) that is produced (or destroyed) inside some volume}}$ is proportional to the $\color{blue}{\text{change of the quantity}}$ plus the $\color{magenta}{\text{total amount that flows in minus the amount that flows out of the volume}}$. | + | The continuity equation states that the total $\color{blue}{\text{change of some quantity}}$ is equal to the $\color{red}{\text{amount that gets produced}}$ minus the amount that $\color{magenta}{\text{flows out of the volume}}$. |
| - | + | ||
| - | Or formulated differently, the total $\color{blue}{\text{change of some quantity}}$ is equal to the $\color{red}{\text{amount that gets produced}}$ plus the amount that $\color{magenta}{\text{flows in minus the amount that flows out of the volume}}$. | + | |
| [{{ :equations:venturi.gif?nolink |Image by Thierry Dugnolle}}] | [{{ :equations:venturi.gif?nolink |Image by Thierry Dugnolle}}] | ||
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| $$ \nabla \cdot J + \frac { \partial ( \nabla \cdot D ) } { \partial t } = 0. | $$ \nabla \cdot J + \frac { \partial ( \nabla \cdot D ) } { \partial t } = 0. | ||
| $$ | $$ | ||
| - | Finally, we use another [[equations:maxwell_equations|Maxwell equation]], namely [[equations:yang_mills_equations:gauss_law|Gauss law]], | + | Finally, we use another [[equations:maxwell_equations|Maxwell equation]], namely [[formulas:gauss_law|Gauss law]], |
| $$\nabla \cdot D = \rho | $$\nabla \cdot D = \rho | ||
| $$ | $$ | ||
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| where $S$ denotes the surface of our volume $V$. This is the integral form of the continuity equation. | where $S$ denotes the surface of our volume $V$. This is the integral form of the continuity equation. | ||
| + | * The first term simply describes the total amount of the quantity, e.g. electric charge or mass, inside our volume. | ||
| + | * The second term describes the amount that flows into the surface minus the amount that flows out of the surface. | ||
| <tabbox Abstract> | <tabbox Abstract> | ||
equations/continuity_equation.1524125171.txt.gz · Last modified: 2018/04/19 08:06 (external edit)
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