Critical Magnetic Thermal Rayleigh Number Research Articles

Effect of anisotropy pores on thermohaline convective instability in micropolar ferrofluid

This work deals with the thermohaline convective instability on horizontal micropolar ferrofluid layer in anisotropic porous effect subjected to a transverse uniform magnetic field. The linear stability analysis is used to omit the non-linear terms on governing equations, normal mode analysis is taken to the study and free boundary conditions are applied. The critical magnetic thermal Rayleigh number N SC is calculated analytically for sufficient large values of M 1 The principle of exchange of stabilities is satisfied for micropolar ferrofluid in an anisotropic porous medium without N 1, N ’ 3, N ’ 5 and τ. The sufficient conditions for the non-existent of overstability are also examined. Moreover, in this work, we tried to investigate the anisotropy effect on porous medium on the system. The parameters N 1and N 5 ’ dominate the system and the large porous medium is taken into account.

ANALYTICAL STUDY ON THERMOHALINE CONVECTIVE INSTABILITY IN A MICROPOLAR FERROFLUID

The present investigation is on linear analytical study of thermohaline convective instability in micropolar ferrofluid using perturbation technique. The fluid layer is heated from below and salted from above. The theory of linear stability is used to reduce the non-linear effects on governing equations and normal mode analysis is taken to study. The critical magnetic thermal Rayleigh number NSC is obtained for sufficient large value of M1 and an oscillatory instability is determined. The parameters and are analyzed for stabilizing behavior and, and M3 give the destabilizing behavior. The results are depicted graphically.

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.

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.

Linear stability of triple-diffusive convection in micropolar ferromagnetic fluid saturating porous medium

The triple-diffusive convection in a micropolar ferromagnetic fluid layer heated and soluted from below is considered in the presence of a transverse uniform magnetic field. An exact solution is obtained for a flat fluid layer contained between two free boundaries. A linear stability analysis and a normal mode analysis method are carried out to study the onset convection. For stationary convection, various parameters such as the medium permeability, the solute gradients, the non-buoyancy magnetization, and the micropolar parameters (i.e., the coupling parameter, the spin diffusion parameter, and the micropolar heat conduction parameter) are analyzed. The critical magnetic thermal Rayleigh number for the onset of instability is determined numerically for a sufficiently large value of the buoyancy magnetization parameter M1. The principle of exchange of stabilities is found to be true for the micropolar fluid heated from below in the absence of the micropolar viscous effect, the microinertia, and the solute gradients. The micropolar viscous effect, the microinertia, and the solute gradient introduce oscillatory modes, which are non-existent in their absence. Sufficient conditions for the non-existence of overstability are also obtained.

Triple -Diffusive Convection in a Magnetized Ferrofluid With MFD Viscosity: A Nonlinear Stability Analysis

A nonlinear stability analysis is performed for a triple- diffusive convection in a magnetized ferrofluid with magnetic field -dependent viscosity (MFD) for stress- free boundaries. The major mathematical emphasis is on how to control the non-linear terms caused by magnetic body force and inertia forces. A suitable generalized energy functional is introduced to perform the nonlinear energy stability analysis. It is found that nonlinear critical stability magnetic thermal Rayleigh number does not coincide with that of linear instability, and thus indicate that the subcritical instabilities are possible. However, it is noted that in case of non-ferrofluid global nonlinear stability Rayleigh number is exactly same as that of linear instability. For lower values of magnetic parameters, this coincidence is immediately lost. The effects of magnetic parameter �� 3 , solute gradients �� 1 & �� 2and MFD viscosity parameter �� , on the subcritical instability region have also been analyzed.

Effect of Rotation on Double-Diffusive Convection in a Magnetized Ferrofluid with Internal Angular Momentum

This paper deals with the theoretical investigation of the effect of rotation in a magnetized ferrofluid with internal angular momentum, heated and soluted from below subjected to a transverse uniform magnetic field. 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 carried out to study the onset of convection. The influence of various parameters like rotation, solute gradient, magnetization and internal angular momentum parameters (i.e. coupling parameter, spin diffusion parameter and 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 values of buoyancy magnetization parameter 1 M and results are depicted graphically. The principle of exchange of stabilities is found to hold true for the ferrofluid with internal angular momentum heated from below in the absence of rotation, coupling between vorticity and spin, microinertia and solute gradient. The oscillatory modes are introduced due to the presence of the rotation, coupling between vorticity and spin, microinertia and solute gradient, 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 nonlinear stability analysis of a rotating double-diffusive magnetized ferrofluid

