Thermal instability of a porous medium with relaxation and inertia in the presence of Hall effects

Summary  The thermal instability of a Rivlin–Ericksen fluid in a porous medium is considered in the presence of a uniform vertical magnetic field to include the effect of Hall currents. For the case of stationary convection, the magnetic field has a stabilizing effect on the system, whereas the Hall current has a destabilizing effect on the system. The medium permeability has both stabilizing and destabilizing effects, depending on the Hall parameter M. The kinematic viscoelasticity has no effect on stationary convection. Graphs have been plotted by giving numerical values to the parameters, to depict the stability characteristics. The magnetic field (and corresponding Hall currents) introduces oscillatory modes in the system, which would be nonexistent in their absence. The sufficient conditions for the nonexistence of overstability are also obtained. Received 20 May 1999; accepted for publication 8 March 2000     

The Effect of Magnetic Field Dependent Viscosity on Thermosolutal Convection in a Ferromagnetic Fluid Saturating a Porous Medium

The effect of magnetic field dependent viscosity on thermosolutal convection in a ferromagnetic fluid saturating a porous medium is considered for a fluid layer heated and soluted from below in the presence of uniform magnetic field. Using linearized stability theory and normal mode analysis, an exact solution is obtained for the case of two free boundaries. For case of stationary convection, medium permeability has a destabilizing effect, whereas a stable solute gradient and magnetic field dependent viscosity have a stabilizing effect on the system. In the absence of magnetic field dependent viscosity, the destabilizing effect of non-buoyancy magnetization is depicted but in the presence of magnetic field dependent viscosity non-buoyancy magnetization may have a destabilizing or stabilizing effect on the onset of instability. The critical wave number and the critical magnetic thermal Rayleigh number for the onset of instability are also determined numerically for sufficiently large values of buoyancy magnetization parameter M₁ and the results are depicted graphically. The principle of exchange of stabilities is found to hold true for the ferromagnetic fluid saturating a porous medium heated from below in the absence of stable solute gradient. The oscillatory modes are introduced due to the presence of the stable solute gradient, which were non-existent in its absence. A sufficient condition for the non-existence of overstability is also obtained. The paper also reaffirms the qualitative findings of earlier investigations which are, in fact, limiting cases of the present study.     

Rayleigh–Taylor Instability of Rotating Oldroydian Viscoelastic Fluids in Porous Medium in Presence of a Variable Magnetic Field

The instability of the plane interface between two oldroydian viscoelastic superposed fluids in the presence of uniform rotation and variable magnetic field in porous medium is considered. For potentially stable configuration, the system is found to be stable for disturbances of all wave numbers. The magnetic field succeeds in stabilizing certain wave-number range, which were unstable in the absence of magnetic field and rotation for the potentially unstable configuration. Sub cases of magnetic free and rotation free configurations are also considered, separately.     

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 M ₁. 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.     

Instabilities of a thin viscoelastic liquid film flowing down an inclined plane in the presence of a uniform electromagnetic field

The effect of a uniform electromagnetic field on the stability of a thin layer of an electrically conducting viscoelastic liquid flowing down on a nonconducting inclined plane is studied under the induction-free approximation. Long-wave expansion method is used to obtain the surface evolution equation. The stabilizing role of the magnetic parameter M and the destabilizing role of the viscoelastic parameter Γ as well as the electric parameter E on this flow field are established. A novel result which emerges from our analysis is that the stabilizing effect of M holds no longer true for both viscous and viscoelastic fluids in the presence of electromagnetic field. It is found that when E exceeds a certain critical value depending on Γ, magnetic field exhibits the destabilizing effect on this flow field. Indeed, this critical value decreases with the increase of the viscoelastic parameter Γ since it has a destabilizing effect inherently. Another noteworthy result which arises from the weakly nonlinear stability analysis is that both the subcritical unstable and supercritical stable zones are possible together with the unconditional stable and explosive zones for different values of Γ depending on the wave number k.     

The Onset of Double Diffusive Convection in a Couple Stress Fluid Saturated Anisotropic Porous Layer

The double diffusive convection in a horizontal couple stress fluid saturated anisotropic porous layer, which is heated and salted from below, is studied analytically. The modified Darcy equation that includes the time derivative term is used to model the momentum equation. The critical Rayleigh number, wavenumber for stationary and oscillatory modes, and frequency of oscillations are obtained analytically using linear theory. The effect of anisotropy parameter, solute Rayleigh number, Lewis number, couple stress parameter, and Vadasz number on the stationary, oscillatory, and finite amplitude convection is shown graphically. It is found that the thermal anisotropy parameter, couple stress parameter, and solute Rayleigh number have stabilizing effect on the stationary, oscillatory, and finite amplitude convection. The mechanical anisotropy parameter has destabilizing effect on stationary, oscillatory, and finite amplitude convection. The Lewis number has stabilizing effect in the case of stationary and finite amplitude modes, with dual effect in the case of oscillatory convection. Vadasz number advances the onset of oscillatory convection. The heat and mass transfer decrease with an increase in the values of couple stress parameter, while both increase with an increase in the value of solute Rayleigh number and mechanical anisotropy parameter. The thermal anisotropy parameter and Lewis number have contrasting effect on the heat mass transfer.     

