Uniform Vertical Rotation Research Articles

Magneto-Rotatory Convection in Couple-Stress Fluid

Background: Thermal convection is the most convective instability when crystals are produced from a single element like silicon and the thermal instability of a fluid layer heated from below plays an important role in geophysics, oceanography, atmospheric physics, etc. The flow through porous media is of considerable interest for petroleum engineers, for geophysical fluid dynamicists and has importance in chemical technology and industry. Many of the flow problems in fluids with couple-stresses indicate some possible experiments, that could be used for determining the material constants, and the results are found to differ from those of Newtonian fluid. Keeping this in view, the present work was to study the effect of a uniform vertical magnetic field on the couple-stress fluid heated from below in the presence of a uniform vertical rotation through permeable media. Methodology: The present problem is studied using the linearized stability theory, Boussinesq approximation, normal mode analysis, and the dispersion relation is obtained. Results: The stationary convection, stability of the system, and oscillatory modes are discussed. In the case of stationary convection, the rotation postpones the onset of convection. The magnetic field and couple-stress may hasten the onset of convection in the presence of rotation while in the absence of rotation; they always postpone the onset of convection. The medium permeability hastens the onset of convection in the absence of rotation while in the presence of rotation, it may postpone the onset of convection. The rotation and magnetic field are found to introduce oscillatory modes in the system which was non-existent in their absence. A sufficient condition for the non-existence of overstability is also obtained.

Linear and weakly non-linear stability analysis of oscillatory convection in rotating ferrofluid layer

In the present paper, the problem of ferroconvection in the presence of uniform vertical rotation is considered to investigate the linear and weakly non-linear oscillatory stability. Two-dimensional convective roll instability is discussed with finite-amplitude disturbances. For linear stability, as a first-order problem, the expressions for Rayleigh numbers for stationary and oscillatory convection are derived and the effects of Coriolis force and magnetic parameters on the onset of ferromagnetic convection are studied, numerically. In weakly nonlinear oscillatory analysis, the second-order and third-order stability problems are discussed and the complex Ginzburg-Landau equation describing the amplitude of convection cell in rotating ferrofluid is derived and consequently, the expression for Nusselt number representing the heat transfer rate is obtained. From the present analysis, we observed that the rotation has the usual stabilizing effect on the linear stability in ferroconvection, however, the magnetic number (M1) and the measure of nonlinearity of magnetization (M3) both have a destabilizing effect on the onset of linear instability. Also, we found that for non-linear convection, the heat transfer rate (the Nusselt number) increases with increasing values of Taylor number, magnetic number, and the measure of nonlinearity of magnetization.

Effect of anisotropy on the onset of convection in rotating bi-disperse Brinkman porous media

Thermal convection in a horizontally isotropic bi-disperse porous medium (BDPM) uniformly heated from below is analysed. The combined effects of uniform vertical rotation and Brinkman law on the stability of the steady state of the momentum equations in a BDPM are investigated. Linear and nonlinear stability analysis of the conduction solution is performed, and the coincidence between linear instability and nonlinear stability thresholds in the L^2-norm is obtained.

Numerical Investigations on Rotatry Micropolar Fluid in Hydromagnetics Permeated with Suspended Particles Saturating Porous Medium

This paper deals with the convection of micropolar fluids heated and soluted from below in the presence of suspended particles (fine dust) and uniform vertical rotation and uniform vertical magnetic field in a porous medium. Using the Boussinesq approximation, the linearized stability theory and normal mode analysis, the exact solutions are obtained for the case of two free boundaries. It is found that the presence of the suspended particles number density, the rotation parameter, stable solute, magnetic field intensity and medium permeability bring oscillatory modes which were non–existent in their absence. It is found that the presence of coupling between thermal and micropolar effects, rotation parameter, solute parameter and suspended particles may introduce overstability in the system. Graphs have been plotted by giving numerical values to the parameters accounting for rotation parameter , magnetic field solute parameter, the dynamic microrotation viscosity and coefficient of angular viscosity to depict the stability characteristics, for both the cases of stationary convection and overstability. It is found that Rayleigh number for the case of overstability and stationary convection increases with increase in rotation parameter, as well as with magnetic field intensity, solute parameter and decreases with increase in micropolar coefficients and medium permeability, for a fixed wave number, implying thereby the stabilizing effect of rotation parameter, magnetic field intensity ,solute parameter and destabilizing effect of micropolar coefficients and medium permeability on the thermosolutal convection of micropolar fluids.