A nonlinear stability result for a double-diffusive magnetized ferrofluid layer rotating about a vertical axis for stress-free boundaries is derived via generalized energy method. The mathematical emphasis is on how to control the nonlinear terms caused by magnetic body and inertia forces. The result is compared with the result obtained by linear instability theory. The critical magnetic thermal Rayleigh number given by energy theory is slightly less than those given by linear theory and thus indicates the existence of subcritical instability for ferrofluids. For non-ferrofluids, it is observed that the nonlinear critical stability thermal Rayleigh number coincides with that of linear critical stability thermal Rayleigh number. For lower values of magnetic parameters, this coincidence is immediately lost. The effect of magnetic parameter, M3, solute gradient, S1, and Taylor number, TA1, on subcritical instability region have been analyzed. We also demonstrate coupling between the buoyancy and magnetic forces in the nonlinear stability analysis.

EFFECT OF SUSPENDED PARTICLES ON MARGINAL STABILITY OF A DOUBLE-DIFFUSIVE MAGNETIZED FERROFLUID WITH INTERNAL ANGULAR MOMENTUM

This article deals with the theoretical investigation of the effect of suspended particles on the marginal stability of a double-diffusive ferrofluid layer with internal angular momentum heated from below subjected to a transverse uniform magnetic field. For a flat fluid layer contained between two free boundaries, an exact solution is obtained using a linear stability analysis theory and normal mode analysis method. For the case of stationary convection, the effect of various parameters like suspended particles, solute gradient, magnetization, coupling parameter, spin diffusion parameter, and heat conduction parameter has been analyzed. The critical magnetic thermal Rayleigh number for the onset of instability is also determined numerically for sufficiently large values of magnetic parameter M1 and results are depicted graphically. It is observed that the critical magnetic thermal Rayleigh number is reduced because the heat capacity of clean fluid is supplemented by that of the suspended particles. The principle of exchange of stabilities is found to hold true for the ferrofluid with internal angular momentum in the absence of suspended particles, solute gradient, coupling parameter, and microinertia. Oscillatory modes are introduced due to the presence of the suspended particles, solute gradient, coupling parameter, and microinertia, which were nonexistent in their absence. The sufficient conditions for the nonexistence of overstability are also obtained.

The Effect of Suspended Particles on Marginal Stability of Magnetized Ferrofluid with Internal Angular Momentum

This paper deals with the theoretical investigation of the effect of dust particles on the marginal stability of a ferrofluid layer with internal angular momentum heated from below subjected to a transverse uniform magnetic field. For a flat fluid layer contained between two free boundaries, an exact solution is obtained using a linear stability analysis theory and normal mode analysis method. For the case of stationary convection, the effect of various parameters like dust particles, magnetization, coupling parameter, spin diffusion parameter, and heat conduction parameter has been analyzed. The critical magnetic thermal Rayleigh number for the onset of instability is also determined numerically for sufficiently large values of the magnetic parameter M1 and results are depicted graphically. It is observed that the critical magnetic thermal Rayleigh number is reduced because the heat capacity of clean fluid is supplemented by that of dust particles. The principle of exchange of stabilities is found to hold true for the ferrofluid with internal angular momentum in the absence of dust particles, coupling parameter, and microinertia. The oscillatory modes are introduced due to the presence of dust particles, coupling parameter, and microinertia, which were nonexistent in their absence. The sufficient conditions for the nonexistence of overstability are also obtained.

Effect of dust particles on a layer of micropolar ferromagnetic fluid heated from below saturating a porous medium

The problem of the effect of dust particles on the thermal convection in micropolar ferromagnetic fluid saturating a porous medium subject to a transverse uniform magnetic field has been investigated theoretically. Linear stability analysis and normal mode analysis methods are used to find an exact solution for a flat micropolar ferromagnetic fluid layer contained between two free boundaries. In case of stationary convection, the effect of various parameters like medium permeability, dust particles, non-buoyancy magnetization, coupling parameter, spin-diffusion parameter and micropolar heat conduction parameter are analyzed. For sufficiently large values of magnetic parameter M 1 , the critical magnetic thermal Rayleigh number for the onset of instability is determined numerically and results are depicted graphically. It is also observed that the critical magnetic thermal Rayleigh number is reduced solely because the heat capacity of clean fluid is supplemented by that of the dust particles. The principle of exchange of stabilities is found to hold true for the micropolar ferromagnetic fluid saturating a porous medium heated from below in the absence of micropolar viscous effect, microinertia and dust particles.

