Continuity equation

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The mass continuity equation is the first of the MHD equations . It essentially states that the mass flowing into a volume must equal the increase in mass of that volume:

\frac{\partial\rho}{\partial t}+\nabla\cdot\left(\rho\mathbf{v}\right)=0

It is also written as particle, rather than mass, conservation:

\frac{\partial n}{\partial t}+\nabla\cdot\left(n\mathbf{v}\right)=0

We know that the change in mass of a volume V must be equal to the net flux through its surface S:

\frac{\partial}{\partial t}\left[\int_{V}\rho dV\right]=-\int_{S}\rho\mathbf{v}\cdot d\mathbf{s}

(here our surface vector points outward, so the negative is the inward flux). By Stokes's theorem, and the commutation of the time derivative with the spatial integral this is equivelent to:

\int_{V}\frac{\partial\rho}{\partial t}dV=-\int_{V}\nabla\cdot\left(\rho\mathbf{v}\right)dV

We can choose V to be whatever we want:

\frac{\partial\rho}{\partial t}+\nabla\cdot\left(\rho\mathbf{v}\right)=0

Which is sometimes written using the total derivative :

\frac{d\rho}{dt}+\rho\nabla\cdot\mathbf{v}=0
    • Continuity equation for continuity of particles

This page was recovered in October 2009 from the Plasmagicians page on Continuity_equation dated 19:33, 8 June 2006.

Retrieved from "https://qed.princeton.edu/main/PlasmaWiki/Continuity_equation"
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