Stability of a Hartmann boundary layer under the influence of a parallel magnetic field

1968 ◽  
Vol 32 (4) ◽  
pp. 721-735 ◽  
Author(s):  
S. Abas
Keyword(s):  
Magnetic Field ◽  
Boundary Layer ◽  
Reynolds Number ◽  
Asymptotic Methods ◽  
Transverse Magnetic ◽  
Magnetic Reynolds Number ◽  
Power Series Solution ◽  
Neutral Stability ◽  
Two Dimensional ◽  
Parallel Magnetic Field

Stability to infinitesimal disturbances—when a parallel magnetic field is imposed—is investigated for the flow in the boundary layer set up by two-dimensional motion between parallel planes of a viscous, incompressible, electrically conducting fluid under the influence of a transverse magnetic field. The flow is assumed to take place at low magnetic Reynolds number. The usual asymptotic methods are employed for the solution, but, apart from the Tollmientype power series solution, an exact solution of the inviscid equation is obtained in terms of the hypergeometric function and its analytic continuation. Curves of neutral stability for two-dimensional disturbances are calculated and the results for critical Reynolds number modified to take into account three-dimensional disturbances. The parallel magnetic field is found to have a strong stabilizing influence.

The effect of a parallel magnetic field on the stability of free boundary-layer type flows of low magnetic Reynolds number

1969 ◽  
Vol 38 (2) ◽  
pp. 243-253 ◽  
Author(s):  
S. Abas
Keyword(s):  
Magnetic Field ◽  
Boundary Layer ◽  
Reynolds Number ◽  
Free Boundary ◽  
Magnetic Reynolds Number ◽  
Wave Number ◽  
Neutral Stability ◽  
Parallel Magnetic Field ◽  
Limiting Profile ◽  
Small Wave Number

Stability to infinitesimal disturbances, when a parallel magnetic field is imposed, is investigated for the free boundary-layer type flows, of low magnetic Reynolds number, between two unbounded parallel streams of a viscous, incompressible, electrically conducting fluid. Neutral stability curves are calculated for small wave-number making use of the limiting profile: previous results by another author are found to be incomplete. A qualitative neutral stability picture is conjectured for other values of the wave-number and, granted a certain part of this conjecture, the conclusion is that the critical Reynolds number remains zero until the parameter Q/R exceeds the value (Q/R)crit [eDot ] 0·0233. It is suggested that a sufficiently strong magnetic field can stabilize a flow of any finite Reynolds number.

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

1967 ◽  
Vol 30 (1) ◽  
pp. 65-82 ◽  
Author(s):  
H. K. Moffatt ◽  
J. Toomre
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Transverse Magnetic Field ◽  
Magnetic Interaction ◽  
Transverse Magnetic ◽  
Magnetic Reynolds Number ◽  
Two Dimensional ◽  
Dimensionless Numbers ◽  
Order Unity ◽  
The 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 ρQ0 and its lateral scale d at an initial section x = 0. The three dimensionless numbers characterizing the problem are a Reynolds number R = Q0/ν, a magnetic Reynolds number Rm = μσQ0, and a magnetic interaction parameter N = σB20d2/ρQ0, where ρ represents density, σ conductivity, μ permeability and B0 applied field strength, and it is assumed that \[ R_m \ll 1,\quad R\gg 1,\quad N\ll 1. \] It is shown that when M2 = 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 M2 [Lt ] 1, it is shown that the viscous similarity solution obtained by Moreau (1963a, b) is relevant. This solution is re-derived and re-interpreted; it implies that the jet momentum is destroyed within a distance of order $R^{\frac{1}{4}}N^{-\frac{3}{4}}$ 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 Rm 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.

