scholarly journals WKB thresholds of standard, helical, and azimuthal magnetorotational instability

2012 ◽  
Vol 8 (S290) ◽  
pp. 233-234 ◽  
Author(s):  
Oleg Kirillov ◽  
Frank Stefani
Keyword(s):  
Magnetic Field ◽  
External Magnetic Field ◽  
Prandtl Number ◽  
Local Stability ◽  
Rotating Flows ◽  
Rossby Number ◽  
Magnetorotational Instability ◽  
Electrically Conducting ◽  
Magnetic Prandtl Number ◽  
Wavelength Approximation

AbstractWe consider rotating flows of an electrically conducting, viscous and resistive fluid in an external magnetic field with arbitrary combinations of axial and azimuthal components. Within the short-wavelength approximation, the local stability of the flow is studied with respect to perturbations of arbitrary azimuthal wavenumbers. In the limit of vanishing magnetic Prandtl number (Pm) we find that the maximum critical Rossby number (Ro) for the occurrence of the magnetorotational instability (MRI) is universally governed by the Liu limit ${\rm Ro}_{Liu}=2-2\sqrt{2}\approx -0.828$ which is below the value for Keplerian rotation RoKepler = −0.75.

scholarly journals Induced magnetic field with radiating fluid over a porous vertical plate: analytical study

2011 ◽  
Vol 7 (2) ◽  
pp. 61-72 ◽  
Author(s):  
Sahin Ahmed
Keyword(s):  
Magnetic Field ◽  
Prandtl Number ◽  
Porous Plate ◽  
Similarity Solutions ◽  
Convection Flow ◽  
Induced Magnetic Field ◽  
Electrically Conducting ◽  
Magnetic Prandtl Number ◽  
Body Parameter ◽  
Mhd Propulsion

The objective of this investigation is to study the influence of thermal radiation and magnetic Prandtl number on the steady MHD heat and mass transfer by mixed convection flow of a viscous, incompressible, electrically-conducting, Newtonian fluid which is an optically thin gray gas over a vertical porous plate taking into account the induced magnetic field. The similarity solutions of the transformed dimensionless governing equations are obtained by series solution. It is found that, velocity is reduced considerably with a rise in conduction-radiation parameter (R) or Hartmann number (M) whereas the skin friction is found to be markedly boosted with an increase in M or Magnetic Prandtl number (Pm). An increase in magnetic body parameter (M) or Magnetic Prandtl number (Pm) is found to escalate induced magnetic field whereas an increase in R is shown to exert the opposite effect. Applications of the study include laminar magneto-aerodynamics, materials processing and MHD propulsion thermo-fluid dynamics.DOI: 10.3329/jname.v7i2.5662

scholarly journals Self-consistent single mode investigations of the quasi-geostrophic convection-driven dynamo model

2018 ◽  
Vol 84 (4) ◽  
Author(s):  
Meredith Plumley ◽  
Michael A. Calkins ◽  
Keith Julien ◽  
Steven M. Tobias
Keyword(s):  
Magnetic Field ◽  
Heat Transfer ◽  
Prandtl Number ◽  
Single Mode ◽  
Dynamo Model ◽  
Rossby Number ◽  
Time Averaging ◽  
Heat Transfer Efficiency ◽  
Magnetic Prandtl Number ◽  
Self Consistent

