A Textbook of Magnetohydrodynamics

Shercliff, J. A.; Scott, F. R.

doi:10.1119/1.1972705

Cited by 100 publications

(135 citation statements)

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“…The starting point for the next-order calculations is the dimensionless MHD equations of motion ͑in the familiar ''Alfvenic'' units͒ for a uniform-density, incompressible, conducting, steady-state fluid, 16 Ohm's law, Ampère's law, and Faraday's law, as given in I:…”

Section: Zeroth-order Revisitedmentioning

confidence: 99%

Toroidal flows in resistive magnetohydrodynamic steady states

1998

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We consider the resistive steady states of a uniformly conducting magnetofluid inside a toroidal boundary. The problem becomes tractable in the limit of slow flow: i.e., low Reynolds number, which may be in turn justified when the viscous Lundquist number is small. Previous calculations are extended to apprehend the toroidal component of the necessary flow. The emerging pattern is one of helical vortices which seem likely to be ubiquitous in toroidal geometry, and which disappear in the ''straight-cylinder approximation.''

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Section: Zeroth-order Revisitedmentioning

confidence: 99%

Toroidal flows in resistive magnetohydrodynamic steady states

1998

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“…where the induced magnetic field is neglected (which is justified for MHD flow at small magnetic Reynolds number by Shercliff [19]). It is also assumed that the external electric field is zero and the electric field due to polarization of charges is negligible.…”

Section: Flow Analysismentioning

confidence: 99%

Heat Transfer due to Magnetohydrodynamic Stagnation-Point Flow of a Power-Law Fluid towards a Stretching Surface in the Presence of Thermal Radiation and Suction/Injection

Mahapatra

Mondal²,

Pal

2012

ISRN Thermodynamics

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An analysis is made on the study of two-dimensional MHD (magnetohydrodynamic) boundary-layer stagnation-point flow of an electrically conducting power-law fluid over a stretching surface when the surface is stretched in its own plane with a velocity proportional to the distance from the stagnation-point in the presence of thermal radiation and suction/injection. The paper examines heat transfer in the stagnation-point flow of a power-law fluid except when the ratio of the free stream velocity and stretching velocity is equal to unity. The governing partial differential equations along with the boundary conditions are first brought into a dimensionless form and then the equations are solved by Runge-Kutta fourth-order scheme with shooting techniques. It is found that the temperature at a point decreases/increases with increase in the magnetic field when free stream velocity is greater/less than the stretching velocity. It is further observed that for a given value of the magnetic parameter M, the dimensionless rate of heat transfer at the surface and |θ (0)| decreases/increases with increase in the power-law index n. Further, the temperature at a point in the fluid decreases with increase in the radiation parameter N R when free stream velocity is greater/less than the stretching velocity.

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“…[39] for an overview of the dimensionless parameters in MHD). A limit of physical interest, hereafter referred to as the inductionless limit [1], is Pm → 0 with B x independent of z, and the Hartmann numbers H x := B x RePm 1/2 and H z := B z RePm 1/2 , measuring the square root of the ratio of Lorentz to viscous forces, non-negligible.…”

Section: Governing Equationsmentioning

confidence: 99%

A spectral Galerkin method for the coupled Orr–Sommerfeld and induction equations for free-surface MHD

Giannakis

Fischer

Rosner

2009

Journal of Computational Physics

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We develop and test spectral Galerkin schemes to solve the coupled Orr-Sommerfeld (OS) and induction equations for parallel, incompressible MHD in free-surface and fixed-boundary geometries. The schemes' discrete bases consist of Legendre internal shape functions, supplemented with nodal shape functions for the weak imposition of the stress and insulating boundary conditions. The orthogonality properties of the basis polynomials solve the matrixcoefficient growth problem, and eigenvalue-eigenfunction pairs can be computed stably at spectral orders at least as large as p = 3,000 with p-independent roundoff error. Accuracy is limited instead by roundoff sensitivity due to non-normality of the stability operators at large hydrodynamic and/or magnetic Reynolds numbers (Re, Rm 4 × 10 4 ). In problems with Hartmann velocity and magnetic-field profiles we employ suitable Gauss quadrature rules to evaluate the associated exponentially weighted sesquilinear forms without error. An alternative approach, which involves approximating the forms by means of Legendre-Gauss-Lobatto (LGL) quadrature at the 2p − 1 precision level, is found to yield equal eigenvalues within roundoff error. As a consistency check, we compare modal growth rates to energy growth rates in nonlinear simulations and record relative discrepancy smaller than 10 −5 for the least stable mode in free-surface flow at Re = 3 × 10 4 . Moreover, we confirm that the computed normal modes satisfy an energy conservation law for free-surface MHD with error smaller than 10 −6 . The critical Reynolds number in free-surface MHD is found to be sensitive to the magnetic Prandtl number Pm, even at the Pm = O(10 −5 ) regime of liquid metals.

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A Textbook of Magnetohydrodynamics

Cited by 100 publications

References 0 publications

Toroidal flows in resistive magnetohydrodynamic steady states

Toroidal flows in resistive magnetohydrodynamic steady states

Heat Transfer due to Magnetohydrodynamic Stagnation-Point Flow of a Power-Law Fluid towards a Stretching Surface in the Presence of Thermal Radiation and Suction/Injection

A spectral Galerkin method for the coupled Orr–Sommerfeld and induction equations for free-surface MHD

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