On Perturbation and Marginal Stability Analysis of Magneto-Convection in Active Mushy Layer

This present study considers the problem of steady magneto-convection in a horizontal mushy layer with variable permeability and an impermeable mush–liquid interface during directional solidification of binary alloys. We model the flow by introducing a uniform magnetic field in the mushy layer which is considered as a porous medium where Darcy’s law holds and the permeability is a function of the local solid volume fraction. Basic-state solutions are obtained analytically using the no-flow condition. With the help of multiple shooting techniques, we obtain numerical solutions to the linear perturbation system for non-magnetic and magnetic cases. Numerical results are presented showing the effects of the magnetic field and the permeability of the layer. These results demonstrate that the application of an external magnetic field has stabilizing effects on the convection and can reduce the tendency for chimney formation in the mushy layer. In addition, variable permeability, which corresponds to an active mushy layer, indicates more stable and realizable flow system as compared to the case of constant permeability.     

Hall effect on MHD free convection boundary layer flow past a vertical flat plate

Effects of Hall current on MHD free convection boundary layer flow of a viscous incompressible electrically conducting fluid past a heated vertical flat plate of finite dimension in the presence of a uniform transverse magnetic field have been studied. An exact solution of the governing equations describing the flow has been obtained. The velocity field, induced magnetic field and bulk temperature distributions in the boundary layer flow have been discussed. It is found that the velocity components increase with an increase in Hall parameter. It is noticed that the induced magnetic field components are radically influenced by the Hall parameter. It is also found that the magnitude of bulk temperature in the x-direction decreases with an increase in either Hall parameter or magnetic parameter. On the other hand, the magnitude of the bulk temperature in the z-direction increases with an increase in Hall parameter whereas it decreases with an increase in magnetic parameter.     

Ferromagnetic Convection in a Rotating Ferrofluid Saturated Porous Layer

The effect of Coriolis force on the onset of ferromagnetic convection in a rotating horizontal ferrofluid saturated porous layer in the presence of a uniform vertical magnetic field is studied. The boundaries are considered to be either stress free or rigid. The modified Brinkman–Forchheimer-extended Darcy equation with fluid viscosity different from effective viscosity is used to characterize the fluid motion. The condition for the occurrence of direct and Hopf bifurcations is obtained analytically in the case of free boundaries, while for rigid boundaries the eigenvalue problem has been solved numerically using the Galerkin method. Contrary to their stabilizing effect in the absence of rotation, increasing the ratio of viscosities, Λ, and decreasing the Darcy number Da show a partial destabilizing effect on the onset of stationary ferromagnetic convection in the presence of rotation, and some important observations are made on the stability characteristics of the system. Moreover, the similarities and differences between free–free and rigid–rigid boundaries in the presence of buoyancy and magnetic forces together or in isolation are emphasized in triggering the onset of ferromagnetic convection in a rotating ferrofluid saturated porous layer. For smaller Taylor number domain, the stress-free boundaries are found to be always more unstable than in the case of rigid boundaries. However, this trend is reversed at higher Taylor number domain because the stability of the stress-free case is increased more quickly than the rigid case.     

Ferromagnetic Convection in a Heterogeneous Darcy Porous Medium Using a Local Thermal Non-equilibrium (LTNE) Model

The combined effects of vertical heterogeneity of permeability and local thermal non-equilibrium (LTNE) on the onset of ferromagnetic convection in a ferrofluid saturated Darcy porous medium in the presence of a uniform vertical magnetic field are investigated. A two-field model for temperature representing the solid and fluid phases separately is used. The eigenvalue problem is solved numerically using the Galerkin method for different forms of permeability heterogeneity function Γ(z) and their effect on the stability characteristics of the system has been analyzed in detail. It is observed that the general quadratic variation of Γ(z) with depth has more destabilizing effect on the system when compared to the homogeneous porous medium case. Besides, the influence of LTNE and magnetic parameters on the criterion for the onset of ferromagnetic convection is also assessed.     

