2004/12/13 by Thierry Alboussière, Thierry Alboussiere
Biochemistry, Genetics and Molecular Biology · Engineering · Physics and Astronomy · #Fluid Dynamics and Turbulent Flows #Geomagnetism and Paleomagnetism Studies #Solar and Space Plasma Dynamics #physics.flu-dyn
paper · pdf · doi:10.1017/s0022112004001740
published as Journal of Fluid Mechanics 521 (2004) pp. 125-154
openalex publication_date 2004/12/13 · arxiv created 2005/03/04 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A flow of electrically conducting fluid in the presence of a steady magnetic field has a tendency to become quasi-two-dimensional, i.e. uniform in the direction of the magnetic field, except in thin so-called Hartmann boundary layers. The condition for this tendency is that of a strong magnetic field, corresponding to large values of the dimensionless Hartmann number ( \hbox\it Ha ≫ 1 ). This is analogous to the case of low-Ekman-number rotating flows, with Ekman layers replacing Hartmann layers. This has been at the origin of the homogeneous model for flows in a rotating frame of reference, with its rich structure: geostrophic contours and shear layers. In magnetohydrodynamics, the characteristic surfaces introduced by Kulikovskii ( Isv. Akad. Nauk SSSR Mekh. Zhidk Gaza , vol. 3, 1968, p. 3) play a role similar to that of the geostrophic contours. However, a general theory for quasi-two-dimensional magnetohydrodynamics is lacking. In this paper, a model is proposed which provides a general framework. Not only can this model account for otherwise disconnected past results, but it is also used to predict a new type of shear layer, of typical thickness \hbox\it Ha-1/4 . Two practical cases are then considered: the classical problem of a fringing transverse magnetic field across a circular pipe flow, treated by Holroyd & Walker ( J. Fluid Mech. vol. 84, 1978, p. 471), and the problem of a rectangular cross-section duct flow in a slowly varying transverse magnetic field. For the first problem, the existence of thick shear layers of dimensionless thickness of order of magnitude \hbox\it Ha-1/4 explains why the flow expected at large Hartmann number was not observed in experiments. The second problem exemplifies a situation where an analytical solution had been obtained in the past Walker & Ludford ( J. Fluid Mech. vol. 56, 1972, p. 481) for the so-called ‘M-shaped’ velocity profile, which is here understood as an aspect of general quasi-two-dimensional magnetohydrodynamics.