Astronomy:Chandrasekhar number

The Chandrasekhar number is a dimensionless quantity used in magnetic convection to represent ratio of the Lorentz force to the viscosity. It is named after the India n astrophysicist Subrahmanyan Chandrasekhar. The number's main function is as a measure of the magnetic field, being proportional to the square of a characteristic magnetic field in a system.

Definition

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

[math]\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}, }[/math]

where [math]\displaystyle{ \ \sigma }[/math] is the Prandtl number, and [math]\displaystyle{ \ \zeta }[/math] is the magnetic Prandtl number.

The Chandrasekhar number is thus defined as:

[math]\displaystyle{ {Q}\ =\ \frac{{B_0}^2 d^2}{\mu_0 \rho \nu \lambda} }[/math]

where [math]\displaystyle{ \ \mu_0 }[/math] is the magnetic permeability, [math]\displaystyle{ \ \rho }[/math] is the density of the fluid, [math]\displaystyle{ \ \nu }[/math] is the kinematic viscosity, and [math]\displaystyle{ \ \lambda }[/math] is the magnetic diffusivity. [math]\displaystyle{ \ B_0 }[/math] and [math]\displaystyle{ \ d }[/math] are a characteristic magnetic field and a length scale of the system respectively.

It is related to the Hartmann number, [math]\displaystyle{ \ Ha }[/math], by the relation:

[math]\displaystyle{ Q\ {=}\ Ha^2\ }[/math]

See also

• Rayleigh number
• Taylor number