Closed-form linear stability conditions for magneto-convection (original) (raw)
Related papers
The effect of uniform vertical magnetic field on the onset of steady Rayleigh-Bènard-Marangoni convection in a relatively hotter or cooler layer of electrically conducting liquid is studied theoretically by means of modified linear stability analysis. The top surface of the layer is non-deformable free where surface tension gradients arise on account of variation of temperature, and the bottom surface is rigid and thermally conducting. The eigenvalue equation is obtained by using the Fourier series method. Numerical results are obtained and presented. The results of this analysis indicate that the critical eigenvalues in the presence of magnetic field are greater in a relatively hotter layer of liquid than a cooler one under identical conditions otherwise. A detail description of marginal stability curves showing the influence of the magnetic field is presented and discussed. The asymptotic behavior of the magnetic field for large values of Chandrasekhar number is also obtained
Mathematical Problems in Engineering, 2013
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.
Rayleigh-Bénard convection with uniform vertical magnetic field
Physical review. E, Statistical, nonlinear, and soft matter physics, 2014
We present the results of direct numerical simulations of Rayleigh-Bénard convection in the presence of a uniform vertical magnetic field near instability onset. We have done simulations in boxes with square as well as rectangular cross sections in the horizontal plane. We have considered the horizontal aspect ratio η=L(y)/L(x)=1 and 2. The onset of the primary and secondary instabilities are strongly suppressed in the presence of the vertical magnetic field for η=1. The Nusselt number Nu scales with the Rayleigh number Ra close to the primary instability as [{Ra-Ra(c)(Q)}/Ra(c)(Q)](0.91), where Ra(c)(Q) is the threshold for onset of stationary convection at a given value of the Chandrasekhar number Q. Nu also scales with Ra/Q as (Ra/Q)(μ). The exponent μ varies in the range 0.39≤μ≤0.57 for Ra/Q≥25. The primary instability is stationary as predicted by Chandrasekhar. The secondary instability is temporally periodic for Pr=0.1 but quasiperiodic for Pr=0.025 for moderate values of Q. ...
Stability analysis of magnetic fluids in the presence of an oblique field and mass and heat transfer
MATEC Web of Conferences
In this paper, we investigate an analysis of the stability of a basic flow of streaming magnetic fluids in the presence of an oblique magnetic field is made. We have use the linear analysis of modified Kelvin-Helmholtz instability by the addition of the influence of mass transfer and heat across the interface. Problems equations model is presented where nonlinear terms are neglected in model equations as well as the boundary conditions. In the case of a oblique magnetic field, the dispersion relation is obtained and discussed both analytically and numerically and the stability diagrams are also obtained. It is found that the effect of the field depends strongly on the choice of some physical parameters of the system. Regions of stability and instability are identified. It is found that the mass and heat transfer parameter has a destabilizing influence regardless of the mechanism of the field.
Effect of a vertical magnetic field on turbulent Rayleigh-Bénard convection
Physical review, 2000
The effect of a vertical uniform magnetic field on Rayleigh-Bénard convection is investigated experimentally. We confirm that the threshold of convection is in agreement with linear stability theory up to a Chandrasekhar number QӍ4ϫ10 6 , higher than in previous experiments. We characterize two convective regimes influenced by MHD effects. In the first one, the Nusselt number Nu proportional to the Rayleigh number Ra, which can be interpreted as a condition of marginal stability for the thermal boundary layer. For higher Ra, a second regime NuϳRa 0.43 is obtained.
Benard convection in a non-linear magnetic fluid
Acta Mechanica, 1990
This work examines the convective instability of a horizontal layer of magnetohydrodynamic fluid of variable permeability when subjected to a non-vertical magnetic field. We use a model proposed by P. H. Roberts [9] in the context of neutron stars but the results obtained are aso relevant to the area of ferromagnetic fluids. The presence of the variable permeability has no effect on the development of instabilities through the mechanism of stationary convection but influences the threshold of overstable convection which is often the preferred mechanism in non-terrestrial applications. In the context of ferromagnetic fluids, both stationary and overstable instability can be expected to be realisable possibilities. R----where g is the acceleration of gravity, fl the uniform adverse temperature gradient, d the depth of the layer and oc, v and ~ are respectively the coefficients of volume expansion, kinematic viscosity and thermal diffusivity. If the Rayleigh number for any layer exceeds some critical value then the fluid layer would be unstable if heated from below.