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

The 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 <i>et al</i>. [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 <i>M</i><i>1</i>. 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.

Triple-diffusive Convection in a Micropolar Rotating Ferrofluid

This paper deals with the theoretical investigation of the triple-diffusive convection in a micropolar ferrofluid layer heated and soluted below subjected to a transverse uniform magnetic field in the presence of uniform vertical rotation. For a flat fluid layer contained between two free boundaries, an exact solution is obtained. A linear stabilityanalysis theory and normal mode analysis method have been carried out to study the onset convection. The influence of various parameters like rotation, solute gradients, and micropolar parameters (i.e. coupling parameter, spindiffusion parameter and micropolar heat conduction parameter) has been analyzed on the onset of stationary convection. The critical magnetic thermal Rayleigh number for the onset of instability is also determined numerically for sufficiently large value of buoyancy magnetization parameter M1(ratio of the magnetic to gravitational forces). The principle of exchange of stabilities is found to hold true for the micropolar fluid heated from below in the absence of micropolar viscous effect, microinertia, solute gradient and rotation. The oscillatory modes are introduced due to the presence of the micropolar viscous effect, microinertia, solute gradient and rotation, which were non-existent in their absence.

On Uniform Magneto-Rotatory Double-Diffusive Convection in Compressible Couple-Stress Fluid

The effect of uniform vertical magnetic field and uniform vertical rotation on double-diffusive convection in a layer of an electrically conducting couple-stress fluid heated and soluted from below is considered. For the case of stationary convection, it is clear that stable solute gradient, compressibility, rotation and magnetic field postpone the onset of the convection in compressible couple-stress fluid heated from below in the presence of a magnetic field. Couple-stress parameter has duel character in presence of rotation. While in absence of rotation Couple-Stress parameter has postpones the onset of convection only. Graphs have been plotted by giving numerical values to the parameters to depict the stability characteristics. Further, the solute gradient, magnetic field, rotation is found to introduce oscillatory modes in the systems that were non-existent in their absence. The sufficient conditions for the non-existence of overstability have found, the violation of which does not necessarily imply the occurrence of overstability.

On Rotatory Hydrodynamic Triply Diffusive Convection in Porous Medium: Darcy Model

Upper bounds for the complex growth rate of an arbitrary oscillatory perturbation, which may be neutral or unstable for triply diffusive convection in porous medium (Darcy model) in the presence of a uniform vertical rotation, are obtained. It is further proved that the result obtained herein is uniformly valid for quite general nature of the bounding surfaces.

Triple- Diffusive Convection in a Micropolar Ferrofluid in the Presence of Rotation

This paper deals with the theoretical investigation of the triple-diffusive convection in a micropolar ferrofluid layer heated and soluted below subjected to a transverse uniform magnetic field in the presence of uniform vertical rotation. For a flat fluid layer contained between two free boundaries, an exact solution is obtained. A linear stability analysis theory and normal mode analysis method have been employed to study the onset convection. The influence of various parameters like rotation, solute gradients, and micropolar parameters (i.e., the coupling parameter, spin diffusion parameter and micropolar heat conduction parameter) on the onset of stationary convection has been analyzed. The critical magnetic thermal Rayleigh number for the onset of instability is also determined numerically for sufficiently large value of the buoyancy magnetization parameter M1 (ratio of the magnetic to gravitational forces). The principle of exchange of stabilities is found to hold true for the micropolar fluid heated from below in the absence of micropolar viscous effect, microinertia, solute gradient and rotation. The oscillatory modes are introduced due to the presence of the micropolar viscous effect, microinertia , solute gradient and rotation, which were non-existent in their absence. In this paper, an attempt is also made to obtain the sufficient conditions for the non-existence of overstability.

A Semi-Circle Theorem in Couple-Stress Fluid in the Presence of Rotation in a Porous Medium

The thermal instability of a couple-stress fluid acted upon by uniform vertical rotation and heated from below in a porous medium is investigated. Following the linearized stability theory and normal mode analysis, the paper through mathematical analysis of the governing equations of couple-stress fluid convection with a uniform vertical rotation in porous medium, for the case of rigid boundaries shows that the complex growth rate σ of oscillatory perturbations, neutral or unstable for all wave numbers, must lie inside a semi-circle , in the right half of a complex σ-plane, where TA is the Taylor number, Pl is the dimensionless medium permeability of the porous medium and F is the couple-stress parameter, which prescribes the upper limits to the complex growth rate of arbitrary oscillatory motions of growing amplitude in a rotatory couple-stress fluid in porous medium heated from below. Further, It is established that the existence of oscillatory motions of growing amplitude in the present configuration, depends crucially upon the magnitude of the non-dimensional number , in the sense so long as , no such motions are possible, and in particular PES is valid.