Effect of rotation on ferromagnetic porous convection with a thermal non-equilibrium model

The effect of rotation on the onset of thermal convection in a horizontal layer of ferrofluid saturated Brinkman porous medium is investigated in the presence of a uniform vertical magnetic field using a local thermal non-equilibrium (LTNE) model. A two-field model for temperature representing the solid and fluid phases separately is used for energy equation. The condition for the occurrence of stationary and oscillatory convection is obtained analytically. The stability of the system has been analyzed when the magnetic and buoyancy forces are acting together as well as in isolation and the similarities as well as differences between the two are highlighted. In contrast to the non-rotating case, it is shown that decrease in the Darcy number Da and an increase in the ratio of effective viscosity to fluid viscosity Λ is to hasten the onset of stationary convection at high rotation rates and a coupling between these two parameters is identified in destabilizing the system. Asymptotic solutions for both small and large values of scaled interphase heat transfer coefficient H _t are presented and compared with those computed numerically. Besides, the influence of magnetic parameters and also parameters representing LTNE on the stability of the system is discussed and the veracity of LTNE model over the LTE model is also analyzed.     

Instability of two streaming conducting and dielectric bounded fluids in porous medium under time-varying electric field

In this paper, we consider the instability of the interface between two superposed streaming conducting and dielectric fluids of finite depths through porous medium in a vertical electric field varying periodically with time. A damped Mathieu equation with complex coefficients is obtained. The method of multiple scales is used to obtain an approximate solution of this equation, and then to analyze the stability criteria of the system. We distinguish between the non-resonance case, and the resonance case, respectively. It is found, in the first case, that both the porosity of porous medium, and the kinematic viscosities have stabilizing effects, and the medium permeability has a destabilizing effect on the system. While in the second case, it is found that each of the frequency of the electric field, and the fluid velocities, as well as the medium permeability, has a stabilizing effect, and decreases the value of the resonance point, while each of the porosity of the porous medium, and the kinematic viscosities has a destabilizing effect, and increases the value of the resonance point. In the absence of both streaming velocities and porous medium, we obtain the canonical form of the Mathieu equation. It is found that the fluid depth and the surface tension have a destabilizing effect on the system. This instability sets in for any value of the fluid depth, and by increasing the depth, the instability holds for higher values of the electric potential; while the surface tension has no effect on the instability region for small wavenumber values. Finally, the case of a steady electric field in the presence of a porous medium is also investigated, and the stability conditions show that each of the fluid depths and the porosity of the porous medium ɛ has a destabilizing effect, while the fluid velocities have stabilizing effect. The stability conditions for two limiting cases of interest, the case of purely fluids), and the case of absence of streaming, are also obtained and discussed in detail.     

Unsteady Flow Through a Highly Porous Medium in the Presence of Radiation

The unsteady natural convection flow of a viscous and incompressible fluid through a porous medium with high porosity bounded by a vertical infinite stationary plate in the presence of radiation has been investigated. The fluid is assumed to be a gray, emitting and absorbing radiation, but non-scattering medium. The effects of the radiation parameter, Grashof number and permeability parameter of the medium on the velocity field as well as the effects of the radiation parameter and Prandtl number on the temperature field have been included in the analysis.     

Double diffusive convection in a porous medium with modulated temperature on the boundaries

The effect of temperature modulation on the onset of double diffusive convection in a sparsely packed porous medium is studied by making linear stability analysis, and using Brinkman-Forchheimer extended Darcy model. The temperature field between the walls of the porous layer consists of a steady part and a time dependent periodic part that oscillates with time. Only infinitesimal disturbances are considered. The effect of permeability and thermal modulation on the onset of double diffusive convection has been studied using Galerkin method and Floquet theory. The critical Rayleigh number is calculated as a function of frequency and amplitude of modulation, Vadasz number, Darcy number, diffusivity ratio, and solute Rayleigh number. Stabilizing and destabilizing effects of modulation on the onset of double diffusive convection have been obtained. The effects of other parameters are also discussed on the stability of the system. Some results as the particular cases of the present study have also been obtained. Also the results corresponding to the Brinkman model and Darcy model have been compared.     

