Ferrofluid Saturated Porous Layer Research Articles

Thermoconvective instability in a ferrofluid saturated porous layer

The Forchheimer-extended Brinkman’s Darcy-flow model was used to investigate the initiation of ferroconvection in a flat porous layer while accounting for effective viscosity. The rigid ferromagnetic, rigid paramagnetic and stress-free isothermal boundary conditions are the three categories. The eigenvalue issue can be properly addressed for stress-free boundaries; the Galerkin approach is utilized to find the critical stability constraints quantitatively for other barriers. It was discovered that the boundary types had a strong influence on the system’s stabilization. Ferromagnetic boundaries are less preferred than paramagnetic boundaries in control of convection. The dependence of many physical limitations on the linear stability of the system is intentionally given, and it is demonstrated that increasing the value of the viscosity ratio delays the beginning of convection.

Comment on the paper “Effect of temperature-dependent viscosity on the onset of Bénard–Marangoni ferroconvection in a ferrofluid saturated porous layer, C. E. Nanjundappa, B. Savitha, B. Arpitha Raju, I. S. Shivakumara, Acta Mechanica 225 (2014) 835–850”

Comment on the paper “Effect of temperature-dependent viscosity on the onset of Bénard–Marangoni ferroconvection in a ferrofluid saturated porous layer, C. E. Nanjundappa, B. Savitha, B. Arpitha Raju, I. S. Shivakumara, Acta Mechanica 225 (2014) 835–850”

Thermoconvective Instability in a Ferrofluid Saturated Porous Layer

Thermoconvective Instability in a Ferrofluid Saturated Porous Layer

Penetrative Brinkman ferroconvection via internal heating in high porosity anisotropic porous layer: influence of boundaries

The penetrative ferrothermal convection (FTC) in a ferrofluid (FF) saturated high porosity anisotropic porous layer via uniform internal heating is investigated. The Brinkman-extended Darcy equation is applied to describe the flow in the porous medium. The permeability in the vertical direction is taken to be twice that of the permeability in the horizontal direction while the ratio of horizontal to vertical effective thermal diffusivity is allowed to vary. The Galerkin method is applied to solve numerically the stability eigenvalue problem for different boundary combinations namely, (i) rigid-paramagnetic (R–P) with large and low magnetic susceptibility, (ii) rigid-ferromagnetic (R–F), and (iii) free-ferromagnetic (F–F). The R–P boundaries with large magnetic susceptibility offer most, while F–F boundaries offer least stabilizing effect against FTC. Besides, the effect of increasing the magnetic number, non-linearity of fluid magnetization parameter, Darcy number and internal heat source strength is to speed up FTC, while the thermal anisotropy and magnetic susceptibility parameter indict a contradictory effect on FTC.

Effect of Dusty Particles on Darcy-Brinkman Gravity-Driven Ferro-Thermal-Convection in a Ferrofluid Saturated Porous Layer with Internal Heat Source: Influence of Boundaries

In the present work, penetrative ferro-thermal-convection (FTC) via internal heating in a ferrofluid-saturated porous layer for different types of velocity, temperature and magnetic potential bounding surfaces, is investigated theoretically. A Galerkin-type based on weighted residual method has been used to obtain the eigenvalue problem. The stability of the system is notably influenced on porous medium via internal heating. It is demonstrated that paramagnetic surface with large magnetic susceptibility $$ (\chi > 0) $$ delay the FTC as compared to very low magnetic susceptibility $$ (\chi = 0) $$ as well as ferromagnetic ones. Increase in heat source strength, non-linearity of magnetization, dust particles and magnetic number hasten, while increase in viscosity ratio, concentration parameter and inverse Darcy number delay the onset. The rigid paramagnetic surfaces induct more stabilizing effect on FTC compared to very low magnetic susceptibility as well as ferromagnetic surfaces. In limiting cases, some results published previously are recovered from the present results.

Linear and weakly nonlinear analysis of a ferrofluid layer for an LTNE model with variable gravity and internal heat source

A ferrofluid saturated porous layer convection problem is studied in the variable gravitational field for a local thermal nonequilibrium (LTNE) model. Internal heating and variation (increasing or decreasing) in gravity with distance through the layer affected the stability of the convective system. The Darcy model is employed for the momentum equation and the LTNE model for the energy equation. The boundaries are considered to be rigid-isothermal and paramagnetic. For the linear stability analysis of the three-dimensional problem, the normal mode has been applied and the eigenvalue problem is solved numerically using Chebyshev pseudospectral method, while weakly nonlinear analysis is carried out with a truncated Fourier series. The effect of different dimensionless parameters on the Rayleigh number has also been studied. We found that the system becomes unstable on the increasing value of nonlinearity index of magnetization ( $$M_3$$ ), porosity-modified conductivity ratio ( $$\beta $$ ), and internal heat parameter ( $$\xi $$ ). It is observed that the system is stabilized by increasing the value of the Langevin parameter ( $$\alpha _\mathrm{{L}}$$ ), variable gravity coefficient ( $$\delta $$ ), and effective heat transfer parameter ( $$H_1$$ ). Runge–Kutta–Gill method has been used for solving the finite-amplitude equations to study the transient behavior of the Nusselt number. Streamlines and isotherms patterns are determined for the steady case and are presented graphically.