A semi-circle theorem in couple-stress fluid in the presence of rotation

The thermal instability of a couple-stress fluid acted upon by uniform vertical rotation and heated from below is investigated. Following the linearized stability theory and normal mode analysis, the paper through mathematical analysis of the governing equations of couple-stress fluid convection with a uniform vertical rotation, for the case of rigid boundaries shows that the complex growth rate μ of oscillatory perturbations, neutral or unstable for all wave numbers, must lie inside a semi-circle in the right half of a complex σ-plane, where T A is the Taylor number and F is the couple-stress parameter, which prescribes the upper limits to the complex growth rate of arbitrary oscillatory motions of growing amplitude in a rotatory couple-stress fluid heated from below. Further, It is established that the existence of oscillatory motions of growing amplitude in the present configuration, depends crucially upon the magnitude of the non-dimensional number , in the sense so long as , no such motions are possible, and in particular PES is valid.

Thermosolutal Magneto - Rotatory Convection In Couple - Stress Fluid through Porous Medium

The thermosolutal convection in a couple-stress fluid layer heated and soluted from below in porous medium is considered to include the effects of uniform vertical magnetic field and uniform vertical rotation. Following the linearized stability theory and normal mode analysis, the dispersion relation is obtained. For the case of stationary convection, the stable solute gradient and rotation have stabilizing effects on the system. In the presence of rotation, the medium permeability has a destabilizing (or stabilizing) effect whereas magnetic field and couple-stress parameter have stabilizing (or destabilizing) effect on the system. On the other hand, in the absence of rotation, medium permeability has a destabilizing effect whereas magnetic field and couple-stress parameter have a stabilizing effect. The dispersion relation is also analyzed numerically. The stable solute gradient, rotation and magnetic field introduce oscillatory modes in the system, which were nonexistent in their absence. A condition for the system to be stable is obtained by using Rayleigh-Ritz inequality.

English

The thermal instability of a Rivlin-Ericksen viscoelastic fluid, acted upon by uniform vertical rotation and heated from below, is investigated. Following linearized stability theory and normal mode analysis, the mathematical analysis of the governing equations of Rivlin-Ericksen viscoelastic fluid convection with a uniform vertical rotation is performed. It is shown that for the cases of rigid boundaries the complex growth rate  of oscillatory perturbations, neutral or unstable for all wave numbers, must lie inside a semi-circle, in the right-hand half of a complex  -plane with the center at the origin. This prescribes the upper limits to the complex growth rate of arbitrary oscillatory motions of growing amplitude in a rotatory Rivlin-Ericksen viscoelastic fluid heated from below. Furthermore, the conditions necessary for the existence of oscillatory motions of growing amplitude in the present configuration and the sufficient condition for the validity of the Principle of Exchange of Stabilities are established.

Rotatory Thermosolutal Convection in a Couple-Stress Fluid

Abstract The thermosolutal instability of couple-stress fluid in the presence of uniform vertical rotation is considered. Following the linearized stability theory and normal mode analysis, the dispersion is obtained. For the case of stationary convection, the stable solute gradient and rotation have stabilizing effects on the system, whereas the couple-stress has both stabilizing and destabilizing effects. The dispersion relation is also analyzed numerically. The stable solute gradient and the rotation introduce oscillatory modes in the system, which did not occur in their absence. The sufficient conditions for the non-existence of overstability are also obtained.