A physical description of magnetic helicity evolution in the presence of reconnection lines

1991 ◽  
Vol 46 (1) ◽  
pp. 179-199 ◽  
Author(s):  
Andrew N. Wright ◽  
Mitchell A. Berger
Keyword(s):  
Magnetic Field ◽  
Reynolds Number ◽  
Magnetic Reynolds Number ◽  
Magnetic Helicity ◽  
Two Dimensional ◽  
Flux Tubes ◽  
Physical Description ◽  
Reconnection Region ◽  
Field Geometry ◽  
Field Lines

The dissipation of relative magnetic helicity due to the presence of a resistive reconnection region is considered. We show that when the reconnection region has a vanishing cross-section, helicity is conserved, in agreement with previous studies. It is also shown that in two-dimensional systems reconnection can produce highly twisted reconnected flux tubes. Reconnection at a high magnetic Reynolds number generally conserves helicity to a good approximation. However, reconnection with a small Reynolds number can produce significant dissipation of helicity. We prove that helicity dissipation in two-dimensional configurations is associated with the retention of some of the inflowing magnetic flux by the reconnection region, vr. When the reconnection site is a simple Ohmic conductor, all of the magnetic field parallel to the reconnection line that is swept into vr is retained. (In contrast, the inflowing magnetic field perpendicular to the line is annihilated.) We are able to relate the amount of helicity dissipation to the retained flux. A physical interpretation of helicity dissipation is developed by considering the diffusion of magnetic field lines through vr. When compared with helicity-conserving reconnection, the two halves of a reconnected flux sheet appear to have slipped relative to each other parallel to the reconnection line. This provides a useful method by which the reconnected field geometry can be constructed: the incoming flux sheets are ‘cut’ where they encounter vr, allowed to slip relative to each other, and then ‘pasted’ together to form the reconnected flux sheets. This simple model yields estimates for helicity dissipation and the flux retained by vr in terms of the amount of slippage. These estimates are in agreement with those expected from the governing laws.

The magneto-hydrodynamic boundary layer in the two-dimensional steady flow past a semi-infinite flat plate. I. Uniform conditions at infinity

1963 ◽  
Vol 273 (1355) ◽  
pp. 496-508 ◽  
Keyword(s):  
Magnetic Field ◽  
Boundary Layer ◽  
Reynolds Number ◽  
Drag Coefficient ◽  
Flat Plate ◽  
Magnetic Reynolds Number ◽  
Linear Ordinary Differential Equations ◽  
Electrically Conducting ◽  
The Magnetic Field ◽  
Conditions At Infinity

The problem investigated is the flow of a viscous, electrically conducting liquid past a fixed, semi-infinite, unmagnetized but conducting flat plate. The liquid flow U and also the magnetic field H 0 at a distance from the plate are both assumed to be uniform and parallel to the plate. It is assumed that the Reynolds number R and magnetic Reynolds number R m are large enough for momentum and magnetic boundary layers to have developed. The standard boundary-layer techniques as used in the Blasius solution then apply and the problem reduces to the solution of two simultaneous non-linear ordinary differential equations. These are examined by the use of an iteration method suggested in the non ­ magnetic problem by Weyl and a solution of reasonable accuracy has been obtained for the drag coefficient. This confirms a similar result obtained in a different way by Carrier & Greenspan. The principal result of the paper is that the boundary layer thickens and drag coefficient diminishes steadily as the parameter S = µH 2 0 / 4πρU 2 increases. When S attains the finite value of unity the drag coefficient obtained here actually vanishes with the flow having been reduced to rest by the action of the magnetic field. This result might be inferred qualitatively since a finite amount of work has to be done in conveying liquid particles across the lines of magnetic force.

