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