Thermal convection of micropolar fluid in the presence of suspended particles in rotation

A study has been made of the convection of micropolar fluids heated from below in the presence of suspended particles (fine dust) and uniform vertical rotation Ω ­(0; 0;­ Ω ). The effect of Coriolis forces on the stability is chosen along the direction of the gravitational field. It is found that the presence of coupling between thermal and micropolar effects, rotation parameter and suspended particles may introduce overstability in the system. Using the Boussinesq approximation, the linearized stability theory and normal mode analysis, the exact solutions are obtained for the case of two free boundaries. Graphs have been plotted by giving numerical values to the parameters accounting for rotation Ω­ (0; 0;­ Ω ­) and the dynamic microrotation viscosity kappa and coefficient of angular viscosity γ' to depict the stability characteristics, for both the cases of stationary convection and overstability. It is found that Rayleigh number for the case of overstability and stationary convection increases with increase in rotation parameters and decreases with increase in micropolar coefficients, for a fixed wave number, showing thereby the stabilizing effect of rotation parameters and destabilizing effect of micropolar coefficients on the thermal convection of micropolar fluids. Thus there is a competition between the stabilizing effect of rotation parameters and destabilizing effect of micropolar coeffcients and the suspended particles. It is also found from the graphs that the Rayleigh number for the case of overstability is always smaller than the Rayleigh number for the case of stationary convection, for a fixed wave number.

Thermal convection in micropolar ferrofluid in the presence of rotation

A linear stability analysis is performed for a micropolar ferrofluid layer, heated from below subjected to a transverse uniform magnetic field in the presence of uniform vertical rotation. For a flat fluid layer contained between two free boundaries, an exact solution is obtained using normal mode analysis method. For the case of stationary convection, the effect of various parameters like rotation, magnetic parameter M 3 (measures the departure of linearity in the magnetic equation of state), micropolar parameters (coupling, spin diffusion and micropolar heat conduction) has been analysed. The critical magnetic thermal Rayleigh number for the onset of instability is also determined numerically for sufficiently large values of magnetic parameter M 1 (ratio of the magnetic to gravitational forces) and results are depicted graphically. The principle of exchange of stabilities is found to hold true for the micropolar ferrofluid heated from below in the absence of micropolar viscous effect, microinertia and rotation. The oscillatory modes are introduced due to the presence of the micropolar viscous effect, microinertia and rotation, which were non-existent in their absence. The sufficient conditions for the non-existence of overstability are also obtained.

Rayleigh-Taylor Instability of a Partially Ionized Rotating Plasma in the Presence of a Variable Horizontal Magnetic Field in Porous Medium

Abstract The Rayleigh-Taylor instability of a partially ionized plasma in a porous medium is considered in the presence of a variable horizontal magnetic field and uniform vertical rotation. The case of exponentially varying density, viscosity, magnetic field, and neutral particle number density is considered. In this case, the magnetic field succeeds in stabilizing waves in a certain wavenumber range which were unstable in the absence of magnetic field, whereas the system is found to be stable for potentially stable configuration/stable stratifications. The medium permeability and rotation do not have any qualitative effect on the nature of the stability or instability.

Some characterization theorems in rotatory magneto thermohaline convection

The present paper extends the results of Banerjeeet al [2] for the hydromagnetic thermohaline convection problems of Veronis’ [9] and Stern’s [8] types to include the effect of a uniform vertical rotation.

On Rayleigh-B�nard convection with rotation and magnetic field

A sufficient condition for the validity of the principle of exchange of stabilities in the Rayleigh-Benard convection problem under the simultaneous action of a uniform vertical rotation and magnetic field is derived. Furthermore the complex growth rate of an arbitrary oscillatory perturbation, neutral or unstable, is shown to lie inside a semicircle in the right half of the complexp-plane. These results are uniformly valid for all combinations of free insulating, rigid perfectly conducting and rigid insulating boundaries.

Effect of modulation on the onset of thermal convection in a rotating fluid

In the present paper, the problem of ferroconvection in the presence of uniform vertical rotation is considered to investigate the linear and weakly non-linear oscillatory stability. Two-dimensional convective roll instability is discussed with finite-amplitude disturbances. For linear stability, as a first-order problem, the expressions for Rayleigh numbers for stationary and oscillatory convection are derived and the effects of Coriolis force and magnetic parameters on the onset of ferromagnetic convection are studied, numerically. In weakly nonlinear oscillatory analysis, the second-order and third-order stability problems are discussed and the complex Ginzburg-Landau equation describing the amplitude of convection cell in rotating ferrofluid is derived and consequently, the expression for Nusselt number representing the heat transfer rate is obtained. From the present analysis, we observed that the rotation has the usual stabilizing effect on the linear stability in ferroconvection, however, the magnetic number (M1) and the measure of nonlinearity of magnetization (M3) both have a destabilizing effect on the onset of linear instability. Also, we found that for non-linear convection, the heat transfer rate (the Nusselt number) increases with increasing values of Taylor number, magnetic number, and the measure of nonlinearity of magnetization.