📂Fluid Mechanics

Definition of Hartmann Number

Definition ^{1 2}

Primarily for conductive fluids, the following dimensionless quantity expressed through magnetic field strength $B$, characteristic length $L$, electrical conductivity $\sigma$, and kinematic viscosity $\mu = \rho \nu$ is called the Hartmann number. $$ \mathrm{Ha} := B L \sqrt{\frac{ \sigma }{ \mu }} $$

Explanation ³

The Hartmann number appears when the magnetic field must be taken into account for a flowing fluid. If the fluid type does not change, $\sigma$ and $\mu$ are fixed, and if the system is specified, $L$ is fixed; therefore, in practice it effectively represents the magnetic field strength $B$.

For example, imagine a wire passing between two concentric cylinders with a hole in the middle as shown above. If we run current through the wire, heat will be generated by Joule heating, and we want to cool it using an efficient heat-transfer medium. Meanwhile, the current in the wire will produce a magnetic field.

Generally, a stronger magnetic field reduces heat-transfer efficiency, but reducing the current to improve efficiency conflicts with the wire’s primary purpose of carrying current. In other words, the magnetic field $B$ is less a tunable system parameter and more a constraint that naturally accompanies the use of electricity, and it is reasonable to regard $\mathrm{Ha}$ as expressing this.