Thermoconvective magnetized ferrofluid with internal angular momentum saturating a porous medium: A nonlinear stability analysis

The generalized energy method is developed to study the nonlinear stability analysis for a magnetized ferrofluid layer with intrinsic rotation of particles, heated from below saturating a porous medium, in the stress-free boundary case. The mathematical emphasis is on how to control the nonlinear terms caused by magnetic body force and body couple on a fluid element. By introducing a suitable generalized energy functional, we perform a nonlinear energy stability (conditional) analysis. It is found that the nonlinear critical stability magnetic thermal Rayleigh number does not coincide with that of linear instability analysis, and thus indicates that the subcritical instabilities are possible. However, it is noted that, in case of non-ferrofluid, global nonlinear stability Rayleigh number is exactly the same as that for linear instability. For lower values of magnetic parameters, this coincidence is immediately lost. The effect of magnetic parameter M 3, medium permeability Da, coupling parameter N 1, and spin diffusion parameter N 3 ′ , on subcritical instability region has also been analyzed. It is shown that with the increase of magnetic parameter ( M 3) and Darcy number ( Da), the subcritical instability region between the two theories decreases quickly while with the increase of N 1 and N 3 ′ , the subcritical instability region between the two theories increases. We also demonstrate coupling between the buoyancy and magnetic forces in nonlinear energy stability analysis as well as in linear instability analysis.

A nonlinear stability analysis for rotating magnetized ferrofluid heated from below saturating a porous medium

A nonlinear (energy) stability analysis is performed for a rotating magnetized ferrofluid layer heated from below saturating a porous medium, in the stress-free boundary case. By introducing a generalized energy functional, a rigorous nonlinear stability result for a thermoconvective rotating magnetized ferrofluid is derived. The mathematical emphasis is on how to control the nonlinear terms caused by magnetic body force. It is found that the nonlinear critical stability magnetic thermal Rayleigh number does not coincide with that of linear instability analysis, and thus indicates that the subcritical instabilities are possible. However, it is noted that, in case of non-ferrofluid, global nonlinear stability Rayleigh number is exactly the same as that for linear instability. For lower values of magnetic parameters, this coincidence is immediately lost. The effect of magnetic parameter, M 3, medium permeability, D a , and rotation, $$T_{A_1}$$ , on subcritical instability region has also been analyzed. It is shown that with the increase of magnetic parameter, M 3, and Darcy number, D a , the subcritical instability region between the two theories decreases quickly while with the increase of Taylor number, $$T_{A_1} $$ , the subcritical region expands. We also demonstrate coupling between the buoyancy and magnetic forces in the presence of rotation in nonlinear energy stability analysis as well as in linear instability analysis.

A nonlinear stability analysis for rotating magnetized ferrofluid heated from below

A nonlinear (energy) stability analysis is performed for a rotating magnetized ferrofluid layer heated from below, in the stress-free boundary case. By introducing a generalized energy functional, a rigorous nonlinear stability result for a thermoconvective rotating magnetized ferrofluid is derived. The mathematical emphasis is on how to control the nonlinear terms caused by magnetic body and inertia forces. It is found that the nonlinear critical stability magnetic thermal Rayleigh number does not coincide with that of linear instability analysis, and thus indicates that the subcritical instabilities are possible. However, it is noted that, in case of non-ferrofluid, global nonlinear stability Rayleigh number is exactly the same as that for linear instability. For lower values of magnetic parameters, this coincidence is immediately lost. The effect of magnetic parameter, M 3, and rotation parameter, T A 1 , on subcritical instability region has also been analyzed. It is shown that with the increase of magnetic parameter, M 3, the subcritical instability region between the two theories decreases quickly while with the increase of Taylor number, T A 1 , the subcritical region expands a little for small values of T A 1 and expands significantly for large values of T A 1 . We also demonstrate coupling between the buoyancy and magnetic forces in the presence of rotation in the nonlinear energy stability analysis as well as in linear instability analysis.

A Nonlinear Stability Analysis for Thermoconvective Magnetized Ferrofluid Saturating a Porous Medium

The generalized energy method is developed to study the nonlinear stability analysis for a magnetized ferrofluid layer heated from below saturating a porous medium, in the stress-free boundary case. The mathematical emphasis is on how to control the nonlinear terms caused by magnetic body force. By introducing a suitable generalized energy functional, we perform a nonlinear energy stability (conditional) analysis. It is found that the nonlinear critical stability magnetic thermal Rayleigh number does not coincide with that of linear instability analysis, and thus indicates that the subcritical instabilities are possible. However, it is noted that, in case of non-ferrofluid, global nonlinear stability Rayleigh number is exactly the same as that for linear instability. For lower values of magnetic parameters, this coincidence is immediately lost. The effect of magnetic parameter, M 3, and medium permeability, Da, on subcritical instability region has also been analyzed. It is shown that with the increase of magnetic parameter (M 3) and Darcy number (Da), the subcritical instability region between the two theories decreases quickly. We also demonstrate coupling between the buoyancy and magnetic forces in nonlinear energy stability analysis as well as in linear instability analysis.