The quasi-geostrophic dynamo model (QGDM) is a multiscale, fully nonlinear Cartesian dynamo model that is valid in the asymptotic limit of low Rossby number. In the additional limit of small magnetic Prandtl number investigated here, the QGDM is a self-consistent, asymptotically exact form of an $\unicode[STIX]{x1D6FC}^{2}$ large-scale dynamo. This article explores methods for simulating the multiscale QGDM and investigates how convection is altered by the magnetic field in the planetary regime of small Rossby number and small magnetic Prandtl number. At present, this combination is beyond the reach of direct numerical simulations. We use a simplified class of solutions whose horizontal structure is restricted to a periodic hexagonal lattice characterized by a single horizontal wavenumber (single mode). In contrast with previous kinematic investigations of the QGDM, the Lorentz force is included to study saturated, self-consistent dynamos. Two methodologies are used to assess handling of the multiple time scales of the QGDM: a stiff, common-in-time approach where all time scales are converted to a single time variable and a heterogeneous multiscale modelling approach employing fast time averaging on the Reynolds, magnetic and buoyancy eddy fluxes that feed back onto the slow scales. These strategies produce consistent results and each illustrates self-similar dynamics as the time-averaging window is increased. The properties of the convection are significantly altered by the dynamo-generated magnetic field. All solutions show a decrease in the overall heat transfer efficiency as compared to non-magnetic convection, suggesting that a change in length scale or flow planform plays a critical role in the enhanced heat transfer efficiency observed in previous dynamo studies. All dynamo solutions show a trend of increasing ohmic dissipation relative to viscous dissipation as the buoyancy forcing is increased.

Magnetohydrodynamic flow constructions with fundamental solutions

1961 ◽  
Vol 10 (3) ◽  
pp. 439-448 ◽  
Author(s):  
Meredith C. Gourdine
Keyword(s):  
Magnetic Field ◽  
Prandtl Number ◽  
Approximate Solutions ◽  
Reynolds Numbers ◽  
Fundamental Solutions ◽  
The Body ◽  
Steady Flows ◽  
Electrically Conducting ◽  
Electrically Conducting Fluid ◽  
Magnetic Prandtl Number

In this paper steady flows of an incompressible, viscous, electrically conducting fluid are constructed from fundamental solutions of magnetohydrodynamics in which the applied magnetic field is parallel to the velocity at infinity. The flat plate and the sphere are considered as examples, and approximate solutions are presented for the limiting cases of large and small Reynolds and magnetic Reynolds numbers. The effects of currents in the body are also considered, and it is found that unless the magnetic Prandtl number is larger than unity, currents in the body have negligible effect on the flow.

Compositional convection in the presence of a magnetic field. II. Cartesian plume

2005 ◽  
Vol 461 (2060) ◽  
pp. 2605-2633 ◽  
Author(s):  
I.A Eltayeb ◽  
E.A Hamza ◽  
J.A Jervase ◽  
E.V Krishnan ◽  
D.E Loper
Keyword(s):  
Magnetic Field ◽  
Prandtl Number ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Single Interface ◽  
Magnetic Prandtl Number ◽  
Infinite Extent ◽  
Chandrasekhar Number ◽  
The Magnetic Field ◽  
Field Inclination

The analysis of part I, dealing with the morphological instability of a single interface in a fluid of infinite extent, is extended to the case of a Cartesian plume of compositionally buoyant fluid, of thickness 2 x 0 , enclosed between two vertical interfaces. The problem depends on six dimensionless parameters: the Prandtl number, σ ; the magnetic Prandtl number, σ m ; the Chandrasekhar number, Q c ; the Reynolds number, Re ; the ratio, B v , of vertical to horizontal components of the ambient magnetic field and the dimensionless plume thickness. Attention is focused on the preferred mode of instability, which occurs in the limit Re ≪1 for all values of the parameters. This mode can be either sinuous or varicose with the wavenumber vector either vertical or oblique , comprising four types. The regions of preference of these four modes are represented in regime diagrams in the ( x 0 ,  σ ) plane for different values of σ m , Q c , B v . These regions are strongly dependent on the field inclination and field strength and, to a lesser extent, on magnetic diffusion. The overall maximum growth rate for any prescribed set of the parameters σ m , Q c , B v , occurs when 1.3< x 0 <1.7, and is sinuous for small σ and varicose for large σ . The magnetic field can enhance instability for a certain range of thickness of the plume. The enhancement of instability is due to the interaction of the field with viscous diffusion resulting in a reverse role for viscosity. The dependence of the helicity and α -effect on the parameters is also discussed.

