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.

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 Large-scale dynamo action of magnetized Taylor–Couette flows

2020 ◽  
Vol 493 (1) ◽  
pp. 1249-1260
Author(s):  
G Rüdiger ◽  
M Schultz
Keyword(s):  
Magnetic Field ◽  
Large Scale ◽  
Mean Field ◽  
Single Mode ◽  
Eddy Diffusivity ◽  
Magnetic Reynolds Number ◽  
Dynamo Model ◽  
Turbulence Energy ◽  
Dynamo Action ◽  
Taylor Couette

ABSTRACT A conducting Taylor–Couette flow with quasi-Keplerian rotation law containing a toroidal magnetic field serves as a mean-field dynamo model of the Tayler–Spruit type. The flows are unstable against non-axisymmetric perturbations which form electromotive forces defining α effect and eddy diffusivity. If both degenerated modes with m = ±1 are excited with the same power then the global α effect vanishes and a dynamo cannot work. It is shown, however, that the Tayler instability produces finite α effects if only an isolated mode is considered but this intrinsic helicity of the single-mode is too low for an α2 dynamo. Moreover, an αΩ dynamo model with quasi-Keplerian rotation requires a minimum magnetic Reynolds number of rotation of Rm ≃ 2000 to work. Whether it really works depends on assumptions about the turbulence energy. For a steeper-than-quadratic dependence of the turbulence intensity on the magnetic field, however, dynamos are only excited if the resulting magnetic eddy diffusivity approximates its microscopic value, ηT ≃ η. By basically lower or larger eddy diffusivities the dynamo instability is suppressed.

scholarly journals Falkner-Skan Flow of a Maxwell Fluid with Heat Transfer and Magnetic Field

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
M. Qasim ◽  
S. Noreen
Keyword(s):  
Magnetic Field ◽  
Heat Transfer ◽  
Prandtl Number ◽  
Comparative Study ◽  
Homotopy Analysis Method ◽  
Maxwell Fluid ◽  
Deborah Number ◽  
Analysis Method ◽  
Flow And Heat Transfer ◽  
Homotopy Analysis

This investigation deals with the Falkner-Skan flow of a Maxwell fluid in the presence of nonuniform applied magnetic fi…eld with heat transfer. Governing problems of flow and heat transfer are solved analytically by employing the homotopy analysis method (HAM). Effects of the involved parameters, namely, the Deborah number, Hartman number, and the Prandtl number, are examined carefully. A comparative study is made with the known numerical solution in a limiting sense and an excellent agreement is noted.

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.

scholarly journals Modeling the Natural Convection Flow in a Square Porous Enclosure Filled with a Micropolar Nanofluid under Magnetohydrodynamic Conditions

2020 ◽  
Vol 10 (5) ◽  
pp. 1633 ◽  
Author(s):  
Nikolaos P. Karagiannakis ◽  
George C. Bourantas ◽  
Eugene D. Skouras ◽  
Vassilios C. Loukopoulos ◽  
Karol Miller ◽  
...  
Keyword(s):  
Magnetic Field ◽  
Heat Transfer ◽  
Volume Fraction ◽  
Solid Volume Fraction ◽  
Convection Flow ◽  
Point Collocation ◽  
Heat Transfer Efficiency ◽  
Micropolar Nanofluid ◽  
Stream Function Formulation ◽  
Porous Enclosure

The laminar, natural convective flow of a micropolar nanofluid in the presence of a magnetic field in a square porous enclosure was studied. The micropolar nanofluid is considered to be an electrically conductive fluid. The governing equations of the flow problem are the conservation of mass, energy, and linear momentum, as well as the angular momentum and the induction equations. In the proposed model, the Darcy–Brinkman momentum equations with buoyancy and advective inertia are used. Experimentally obtained forms of the dynamic viscosity, the thermal conductivity, and the electric conductivity are employed. A meshless point collocation method has been applied to numerically solve the flow and transport equations in their vorticity-stream function formulation. The effects of characteristic dimensionless parameters, such as the Rayleigh and Hartmann numbers, for a range of porosity and solid volume fraction of Al2O3 particles in a water-based micropolar nanofluid on the flow and heat transfer in the cavity are investigated. The results indicate that the intensity of the magnetic field significantly affects both the flow and the temperature distributions. Moreover, the addition of nanoparticles deteriorates the heat-transfer efficiency under specific conditions.

Sign in / Sign up

Export Citation Format

Share Document