Electrohydrodynamic instability of two superposed Walters B´ viscoelastic fluids in relative motion through porous medium

Summary  The electrohydrodynamic Kelvin–Helmholtz instability of the interface between two uniform superposed viscoelastic (B′ model) dielectric fluids streaming through a porous medium is investigated. The considered system is influenced by applied electric fields acting normally to the interface between the two media, at which there are no surface charges present. In the absence of surface tension, perturbations transverse to the direction of streaming are found to be unaffected by either streaming and applied electric fields for the potentially unstable configuration, or streaming only for the potentially stable configuration, as long as perturbations in the direction of streaming are ignored. For perturbations in all other directions, there exists instability for a certain wavenumber range. The instability of this system can be enhanced (increased) by normal electric fields. In the presence of surface tension, it is found also that the normal electric fields have destabilizing effects, and that the surface tension is able to suppress the Kelvin–Helmholtz instability for small wavelength perturbations, and the medium porosity reduces the stability range given in terms of the velocities difference and the electric fields effect. Finally, it is shown that the presence of surface tension enhances the stabilizing effect played by the fluid velocities, and that the kinematic viscoelasticity has a stabilizing as well as a destabilizing effect on the considered system under certain conditions. Graphics have been plotted by giving numerical values to the parameters, to depict the stability characteristics. Received 27 March 2000; accepted for publication 3 May 2001     

Effect of ion‐slip current on the thermal instability of mixed connection in a boundary layer flow

Effect of ion‐slip current on the thermal instability in a boundary layer is studied. The criterion on the position marking the onset of longitudinal vortices is defined in the present paper. The results show that the onset position characterized by the Grashof number depends on the Prandtl number, the Reynolds number, the wave number, the Hall parameter, the ion‐slip parameter, and the Hartmann number. The flow becomes more stable as the magnetic field increases. However, the destabilizing effects are found on the flow when the Hall and ion‐slip currents are presented. The results of the present numerical prediction show reasonable agreement with the experimental data in the case of zero Hartmann number, ion‐slip parameter, and Hall parameter in the open literature. Copyright © 2011 John Wiley & Sons, Ltd.     

Effect of Thermal/Gravity Modulation on the Onset of Convection in a Maxwell Fluid Saturated Porous Layer

The effect of thermal/gravity modulation on the onset of convection in a Maxwell fluid saturated porous layer is investigated by a linear stability analysis. Modified Darcy–Maxwell model is used to describe the fluid motion. The regular perturbation method based on the small amplitude of modulation is employed to compute the critical Rayleigh number and the corresponding wavenumber. The stability of the system characterized by a correction Rayleigh number is calculated as a function of the viscoelastic parameter, Darcy–Prandtl number, normalized porosity, and the frequency of modulation. It is found that the low frequency symmetric thermal modulation is destabilizing while moderate and high frequency symmetric modulation is always stabilizing. The asymmetric modulation and lower wall temperature modulations are, in general, stabilizing while the system becomes unstable for large values of Darcy–Prandtl number and for small frequencies. It is shown that in general the gravity modulation produces a stabilizing effect on the onset of convection for moderate and high frequency. The small frequency gravity modulation is found to have destabilizing effect on the stability of the system.     

Rayleigh Taylor instability of rotating compressible fluid through porous media

The hydromagnetic instability of a stratified horizontal layer of viscous compressible rotating fluid through porous media in the presence of vertical magnetic field is considered. The solution has been obtained through the use of variational principle. The dispersion relation is derived for a layer having exponential density stratification along the vertical direction. It is found that viscosity has stabilizing influence while permeability of porous medium, magnetic resistivity and coriolis forces have destabilizing influence on the system.     

Onset of Double-Diffusive Reaction–Convection in an Anisotropic Rotating Porous Layer

The linear and non-linear stability of a rotating double-diffusive reaction–convection in a horizontal anisotropic porous layer subjected to chemical equilibrium on the boundaries is investigated considering a Darcy model that includes the Coriolis term. The effect of Taylor number, mechanical, and thermal anisotropy parameters, reaction rate, solute Rayleigh number, Lewis number, and normalized porosity on the stability of the system is investigated. We find that the Taylor number has a stabilizing effect, chemical reaction may be stabilizing or destabilizing and that the anisotropic parameters have significant influence on the stability criterion. The effect of various parameters on the stationary, oscillatory, and finite-amplitude convection is shown graphically. A weak nonlinear theory based on the truncated representation of Fourier series method is used to find the finite amplitude Rayleigh number and heat and mass transfer.     

Stability of resonant interfacial waves in the presence of uniform magnetic field

The aim of this work is to study the instability of interacting waves between two immiscible magnetic liquids. The effects of gravitation and a uniform normal magnetic field are taken into account. The method of multiple scales is used to determine the stability criteria of the considered problem. The various stability criteria are discussed both analytically and graphically. According to the numerical examples, we have remarked that the increase of the ratio of the permeability of the liquids appears to be the destabilizing effect of the magnetic field. The short waves below the critical wavenumbers are stable whereas a number of long waves are unstable. The viscosity effect on the stability criteria is a dual-role one, depending on the strength of the applied magnetic field.