ONSET OF INSTABILITY IN HYDROMAGNETIC COUETTE FLOW

2004 ◽  
Vol 02 (02) ◽  
pp. 145-159 ◽  
Author(s):  
ISOM H. HERRON
Keyword(s):  
Magnetic Field ◽  
Prandtl Number ◽  
Boundary Conditions ◽  
Viscous Flow ◽  
Couette Flow ◽  
Axial Magnetic Field ◽  
Narrow Gap ◽  
Rotating Cylinders ◽  
Magnetic Prandtl Number ◽  
The Stability

The stability of viscous flow between rotating cylinders in the presence of a constant axial magnetic field is considered. The boundary conditions for general conductivities are examined. It is proved that the Principle of Exchange of Stabilities holds at zero magnetic Prandtl number, for all Chandrasekhar numbers, when the cylinders rotate in the same direction, the circulation decreases outwards, and the cylinders have insulating walls. The result holds for both the finite gap and the narrow gap approximation.

scholarly journals MAGNETOHYDRODYNAMIC HEAT AND MASS TRANSFER FLOW WITH INDUCED MAGNETIC FIELD AND VISCOUS DISSIPATIVE EFFECTS

2014 ◽  
Vol 44 (1) ◽  
pp. 9-17
Author(s):  
S. AHMED ◽  
A. BATIN
Keyword(s):  
Magnetic Field ◽  
Prandtl Number ◽  
Current Density ◽  
Schmidt Number ◽  
Porous Plate ◽  
Mhd Flow ◽  
Induced Magnetic Field ◽  
Dissipative Effects ◽  
Magnetic Prandtl Number ◽  
Mhd Propulsion

An approximate solution to the problem of steady free convective MHD flow of an incompressible viscous electrically-conducting fluid over an infinite vertical isothermal porous plate with mass convection is presented here. A uniform magnetic field is assumed to be applied transversely to the direction of the flow, taking into account the induced magnetic field with viscous and magnetic dissipations of energy. The dimensionless governing equations are solved by using the series solution method. The induced magnetic field, current density, temperature gradient and flow velocity are studied for magnetohydrodynamic body force, magnetic Prandtl number, Schmidt number and Eckert number. It is observed that the induced magnetic field is found to increase with a rise in magnetic Prandtl number. Current density is strongly reduced with increasing magnetic Prandtl number, but enhanced with Schmidt number. The acquired knowledge in our study can be used by designers to control MHD flow as suitable for a certain applications such as laminar magneto-aerodynamics, and MHD propulsion thermo-fluid dynamics.

scholarly journals The convective instability of a Maxwell–Cattaneo fluid in the presence of a vertical magnetic field

2020 ◽  
Vol 476 (2241) ◽  
pp. 20200494
Author(s):  
I. A. Eltayeb ◽  
D. W. Hughes ◽  
M. R. E. Proctor
Keyword(s):  
Magnetic Field ◽  
Prandtl Number ◽  
Critical Rayleigh Number ◽  
Free Boundaries ◽  
Governing Equations ◽  
Horizontal Scale ◽  
Temperature Relation ◽  
Magnetic Prandtl Number ◽  
Zero Values ◽  
Chandrasekhar Number

We study the instability of a Bénard layer subject to a vertical uniform magnetic field, in which the fluid obeys the Maxwell–Cattaneo (MC) heat flux–temperature relation. We extend the work of Bissell ( Proc. R. Soc. A 472, 20160649 (doi:10.1098/rspa.2016.0649)) to non-zero values of the magnetic Prandtl number p m . With non-zero p m , the order of the dispersion relation is increased, leading to considerably richer behaviour. An asymptotic analysis at large values of the Chandrasekhar number Q confirms that the MC effect becomes important when C Q 1/2 is O (1), where C is the MC number. In this regime, we derive a scaled system that is independent of Q . When CQ 1/2 is large, the results are consistent with those derived from the governing equations in the limit of Prandtl number p  → ∞ with p m finite; here we identify a new mode of instability, which is due neither to inertial nor induction effects. In the large p m regime, we show how a transition can occur between oscillatory modes of different horizontal scale. For Q  ≫ 1 and small values of p , we show that the critical Rayleigh number is non-monotonic in p provided that C  > 1/6. While the analysis of this paper is performed for stress-free boundaries, it can be shown that other types of mechanical boundary conditions give the same leading-order results.

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