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| Hartmann number | |
|---|---|
| Name | Hartmann number |
| Dimension | Dimensionless |
| Related | Reynolds number, Magnetic Reynolds number, Nusselt number |
Hartmann number The Hartmann number is a dimensionless parameter used in magnetohydrodynamics to quantify the influence of a magnetic field on electrically conducting fluids. It appears in analyses of flows affected by imposed magnetic fields and connects material properties, geometric scales, and field strength in problems studied in contexts such as industrial metallurgy, geophysics, and astrophysics. Developed in the mid-20th century, the parameter is widely used alongside other nondimensional groups in research institutions and engineering practice.
The Hartmann number represents the ratio of magnetic Lorentz forces to viscous forces in a conducting fluid and provides a measure of magnetic damping relative to viscous diffusion. In practical terms, Ha indicates how strongly an applied magnetic field, often produced by a device in a laboratory at institutions like the CERN or used in furnaces at companies such as ArcelorMittal, suppresses velocity fluctuations in an electrolyte or liquid metal. For flows in channels or pipes studied at universities like MIT and University of Cambridge, Ha helps determine whether a flow will develop thin boundary layers adjacent to walls, analogous to phenomena observed in experiments at facilities like the Max Planck Institute for Plasma Physics and in planetary interiors modeled by groups at the California Institute of Technology.
Mathematically, the Hartmann number combines the magnetic field magnitude produced by sources such as devices used at Lawrence Livermore National Laboratory with fluid properties measured in laboratories like NIST. It is defined using physical constants familiar in studies performed at institutions including Imperial College London and ETH Zurich. In canonical analyses found in textbooks from publishers such as Cambridge University Press and Springer, Ha appears together with the Reynolds number and the Magnetic Reynolds number to nondimensionalize the Navier–Stokes equations coupled to Maxwell’s equations, governing flows investigated by researchers at Princeton University and the University of Tokyo.
The derivation of the Hartmann number follows from scaling the momentum equation used in classical treatments by authors associated with the Royal Society and mathematical approaches taught at Sorbonne University. By balancing Lorentz and viscous terms, one obtains Ha, and limiting cases correspond to asymptotic regimes employed in analyses at centers such as the Institute of Physics, Chinese Academy of Sciences or the Max Planck Society. In the low-Ha limit, viscous forces dominate and flow resembles canonical solutions studied by researchers at Stanford University; in the high-Ha limit, magnetic damping enforces quasi-two-dimensional structures akin to features explored in numerical work at Los Alamos National Laboratory.
The Hartmann number is central to applications ranging from liquid-metal cooling systems for reactors examined at Argonne National Laboratory to the modeling of stellar interiors pursued by teams at Harvard University and University of Oxford. In industrial processes like continuous casting managed by firms such as ThyssenKrupp, Ha informs magnetic braking and flow control strategies. In geophysical and astrophysical settings studied by groups at NASA and the European Space Agency, the parameter helps characterize magnetically dominated regimes relevant to planetary dynamos and accretion disk dynamics.
Estimating Ha in laboratory experiments involves measurements of magnetic field strength with instruments developed at facilities like National Physical Laboratory (UK), and fluid properties determined in collaborations with laboratories such as Brookhaven National Laboratory. Experiments at institutions like École Polytechnique and Technical University of Munich use Hall probes and velocimetry techniques refined at Caltech to obtain the inputs for Ha computations. Studies reported in journals associated with societies such as the American Physical Society document protocols for uncertainty analysis when measuring Ha in electrolytes and liquid metals.
Numerical investigations of flows with specified Hartmann numbers are produced using codes developed at centers such as Los Alamos National Laboratory, Princeton University, and commercial vendors like ANSYS. Simulations often reproduce classic Hartmann flow between parallel plates studied historically at institutions including University of Göttingen and University of Chicago, and are benchmarked against experiments performed at Johns Hopkins University. Computational approaches range from spectral methods used in projects at ETH Zurich to finite-volume methods in software supported by Siemens research groups.
The Hartmann number is used alongside other dimensionless parameters such as the Reynolds number, Magnetic Reynolds number, Stuart number, and Prandtl number in multidisciplinary studies carried out at research centers including MIT, Imperial College London, and the Max Planck Institute for Meteorology. Comparisons among these numbers guide scaling analyses in contexts from industrial metallurgy at Nippon Steel to planetary magnetism research at University of Alberta.
Category:Dimensionless numbers