Separation and magnetohydrodynamics

1969 ◽  
Vol 38 (3) ◽  
pp. 481-498 ◽  
Author(s):  
John Buckmaster
Keyword(s):  
Magnetic Field ◽  
Boundary Layer ◽  
Stagnation Point ◽  
Linear Equations ◽  
Inviscid Flow ◽  
Transverse Magnetic ◽  
Magnetic Reynolds Number ◽  
Boundary Layer Equations ◽  
Rear Stagnation Point

This paper is an investigation of MHD boundary layers in a transverse magnetic field when the magnetic Reynolds number (Rm) is small. The main purpose is to understand something about the suppression of separation by a strong magnetic field, with particular emphasis on the behaviour near a rear stagnation point. Given anO(1) inviscid flow it is shown that there is a critical value ofN, the interaction parameter, to completely suppress separation. This value is one half that proposed by Leibovich (1967), a discrepancy that is due to the non-regularity of the boundary-layer equations at a rear stagnation point, a possibility that Leibovich did not consider in his solution. Model linear equations suggest the true role of Leibovich's solution. The possibility of a viscous wake leaving the rear stagnation point is considered and it is suggested that one doesnotarise from vorticity generated in the boundary layer.

Flow induced in a cylindrical column by a uniformly rotating magnetic field

1967 ◽  
Vol 297 (1451) ◽  
pp. 558-563
Keyword(s):  
Magnetic Field ◽  
Boundary Layer ◽  
Reynolds Number ◽  
High Frequency ◽  
Fourier Transforms ◽  
Reynolds Numbers ◽  
Magnetic Reynolds Number ◽  
Rotating Magnetic Field ◽  
Cylindrical Column ◽  
Field Lines

Under laboratory conditions, the magnetic Reynolds number is quite small in a conductor, but can be made appreciable if a high frequency rotating field is applied. Moffatt investigated this problem for high magnetic Reynolds numbers and concluded that there existed a magnetic boundary layer due to spiralling of field lines. Applying Fourier transforms and solving the corrected equations, we find that at low magnetic Reynolds numbers the field lines uniformly penetrate the cylindrical column and do not exhibit any appreciable spiralling. The rotation opposes the drift due to conductivity which is evened out as one proceeds from the centre to the surface. This uniform behaviour persists for small magnetic Reynolds number inside and outside. When the magnetic Reynolds number becomes large, of the order of 100 (say), the field lines passing through the axis of the cylinder exhibit spiralling as suggested by Moffatt since the diffusion is unable to counterbalance the rotational effects.

Analysis of an Electromagnetic Boundary Layer Probe for Low Magnetic Reynolds Number Flows

1993 ◽  
Vol 115 (4) ◽  
pp. 726-731 ◽  
Author(s):  
L. S. Langston ◽  
R. G. Kasper
Keyword(s):  
Magnetic Field ◽  
Boundary Layer ◽  
Reynolds Number ◽  
Boundary Layer Flow ◽  
Magnetic Reynolds Number ◽  
Field Size ◽  
Flow Meter ◽  
Displacement Thickness ◽  
Layer Flow ◽  
Reynolds Number Flows

Electromagnetic (EM) flow meters are used to measure volume flow rates of electrically conductive fluids (e.g., low magnetic Reynolds number flows of seawater, milk, etc.) in pipe flows. The possibility of using a modified form of EM flow meter to nonobtrusively measure boundary-layer flow characteristics is analytically investigated in this paper. The device, named an electromagnetic boundary layer (EBL) probe, would have a velocity integral-dependent voltage induced between parallel wall-mounted electrodes, as a conductive fluid flows over a dielectric wall and through the probe’s magnetic field. The Shercliff-Bevir integral equation, taken from EM flow meter theory and design, is used as the basis of the analytical model for predicting EBL probe voltage outputs, given a specified probe geometry and boundary layer flow conditions. Predictions are made of the effective range of the nonobtrusive EBL probe in terms of electrode dimensions, the magnetic field size and strength, and boundary layer velocity profile and thickness. The analysis gives expected voltage calibration curves and shows that an array of paired electrodes would be a beneficial feature for probe design. A key result is that the EBL probe becomes a displacement thickness meter, if operated under certain conditions. That is, the output voltage was found to be directly proportional to the boundary layer displacement thickness, δ1, for a given free stream velocity.

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