scholarly journals A Linear Stability Analysis of Thermal Convection in a Fluid Layer with Simultaneous Rotation and Magnetic Field Acting in Different Directions

2013 ◽  
Vol 2013 ◽  
pp. 1-15 ◽  
Author(s):  
Ruben Avila ◽  
Ares Cabello
Keyword(s):  
Magnetic Field ◽  
Thermal Convection ◽  
Fluid Layer ◽  
Complex Problem ◽  
Wave Number ◽  
Critical Parameters ◽  
Onset Of Convection ◽  
Convective Thermal ◽  
Boussinesq Fluid ◽  
Simultaneous Rotation

The onset of thermal convection of a Boussinesq fluid located in an unbounded layer heated from below and subject simultaneously to rotation and magnetic field, whose vectors act in different directions, is presented. To the knowledge of the authors, the convective thermal instability analysis for this complex problem has not been previously reported. In this paper, we use the Tau Chebyshev spectral method to calculate the value of the critical parameters (wave number and Rayleigh number at the onset of convection) as a function of (i) different kinds of boundaries, (ii) angle between the three vectors, and (iii) different values of the Taylor numberT(rate of rotation) and magnetic parameterQ(strength of the magnetic force). For the classical problems previously reported in the literature, we compare our calculations with Chandrasekhar’s variational method results and show that the present method is applicable.

scholarly journals Effects of Magnetic Field and Non-Uniform Basic Temperature Gradient

2002 ◽  
Vol 1 (1) ◽  
pp. 1-14
Author(s):  
S. Pranesh
Keyword(s):  
Magnetic Field ◽  
Temperature Gradient ◽  
Fluid Layer ◽  
Suspended Particles ◽  
Conducting Fluid ◽  
Wave Number ◽  
Uniform Temperature ◽  
Onset Of Convection ◽  
Electrically Conducting ◽  
Electrically Conducting Fluid

The effects of a non-uniform temperature gradient and magnetic field on the onset of convection in a horizontal layer of Boussinesq fluid with suspended particles confined between an upper free/adiabatic boundary and a lower rigid/isothermal boundary have been considered. A linear stability analysis is performed. The microrotation is assumed to vanish at the boundaries. The Galerkin technique is used to obtain the Eigen values. The influence of various parameters on the onset of convection has been analysed. Six different non-uniform temperature profiles are considered and their comparative influence on onset is discussed. It is observed that the electrically conducting fluid layer with suspended particles heated from below is more stable compared to the classical electrically conducting fluid without Suspended particles. The critical wave number is found to be insensitive to the changes in the parameters but sensitive to the changes in the Chandrasekhar number.

The Effect Of A Magnetic Field Dependent Viscosity On The Thermal Convection In A Ferromagnetic Fluid In A Porous Medium

2004 ◽  
Vol 59 (7-8) ◽  
pp. 397-406 ◽  
Author(s):  
◽  
Pavan Kumar Bharti ◽  
Divya Sharma ◽  
R. C. Sharma
Keyword(s):  
Magnetic Field ◽  
Porous Medium ◽  
Thermal Convection ◽  
Fluid Layer ◽  
Wave Number ◽  
Medium Permeability ◽  
Field Dependent ◽  
Stabilizing Effect ◽  
Ferromagnetic Fluid ◽  
Mfd Viscosity

The effect of the magnetic field dependent (MFD) viscosity on the thermal convection in a ferromagnetic fluid in the presence of a uniform vertical magnetic field is considered for a fluid layer in a porous medium, heated from below. For a ferromagnetic fluid layer between two free boundaries an exact solution is obtained, using a linear stability analysis. For the case of stationary convection, the medium permeability has a destabilizing effect, whereas the MFD viscosity has a stabilizing effect. In the absence of MFD viscosity, the destabilizing effect of magnetization is depicted, but in its presence the magnetization may have a destabilizing or stabilizing effect. The critical wave number and critical magnetic thermal Rayleigh number for the onset of instability is determined numerically for sufficiently large values of the magnetic parameter M1. Graphs are plotted to depict the stability characteristics. The principle of exchange of stabilities is valid for a ferromagnetic fluid heated from below and saturating a porous medium.