A model for Rayleigh-Bénard magnetoconvection
The European Physical Journal B, 2015
A model for three-dimensional Rayleigh-Bénard convection in low-Prandtl-number fluids near onset with rigid horizontal boundaries in the presence of a uniform vertical magnetic field is constructed and analyzed in detail. The kinetic energy K, the convective entropy Φ and the convective heat flux (N u − 1) show scaling behaviour with ǫ = r − 1 near onset of convection, where r is the reduced Rayleigh number. The model is also used to investigate various magneto-convective structures close to the onset. Straight rolls, which appear at the primary instability, become unstable with increase in r and bifurcate to three-dimensional structures. The straight rolls become periodically varying wavy rolls or quasiperiodically varying structures in time with increase in r depending on the values of Prandtl number P r. They become irregular in time, with increase in r. These standing wave solutions bifurcate first to periodic and then quasiperiodic traveling wave solutions, as r is raised further. The variations of the critical Rayleigh number Raos and the frequency ωos at the onset of the secondary instability with P r are also studied for different values of Chandrasekhar's number Q.
Russian Journal of Electrochemistry, 2005
Introduction. When a current is passed through a stagnant solution between two horizontal electrodes, two states of the system can be observed: (1) The solution remains stagnant in spite of the variation in its density ρ near the electrodes. The buoyancy forces are balanced by the viscosity forces. (2) The buoyancy force initiates convective instability, the initially stagnant solution starts to flow: a solution with a higher density, which forms near the upper electrode, flows downward, and a solution with a lower density, which forms near the lower electrode, flows upward. For the limiting-current mode, to the approximation of solution electroneutrality, the problem of Rayleigh-Benard instability for a binary electrolyte is equivalent to the problem of heat convection that has been much studied [1, 2, 3, 4, 5]. In this case, only a monotonic convective instability can arise [6]. In a solution with a more complex composition, an oscillatory instability can arise along with the monotonic one. Several works were devoted to the study of monotonic and oscillatory instabilities in electrochemical systems with three types of ions [7, 8]. In these works, approximate solutions of the problem for the cathodic deposition (anodic dissolution) of metal were obtained. Systems with redox reactions were not considered in the literature. Moreover, the Rayleigh numbers, which were used in these studies, differ from the commonly accepted values. The problem of Rayleigh-Benard for heat convection in the magnetic field was studied by Chandrasekhar [1]. In this work, we will theoretically analyze the conditions of the onset of monotonic and oscillatory free-convective instabilities in the solution with three types of ions with concentrations c 1 , c 2 , and c 3 , diffusion coefficients D 1 , D 2 and D 3 , and charges z 1 , z 2 and z 3 , which is placed in the space between two plane horizontal electrodes, taking into account the migration transfer of a supporting electrolyte. The effect of applied magnetic field on the onset of free convective instability is discussed.
Applied Physics Research, 2014
We investigate the combined effects of rotation, magnetic field and helical force on the onset of stationary and oscillatory convection in a horizontal electrically conducting fluid layer heated from below with free-free boundary conditions. For this investigation the linear stability analysis studied in detail by Chandrasekhar (1961) is used. We obtain the condition for the formation of a single large-scale structure. In (Pomalégni et al., 2014) it was shown the existence of a critical value of the intensity of the helical force for which the apparition of two cells at marginal stability for the oscillatory convection is obtained. Then, we have shown here how the increasing of parameter Ta influences this critical value of the helical force intensity.
Journal of Engineering Mathematics, 1999
The onset of steady Benard-Marangoni convection in two horizontal liquid layers of electrically conducting immiscible fluids subjected to a uniform vertical magnetic field and temperature gradient is analysed by means of a combination of analytical and numerical techniques. The free surface can be either deformable or nondeformable and the interface between the fluids is always assumed to be flat. The effect of the lower layer on the critical values of Rayleigh, Marangoni and wave numbers for the onset of steady convection is investigated. When the free surface is nondeformable, the critical parameters for the onset of pure Marangoni convection are increased, whereas for the onset of pure Benard convection they are decreased compared to the single-layer model. The results for a single-layer and for two-layers are qualitatively similar for Benard-Marangoni convection when the free surface is deformable. All disturbances can be stabilized with sufficiently strong magnetic field when t...