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Continuity Equation

\[\frac{\partial \rho}{\partial t} + \nabla\cdot(\rho \mathbf{U}) = 0\]

Source lecture(s): pc368_lec05_mhd

Physical Meaning

Mass (or particle number) is conserved. The continuity equation states that the change in density at a point equals the net flux of fluid into that point.

Variable Definitions

Symbol Definition SI Units
\(\rho\) Mass density kg m\(^{-3}\)
\(\mathbf{U}\) Fluid velocity m s\(^{-1}\)
\(t\) Time s

Assumptions

  • Single-species or multi-species with negligible inter-species diffusion.
  • Fluid description valid (continuum limit).

Derivation

Integrate the zeroth velocity moment of the Boltzmann equation over velocity space:

\[\int \Bigl(\frac{\partial f}{\partial t} + \mathbf{v}\cdot\nabla f + \frac{q}{m}\mathbf{F}\cdot\frac{\partial f}{\partial \mathbf{v}} + C[f]\Bigr) d^3v = 0\]

For collisionless evolution or when collisional terms conserve particles, the integral yields:

\[\frac{\partial}{\partial t}\int f d^3v + \nabla\cdot\int \mathbf{v} f d^3v = 0\]

Define \(n = \int f d^3v\) and \(\Gamma = n\mathbf{U} = \int \mathbf{v} f d^3v\) to obtain the continuity equation.

Applications

  • Mass balance in compressible flows.
  • Source terms for ionization.
  • Radial transport in tokamaks.

Connections to Other Equations