Spherical shell rotating convection in the presence of a toroidal magnetic field

1995 ◽  
Vol 448 (1933) ◽  
pp. 245-268 ◽  
Keyword(s):  
Magnetic Field ◽  
Prandtl Number ◽  
Spherical Shell ◽  
Lorentz Force ◽  
Thermal Convection ◽  
Toroidal Magnetic Field ◽  
Wave Number ◽  
Convective Motions ◽  
Magnetic Convection ◽  
Convection Rolls

Convective instabilities of a self-gravitating, rapidly rotating fluid spherical shell are investigated in the presence of an imposed azimuthal axisymmetric magnetic field in the form of the toroidal decay mode that satisfies electrically insulating boundary conditions and has dipole symmetry. Concentration is on two major questions: how purely thermal convection of the different forms (Zhang 1992, 1994) is affected by the Lorentz force, the strength of which is measured by the Elsasser number ∧, and in what manner purely magnetic instabilities in a spherical shell (Zhang & Fearn 1993, 1994) are associated with magnetic convection. It is found that the two-dimensionality of purely thermal convection (Busse 1970) survives under the influence of a strong Lorentz force. Convective motions always attempt to satisfy the Proudman–Taylor constraint and remain predominantly two-dimensional in the whole range of ∧, 0 ≤ ∧ ≤ ∧ c , where ∧ c ═ O (10) is the critical Elsasser number for purely magnetic instabilities. Though the optimum azimuthal wave number m of convection rolls decreases drastically, from m ~ O ( T 1/6 ) at ∧ ═ 0 to m ═ O (5) at ∧ ═ O (1). We show that there exist no optimum values of ∧ that can give rise to an overall minimum of the (modified) Rayleigh number R *; the optimum value of R * is a monotonically, smoothly decreasing function of ∧, from R * ═ O ( T 1/6 ) at ∧ < O ( T -1/6 ) to R * ═ O (–10) at ∧ ═ 20. We also show that the influence of the magnetic field on thermal convection is crucially dependent on the size of the Prandtl number. At sufficiently small Prandtl number, the Poincaré convection mode (Zhang 1994) is preferred in the region 0 ≤ ∧ < ∧ c , and is only slightly affected by the presence of the toroidal magnetic field. Analytical solutions of the magnetic convection problem are then obtained based on a perturbation analysis, showing a good agreement with the numerical solution.

scholarly journals Effects of Suction–Injection–Combination (SIC) on the onset of Rayleigh–Bénard Electroconvection in a Micropolar Fluid

2015 ◽  
Vol 14 (3) ◽  
pp. 23-42 ◽  
Author(s):  
S Pranesh ◽  
Tarannum Sameena ◽  
Baby Riya
Keyword(s):  
Stability Analysis ◽  
Micropolar Fluid ◽  
Fluid Layer ◽  
Suspended Particles ◽  
Wave Number ◽  
Critical Wave Number ◽  
Onset Of Convection ◽  
Temperature Boundary ◽  
The Stability ◽  
Rayleigh Benard

The effect of Suction – injection combination on the onset of Rayleigh – Bénard electroconvection micropolar fluid is investigated by making a linear stability analysis. The Rayleigh-Ritz technique is used to obtain the eigenvalues for different velocity and temperature boundary combinations. The influence of various parameters on the onset of convection has been analysed. It is found that the effect of Prandtl number on the stability of the system is dependent on the SIC being pro-gravity or anti-gravity. A similar Pe-sensitivity is found in respect of the critical wave number. It is observed that the fluid layer with suspended particles heated from below is more stable compared to the classical fluid layer without suspended particles.

