From a7da57c0736bec58d1fc4ec99d211099c31bb45f Mon Sep 17 00:00:00 2001 From: Preston Pan Date: Wed, 24 Jan 2024 19:26:59 -0800 Subject: new content --- mindmap/continuity equation.org | 24 ++++++++++++++++++++++++ 1 file changed, 24 insertions(+) create mode 100644 mindmap/continuity equation.org (limited to 'mindmap/continuity equation.org') diff --git a/mindmap/continuity equation.org b/mindmap/continuity equation.org new file mode 100644 index 0000000..579e09a --- /dev/null +++ b/mindmap/continuity equation.org @@ -0,0 +1,24 @@ +:PROPERTIES: +:ID: a871e62c-b4a0-4674-9dea-d377de2f780b +:END: +#+title: continuity equation +#+author: Preston Pan +#+html_head: +#+html_head: +#+html_head: +#+options: broken-links:t + +* Derivation and Motivation +In continuum mechanics, the continuity equation is a statement about the inability for particles to teleport between +two different points in space. In other words, each particle must take a path between two points. In particular, if +$Q_{enc} = \int_{V} \rho(r')d\tau$ or $Q_{enc}$ is the total amount of particles inside some surface where $\rho$ is the density: +\begin{align*} +\oint_{S} \vec{J} \cdot d\vec{a} = -\frac{\partial Q_{enc}}{\partial t}, +\end{align*} +or in other words, the amount that the current goes through some closed surface must be proportional to the loss of particles +inside of the enclosure. This is of course because of conservation of mass (which is in and of itself conservation of energy). +Using the [[id:44e65b69-e5d5-464a-b1f3-8a914e1b7e9e][divergence theorem]]: +\begin{align*} +\int_{V}\vec{\nabla} \cdot \vec{J}d\tau = - \int_{V}\frac{\partial\rho}{\partial t}d\tau \\ +\vec{\nabla} \cdot \vec{J} = -\frac{\partial\rho}{\partial t} +\end{align*} -- cgit