The annihilation of a two-dimensional jet by a transverse magnetic field

The effect of an applied transverse magnetic field on the development of a two-dimensional jet of incompressible fluid is examined. The jet is prescribed in terms of its mass flux ρ Q 0 and its lateral scale d at an initial section x = 0. The three dimensionless numbers characterizing the problem are a Reynolds number R = Q 0 /ν, a magnetic Reynolds number R m = μσ Q 0 , and a magnetic interaction parameter N = σ B 2 0 d 2 /ρ Q 0 , where ρ represents density, σ conductivity, μ permeability and B 0 applied field strength, and it is assumed that \[ R_m 1, R 1, N 1. \] It is shown that when M 2 = RN [Gt ] 1, an inviscid treatment is appropriate, and that the effect of the magnetic field is then to destroy the jet momentum within a distance of order N −1 in the downstream direction. A general solution for inviscid development is obtained, and it is shown that a large class of velocity profiles (though not all of them) are self-preserving. When M 2 [Lt ] 1, it is shown that the viscous similarity solution obtained by Moreau (1963 a, b ) is relevant. This solution is re-derived and re-interpreted; it implies that the jet momentum is destroyed within a distance of order R1/4N-3/4 in the downstream direction. Some further aspects of the jet annihilation problem are qualitatively discussed in § 4, viz. the nature of the overall flow field, the effect of the presence of distant boundaries, the effect of increasing R m to order unity and greater, and the effect of oblique injection. Finally the development of a jet of conducting fluid into a nonconducting environment is considered; in this case the jet is not stopped by the magnetic field unless a return path outside the fluid for the induced current is available.