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.

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 Effect of Magnetic Field Dependent Viscosity on Ferromagnetic Convection in a Rotating Sparsely Distributed Porous Medium - Revisited

2020 ◽  
Vol 25 (1) ◽  
pp. 142-158
Author(s):  
J. Prakash ◽  
P. Kumar ◽  
S. Manan ◽  
K.R. Sharma
Keyword(s):  
Magnetic Field ◽  
Porous Medium ◽  
Critical Rayleigh Number ◽  
Wave Number ◽  
Stationary Mode ◽  
Free Boundaries ◽  
Field Dependent ◽  
Mfd Viscosity ◽  
Magnetic Field Dependent Viscosity ◽  
Dependent Viscosity

AbstractThe effect of magnetic field dependent (MFD) viscosity on the thermal convection in a ferrofluid layer saturating a sparsely distributed porous medium has been investigated by using the Darcy-Brinkman model in the simultaneous presence of a uniform vertical magnetic field and a uniform vertical rotation. A correction is applied to the study of Vaidyanathan et al. [11] which is very important in order to predict the correct behavior of MFD viscosity. A linear stability analysis has been carried out for stationary modes and oscillatory modes separately. The critical wave number and critical Rayleigh number for the onset of instability, for the case of free boundaries, are determined numerically for sufficiently large values of the magnetic parameter M1. Numerical results are obtained and are illustrated graphically. It is shown that magnetic field dependent viscosity has a destabilizing effect on the system for the case of stationary mode and a stabilizing effect for the case of oscillatory mode, whereas magnetization has a destabilizing effect.

scholarly journals Two-dimensional magnetohydrodynamic turbulence in the small magnetic Prandtl number limit

2012 ◽  
Vol 703 ◽  
pp. 85-98 ◽  
Author(s):  
David G. Dritschel ◽  
Steven M. Tobias
Keyword(s):  
Magnetic Field ◽  
Prandtl Number ◽  
Spectral Method ◽  
Two Dimensional ◽  
Mhd Turbulence ◽  
Magnetic Prandtl Number ◽  
Wide Range ◽  
Lagrangian Advection ◽  
Pseudo Spectral Method ◽  
Pseudo Spectral

AbstractIn this paper we introduce a new method for computations of two-dimensional magnetohydrodynamic (MHD) turbulence at low magnetic Prandtl number $\mathit{Pm}= \nu / \eta $. When $\mathit{Pm}\ll 1$, the magnetic field dissipates at a scale much larger than the velocity field. The method we utilize is a novel hybrid contour–spectral method, the ‘combined Lagrangian advection method’, formally to integrate the equations with zero viscous dissipation. The method is compared with a standard pseudo-spectral method for decreasing $\mathit{Pm}$ for the problem of decaying two-dimensional MHD turbulence. The method is shown to agree well for a wide range of imposed magnetic field strengths. Examples of problems for which such a method may prove invaluable are also given.

scholarly journals Hydromagnetic dynamos at the low Ekman and magnetic Prandtl numbers

2016 ◽  
Vol 46 (3) ◽  
pp. 221-244 ◽  
Author(s):  
Ján Šimkanin
Keyword(s):  
Magnetic Field ◽  
Magnetic Fields ◽  
Prandtl Number ◽  
Large Scale ◽  
Small Scale ◽  
Polar Regions ◽  
Magnetic Diffusion ◽  
Magnetic Prandtl Number ◽  
Prandtl Numbers ◽  
The Magnetic Field

Abstract Hydromagnetic dynamos are numerically investigated at low Prandtl, Ekman and magnetic Prandtl numbers using the PARODY dynamo code. In all the investigated cases, the generated magnetic fields are dominantly-dipolar. Convection is small-scale and columnar, while the magnetic field maintains its large-scale structure. In this study the generated magnetic field never becomes weak in the polar regions, neither at large magnetic Prandtl numbers (when the magnetic diffusion is weak), nor at low magnetic Prandtl numbers (when the magnetic diffusion is strong), which is a completely different situation to that observed in previous studies. As magnetic fields never become weak in the polar regions, then the magnetic field is always regenerated in the tangent cylinder. At both values of the magnetic Prandtl number, strong polar magnetic upwellings and weaker equatorial upwellings are observed. An occurrence of polar magnetic upwellings is coupled with a regenaration of magnetic fields inside the tangent cylinder and then with a not weakened intensity of magnetic fields in the polar regions. These new results indicate that inertia and viscosity are probably negligible at low Ekman numbers.

Magnetic Prandtl Number and Astrophysical Convection

1981 ◽  
Vol 4 (2) ◽  
pp. 208-209
Author(s):  
J. O. Murphy ◽  
J. M. Lopez
Keyword(s):  
Magnetic Field ◽  
Prandtl Number ◽  
Magnetic Prandtl Number ◽  
Convective Processes

In the astrophysical context, where the effects of a magnetic field on stellar convective processes have to be taken into account, we have ϰ > > η > > ν (Weiss 1977) where ϰ, η and ν are respectively the thermal, magnetic and viscous diffusivities.

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