Experimental and numerical studies of magnetoconvection in a rapidly rotating spherical shell

2007 ◽  
Vol 580 ◽  
pp. 123-143 ◽  
Author(s):  
N. GILLET ◽  
D. BRITO ◽  
D. JAULT ◽  
H. C. NATAF
Keyword(s):  
Magnetic Field ◽  
Spherical Shell ◽  
Scaling Law ◽  
Companion Paper ◽  
Magnetic Study ◽  
Critical Parameters ◽  
Liquid Gallium ◽  
Length Scale ◽  
Onset Of Convection ◽  
Conductive Liquid

Thermal magnetoconvection in a rapidly rotating spherical shell is investigated numerically and experimentally in electrically conductive liquid gallium (Prandtl number P = 0.025), at Rayleigh numbers R up to around 6 times critical and at Ekman numbers E ∼ 10−6. This work follows up the non-magnetic study of convection presented in a companion paper (Gillet et al. 2007). We study here the addition of a z-invariant toroidal magnetic field to the fluid flow. The experimental measurements of fluid velocities by ultrasonic Doppler velocimetry, together with the quasi-geostrophic numerical simulations incorporating a three-dimensional modelling of the magnetic induction processes, demonstrate a stabilizing effect of the magnetic field in the weak-field case, characterized by an Elsasser number Λ < (E/P)1/3. We find that this is explained by the changes of the critical parameters at the onset of convection as Λ increases. As in the non-magnetic study, strong zonal jets of characteristic length scales ℓβ (Rhines length scale) dominates the fluid dynamics. A new characteristic of the magnetoconvective flow is the elongation of the convective cells in the direction of the imposed magnetic field, introducing a new length scale ℓφ. Combining experimental and numerical results, we derive a scaling law $\overline{U} \,{\sim}\, (\widetilde{U}_s \widetilde{U}_{\phi})^{2/3} \,{\sim}\, \widetilde{U}_s{}^{4/3} (\ell_{\phi}/\ell_{\beta})^{2/3}$ where U is the axisymmetric motion amplitude, Ũs and Ũφ are the non-axisymmetric radial and azimuthal motion amplitudes, respectively.

Bifurcations and Hyperchaos in Magnetoconvection of Non-Newtonian Fluids

2018 ◽  
Vol 28 (10) ◽  
pp. 1830034 ◽  
Author(s):  
G. C. Layek ◽  
N. C. Pati
Keyword(s):  
Magnetic Field ◽  
Prandtl Number ◽  
Couple Stress ◽  
Fluid Layer ◽  
Equilibrium Points ◽  
Period Doubling ◽  
Hyperchaotic System ◽  
Onset Of Convection ◽  
Magnetic Diffusivity ◽  
The Magnetic Field

We report the existence of multiple bifurcations, period-doubling route to chaos and hyperchaos in magnetoconvection of a shallow, electrically conducting non-Newtonian couple-stress fluid layer heated from underneath with vertically imposed magnetic field. By adopting low-order approximations, a four-dimensional dissipative hyperchaotic system with six control parameters is obtained. Both fluid couple-stress and magnetic parameters delay the onset of convection. In contrast to plain couple-stress convection, the magnetic field instigates the occurrence of Andronov–Hopf, Bogdanov–Takens bifurcations with double-homoclinic connection at the origin near the onset of magnetoconvection when the magnetic diffusivity ratio (the ratio of fluid Prandtl number to magnetic Prandtl number) is less than one. Thereafter the system exhibits periodic attractors near subcritical Hopf bifurcation at nonzero equilibrium points undergoing chaos through period-doubling process. Owing to the interaction between the magnetic field and fluid couple-stress, the present model system exhibits hyperchaos via a suddenly created instability during stable oscillation for critical values of the parameters.

An oscillatory secondary bifurcation for magnetoconvection and rotating convection at small aspect ratio

2002 ◽  
Vol 467 ◽  
pp. 241-257 ◽  
Author(s):  
A. R. HALFORD ◽  
M. R. E. PROCTOR
Keyword(s):  
Magnetic Field ◽  
Wavelength Modulation ◽  
Vertical Magnetic Field ◽  
Small Aspect Ratio ◽  
Onset Of Convection ◽  
Rapid Rotation ◽  
Long Wavelength ◽  
Cellular Modes ◽  
Secondary Bifurcation ◽  
Boussinesq Fluid

We consider the dynamics of convection in a strong vertical magnetic field, and in the presence of rapid rotation. In both these cases, in circumstances which can be realized in the laboratory, the onset of convection is in the form of tall thin cells. Because of this, the dynamics near onset is characterized by an interaction between the cellular modes and the horizontally averaged temperature profile. The effects on the dynamics are slight in the case of a Boussinesq fluid. However in both cases, when the layer is stratified (non-Boussinesq), the convection can lose stability to oscillations close to onset. Properties of the oscillations and their stability to long-wavelength modulation are extensively investigated.

