A Linear Stability Analysis of Thermal Convection in a Fluid Layer with Simultaneous Rotation and Magnetic Field Acting in Different Directions (original) (raw)
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A New Approach for Investigating the Thermal Instability of a Fluid Layer
ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 1995
A new direct method for solving the several characteristic value problems arising in the linear theory of thermal convection in a layer of fluid heated ,from below in the absence or presence of rotation andlor magnetic field is presented. Necessary and sufficient conditions for the existence of nontrivial solutions of several characteristic value problems are derived in the general case, and then the method is favourably applied to study the thermal instability of a layer confined by any type of boundaries. The method is rigorous, simple to apply, applicable to any type of boundaries: free, rigid, mixed, perfectly conducting or non-conducting, and moreover, it is easily implemented using the available mathematical software packages. Some unsolved convection problems with rotation and magnetic field acting simultaneously are tackled for the first time using this method. MSC (1991): 76E05 3 MURPHY, J. 0.; STEINER, J. M.: The effect of a magnetic field and rotation acting simultaneously on the onset of stationary convection 4 NAKAGAWA, Y.: Experiments on the instability of a layer of mercury heated from below and subject to the simultaneous action of a in a fluid layer with rigid boundaries.
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Instability in temperature modulated rotating Rayleigh–B ´ enard convection
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THERMAL CONVECTION IN THE PRESENCE OF A MAGNETIC FIELD : NUMERICAL SIMULATION
The interaction between thermal convection and magnetic field is of interest in geophysical and astrophysical problems as well as in metallurgical processes such as casting or crystalization. A magnetic field may act in such a way to damp the convective velocity field in the melt or to reorganize the flow aligned with the magnetic field. This ability to manipulate the flow field is of technological importance in industrial processes. In this work, a direct numerical simulation of three-dimensional Boussinesq convection in a horizontal layer of electrically conducting fluid confined between two perfectly conducting horizontal plates heated from below in a gravitational and magnetic field is performed using a spectral element method. Periodic boundary conditions are assumed in the horizontal directions. The numerical model is then used to study the effects of imposing magnetic field.
Advances in Space Research, 1995
We study the linear stability of thermocapillary-driven convection in a planar unbounded layer of an electrically conducting low-Prandtl-number liquid heated from the side and subjected to a transverse magnetic field. The thresholds of convective instability for both longitudinal and oblique disturbances are calculated numerically and also asymptotically by considering the Hartmann and Prandtl numbers as large and small parameters, respectively. The magnetic field has a stabilizing effect on the flow with the critical temperature gradient for the transition from steady to oscillatory convection increasing as square of the field strength, as also does the critical frequency, while the critical wavelength reduces inversely with field strength. These asymptotics develop in a strong enough magnetic field when the instability is entirely due to the jet of the base flow confined in the Hartmann layer at the free surface. In contrast to the base flow, the critical disturbances, having a long wavelength at small Prandtl numbers, extend from the free surface into the bulk of the liquid layer over a distance exceeding the thickness of the Hartmann layer by a factor O(Pr −1/2 ). For Ha . Pr −1/2 the instability is influenced by the actual depth of the layer. For such moderate magnetic fields the instability threshold is sensitive to the thermal properties of the bottom of the layer and the dependences of the critical parameters on the field strength are more complicated. In the latter case, various instability modes are possible depending on the thermal boundary conditions and the relative magnitudes of Prandtl and Hartmann numbers.
Thermal convection in the presence of a vertical magnetic field
Acta Mechanica, 2007
The interaction between thermal convection and an external uniform magnetic field in the vertical is numerically simulated within a computational domain of a horizontally periodic convective box between upper and lower rigid plates. The numerical technique is based on a spectral element method developed earlier to simulate natural thermal convection. In this work, it is extended to a magnetoconvection problem. Its main features are the use of rescaled Legendre-Lagrangian polynomial interpolants in expanding the flow variables except the pressure for which a modal expansion in terms of lower order polynomials is used to avoid the complicated staggered grid approach. The technique is validated in the steady roll and oscillatory convective regimes where various experimental and numerical results are available in the literature. The effect of a vertical magnetic field in such a way to inhibit the convective motions has been demonstrated.
This paper deals with buoyant convection generated by a horizontal gradient of temperature in an infinite fluid layer, which is known as Hadley circulation, and studies the effects induced by applying a rotation around the vertical axis. First, the basic flow profile with rotation is derived and the influence of the rotation is depicted: The original longitudinal velocity profile is decreased in intensity when rotation is applied and its structure is progressively changed, whereas a transverse velocity component is created, which increases with the rotation intensity, overcomes the longitudinal velocity, and eventually decreases. Different asymptotic behaviors for these profiles have also been highlighted. The stability of these flows is then studied. The effects of the Prandtl number, the Taylor number, and the thermal boundary conditions are highlighted for the three types of instability occurring in such a situation (shear, oscillatory, and Rayleigh instabilities). It is observed that they are all stabilized by the rotation and that the increase of the critical thresholds is accompanied by a spinning of the wave vector corresponding to a progressive change of the orientation of the marginal perturbation rolls. Energy budgets are finally used to analyze the instability mechanisms.
International Journal of Engineering and Applied Sciences (IJEAS), 2020
Two dimensional Rayleigh-Bénard convection in a Boussinesq fluid is revisited using DTM-Padé approximation. The stationary and oscillatory instability analysis is obtained using critical Rayleigh numbers. We observe that the effect of increasing the rotational rate is to increase the critical wave number and also the solutions are oscillating more in the horizontal direction as the rotational rate is increased. We have analysed the stability conditions to determine the type of instability at the onset of convection that were dependent on the value of the Taylor number and the Prandtl number. Asymptotic limits reveal that the flow would be comprised of columns of fluid aligned with the rotational axis since as the wave number increases the rotational rate also increases and the critical wave length decreases by setting the onset of convection in the form of tall thin columns. The flow equations are solved to obtain the linearised solutions and Differential Transform Method is applied along with Padé approximation to yield solutions without discretization and linearization and results are displayed in the graphical forms.