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Chandrasekhar number

The Chandrasekhar number is usually denoted by the letter ${\displaystyle \ Q}$, and is motivated by a dimensionless form of the Navier-Stokes equation in the presence of a magnetic force in the equations of magnetohydrodynamics:

${\displaystyle {\frac {1}{\sigma }}\left({\frac {\partial ^{}\mathbf {u} }{\partial t^{}}}\ +\ (\mathbf {u} \cdot \nabla )\mathbf {u} \right)\ =\ -{\mathbf {\nabla } }p\ +\ \nabla ^{2}\mathbf {u} \ +{\frac {\sigma }{\zeta }}{Q}\ ({\mathbf {\nabla } }\wedge \mathbf {B} )\wedge \mathbf {B} ,}$

where ${\displaystyle \ \sigma }$ is the Prandtl number, and ${\displaystyle \ \zeta }$ is the magnetic Prandtl number.

The Chandrasekhar number is thus defined as:^[1]

${\displaystyle {Q}\ =\ {\frac {{B_{0}}^{2}d^{2}}{\mu _{0}\rho \nu \lambda }}}$

where ${\displaystyle \ \mu _{0}}$ is the magnetic permeability, ${\displaystyle \ \rho }$ is the density of the fluid, ${\displaystyle \ \nu }$ is the kinematic viscosity, and ${\displaystyle \ \lambda }$ is the magnetic diffusivity. ${\displaystyle \ B_{0}}$ and ${\displaystyle \ d}$ are a characteristic magnetic field and a length scale of the system respectively.

It is related to the Hartmann number, ${\displaystyle \ Ha}$, by the relation:

${\displaystyle Q\ {=}\ Ha^{2}\ }$

• Rayleigh number
• Taylor number