Natural Convection in an Inclined Fluid Layer With a Transverse Magnetic Field: Analogy With a Porous Medium

1995 ◽  
Vol 117 (1) ◽  
pp. 121-129 ◽  
Author(s):  
P. Vasseur ◽  
M. Hasnaoui ◽  
E. Bilgen ◽  
L. Robillard
Keyword(s):  
Magnetic Field ◽  
Transverse Magnetic Field ◽  
Hartmann Number ◽  
Fluid Layer ◽  
Critical Rayleigh Number ◽  
Integral Form ◽  
Transverse Magnetic ◽  
Horizontal Layer ◽  
Governing Equations ◽  
Onset Of Convection

In this paper the effect of a transverse magnetic field on buoyancy-driven convection in an inclined two-dimensional cavity is studied analytically and numerically. A constant heat flux is applied for heating and cooling the two opposing walls while the other two walls are insulated. The governing equations are solved analytically, in the limit of a thin layer, using a parallel flow approximation and an integral form of the energy equation. Solutions for the flow fields, temperature distributions, and Nusselt numbers are obtained explicitly in terms of the Rayleigh and Hartmann numbers and the angle of inclination of the cavity. In the high Hartmann number limit it is demonstrated that the resulting solution is equivalent to that obtained for a porous layer on the basis of Darcy’s model. In the low Hartmann number limit the solution for a fluid layer in the absence of a magnetic force is recovered. In the case of a horizontal layer heated from below the critical Rayleigh number for the onset of convection is derived in term of the Hartmann number. A good agreement is found between the analytical predictions and the numerical simulation of the full governing equations.

Onset of wall-attached convection in a rotating fluid layer in the presence of a vertical magnetic field

2008 ◽  
Vol 600 ◽  
pp. 427-443
Author(s):  
J. J. SÁNCHEZ-ÁLVAREZ ◽  
E. CRESPO DEL ARCO ◽  
F. H. BUSSE
Keyword(s):  
Magnetic Field ◽  
Rotation Rate ◽  
Magnetic Energy ◽  
Fluid Layer ◽  
Travelling Waves ◽  
Critical Rayleigh Number ◽  
Vertical Axis ◽  
Vertical Magnetic Field ◽  
Onset Of Convection ◽  
Magnetic Energy Density

A horizontal fluid layer heated from below and rotating about a vertical axis in the presence of a vertical magnetic field is considered. From earlier work it is known that the onset of convection in a rotating layer usually occurs in the form of travelling waves attached to the vertical sidewalls of the layer. It is found that this behaviour persists when a vertical magnetic field is applied. When the Elsasser number Λ is kept constant and the sidewall is thermally insulating the critical Rayleigh number Rc increases in proportion to the rotation rate described by the square root of the Taylor number, τ. This asymptotic relationship is found for an electrically highly conducting sidewall as well as for an electrically insulating one. At fixed rotation rate for Q≫τ, Rc grows in proportion to Q when the sidewall is electrically highly conducting, and in proportion to Q3/4 when the sidewall is electrically insulating. Here Q is the Chandrasekhar number which is a measure of the magnetic energy density, and a thermally insulating sidewall has been assumed. Of particular interest is the possibility that the magnetic field counteracts the stabilizing influence of rotation on the onset of sidewall convection in the case of thermally insulating sidewalls. When the sidewall is thermally highly conducting, Rc for the sidewall mode grows in proportion to τ4/3. This asymptotic behaviour is found for both cases of electrical boundary conditions, but it no longer precedes the onset of bulk convection for Λ ≳ 1.

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