Effect of magnetic field dependent viscosity on ferroconvection in the presence of dust particles

This paper deals with the theoretical investigation of the effect of magnetic field dependent (MFD) viscosity on the thermal convection in a ferromagnetic fluid in the presence of dust particles. For a flat ferromagnetic fluid layer contained between two free boundaries, the exact solution is obtained using a linear stability analysis and a normal mode analysis method. For the case of stationary convection, dust particles always have a destabilizing effect, whereas the MFD viscosity has a stabilizing effect on the onset of convection. In the absence of MFD viscosity, the destabilizing effect of magnetization is depicted but in the presence of MFD viscosity, non-buoyancy magnetization may have a destabilizing or a stabilizing effect on the onset of convection. The critical wave number and critical magnetic thermal Rayleigh number for the onset of stationary convection are also determined numerically for sufficiently large values of buoyancy magnetization parameter M 1. Graphs have been plotted by giving numerical values to the parameters to depict the stability characteristics. It is observed that the critical magnetic thermal Rayleigh number is reduced solely because the heat capacity of clean fluid is supplemented by that of the dust particles. The principle of exchange of stabilities is found to hold true for the ferromagnetic fluid heated from below in the absence of dust particles. The oscillatory modes are introduced due to the presence of the dust particles, which were non-existent in their absence. A sufficient condition for the non-existence of overstability is also obtained.

EFFECT OF MAGNETIC FIELD–DEPENDENT VISCOSITY ON THERMAL CONVECTION IN A FERROMAGNETIC FLUID

This article deals with the theoretical investigation of the effect of magnetic field–dependent (MFD) viscosity on a layer of ferromagnetic fluid heated from below subject to a transverse uniform magnetic field. For a flat fluid layer contained between two free boundaries, an exact solution is obtained using a linear stability analysis and normal mode analysis method. For the case of stationary convection, the MFD viscosity has a stabilizing effect, whereas the departure of linearity in the magnetic equation of state has a destabilizing (or stabilizing) effect on the system under certain conditions. The critical wave number and critical magnetic thermal Rayleigh number for the onset of instability are also determined numerically for large values of buoyancy magnetization, and results are depicted graphically. The principle of exchange of stabilities is valid for the ferromagnetic fluid heated from below.

A nonlinear stability analysis for magnetized ferrofluid heated from below

A nonlinear (energy) stability analysis is performed for a magnetized ferrofluid layer heated from below, in the stress-free boundary case. By introducing a suitable generalized energy functional, a rigorous nonlinear stability result is derived for a thermoconvective magnetized ferrofluid. The mathematical emphasis is on how to control the nonlinear terms caused by the magnetic body and inertia forces. It is found that the nonlinear critical stability magnetic thermal Rayleigh number does not coincide with that of the linear instability analysis, and thus indicates that the subcritical instabilities are possible. However, it is noted that, in the case of non-ferrofluid, global nonlinear stability Rayleigh number is exactly the same as that for linear instability. For lower values of magnetic parameters, this coincidence is immediately lost. The effect of magnetic parameter,M3, on subcritical instability region has also been analysed. It is shown that with the increase of magnetic parameter,M3, the subcritical instability region between the two theories decreases quickly. We also demonstrate coupling between the buoyancy and the magnetic forces in the nonlinear energy stability analysis.

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.

Linear stability of double-diffusive convection in a micropolar ferromagnetic fluid saturating a porous medium

This paper deals with the theoretical investigation of the double-diffusive convection in a micropolar ferromagnetic fluid layer heated and soluted from below saturating a porous medium subjected to a transverse uniform magnetic field. 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 carried out to study the onset of convection. The influence of various parameters like medium permeability, solute gradient, non-buoyancy magnetization and micropolar parameters (i.e. coupling parameter, spin diffusion 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 values of buoyancy magnetization parameter M 1 and results are depicted graphically. The principle of exchange of stabilities is found to hold true for the micropolar ferromagnetic fluid saturating a porous medium heated from below in the absence of micropolar viscous effect, microinertia and solute gradient. The oscillatory modes are introduced due to the presence of the micropolar viscous effect, microinertia and solute gradient, 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.