Cattaneo–LTNE porous ferroconvection

PurposeThe onset of convection in a ferrofluid-saturated porous layer has been investigated using a local thermal nonequilibrium (LTNE) model by allowing the solid phase to transfer heat via a Cattaneo heat flux theory while the fluid phase to transfer heat via usual Fourier heat-transfer law. The flow in the porous medium is governed by modified Brinkman-extended Darcy model. The instability of the system is discussed exactly for stress-free boundaries, while for rigid-ferromagnetic/paramagnetic boundaries the results are obtained numerically using the Galerkin method. The presence of Cattaneo effect introduces oscillatory convection as the preferred mode of instability contrary to the occurrence of instability via stationary convection found in its absence. Besides, oscillatory ferroconvection is perceived when the solid thermal relaxation time parameter exceeds a threshold value and increase in its value is to hasten the oscillatory onset. The effect of different boundary conditions on the instability of the system is noted to be qualitatively same. The paper aims to discuss these issues.Design/methodology/approachThe investigators would follow the procedure of Straughan (2013) to obtain the expression for Rayleigh number. The Brinkman-extended Darcy model is used to describe the flow in a porous medium. The investigators have used a Galerkin method to obtain the numerical results for rigid-ferromagnetic/paramagnetic boundaries, while the instability of the system is discussed exactly for stress-free boundaries.FindingsThe Cattaneo–LTNE porous ferroconvection has been analyzed for different velocity and magnetic boundary conditions. The Brinkman-extended Darcy model is used to describe the flow in a porous medium. The effect of different types of velocity and magnetic boundary conditions on the instability of the system has been highlighted. The instability of the system is discussed exactly for stress-free boundaries, while for rigid-ferromagnetic/paramagnetic boundaries the results are obtained numerically using the Galerkin method.Originality/valueThe novelty of the present paper is to combine LTNE and second sound effects in solids on thermal instability of a ferrofluid-saturated porous layer by retaining the usual Fourier heat-transfer law in the ferrofluid. The Brinkman-extended Darcy model is used to describe the flow in a porous medium. The effect of different types of velocity and magnetic boundary conditions on the instability of the system is discussed.

Thermomagnetic convection in a layer of magnetic nanofluid saturating porous medium with magnetic field dependent viscosity

In this paper, the effect of magnetic field dependent (MFD) viscosity on the onset of thermomagnetic convection is examined for a horizontal layer of magnetic nanofluid (or ferrofluid) saturated porous layer. A model that comprises the effect of Brownian diffusion, thermophoresis, magnetophoresis, and Darcy's law is considered. We employed the Chebyshev pseudospectral QZ-method to solve the developed mathematical model and the results are derived for water-based and ester-based magnetic nanofluids. The effects of significant parameters are analysed at the onset of convection in the gravity as well as in the microgravity environment. The influence of MFD viscosity, permeability parameter, the width of the fluid layer, Lewis number, Langevin parameter, particle concentration, and the concentration Rayleigh number at the onset of thermomagnetic convection are graphically illustrated.

Linear Stability Analysis of Penetrative Convection via Internal Heating in a Ferrofluid Saturated Porous Layer

Penetrative convection due to purely internal heating in a horizontal ferrofluid-saturated porous layer is examined by performing linear stability analysis. Four different types of heat supply functions are considered. The Darcy model is used to incorporate the effect of the porous medium. Numerical solutions are obtained by using the Chebyshev pseudospectral method, and the results are discussed for all three boundary conditions: when both boundaries are impermeable and conducting; when both boundaries are conducting with lower boundary impermeable and free upper boundary; and when both boundaries are impermeable with lower boundary conducting and upper with constant heat flux. The effect of the Langevin parameter, width of ferrofluid layer, permeability parameter, and nonlinearity of the fluid magnetization has been observed at the onset of penetrative convection for water- and ester-based ferrofluids. It is seen that the Langevin parameter, width of ferrofluid layer, and permeability parameter have stabilizing effects on the onset of convection, while the nonlinearity of the fluid magnetization advances the onset of convection.

On Exchange of Stabilities in Ferromagnetic Convection in a Rotating Ferrofluid Saturated Porous Layer

In the present paper, first of all, it is proved that the ‘principle of the exchange of stabilities’ is not, in general valid, for the case of free boundaries and then a sufficient condition is derived for the validity of this principle in ferromagnetic convection, for the case of free boundaries, in a horizontal ferrofluid saturated porous layer in the presence of a uniform vertical magnetic field and uniform rotation about the vertical axis.

Effect of temperature-dependent viscosity on the onset of Bénard–Marangoni ferroconvection in a ferrofluid saturated porous layer

The criterion for the onset of Bénard–Marangoni ferroconvection in an initially quiescent magnetized ferrofluid saturated horizontal Brinkman porous layer is investigated in the presence of a uniform vertical magnetic field. The viscosity is considered to be varying exponentially with temperature. The lower rigid boundary and the upper free boundary at which the surface tension effects are accounted for are assumed to be perfectly insulated to temperature perturbations. The eigenvalue problem is solved numerically using the Galerkin technique and analytically by regular perturbation technique with wave number a as a perturbation parameter. It is observed that the analytical and numerical results are very well comparable. The characteristics of stability of the system are strongly dependent on the viscosity parameter B. The effect of B on the onset of Bénard–Marangoni ferroconvection in a porous layer is dual in nature depending on the choices of the physical parameters, and a sublayer starts to form at higher values of B. The nonlinearity of fluid magnetization M 3 is found to have no influence on the onset of ferroconvection, whereas an increase in the value of the magnetic number M 1 and the Darcy number Da is to advance the onset of Bénard–Marangoni ferroconvection in a porous layer.

Penetrative ferroconvection via internal heating in a saturated porous layer with constant heat flux at the lower boundary

A model for penetrative ferroconvection via internal heat generation in a ferrofluid saturated porous layer is explored. The Brinkman–Lapwood extended Darcy equation with fluid viscosity different from effective viscosity is used to describe the flow in the porous medium. The lower boundary of the porous layer is assumed to be rigid- paramagnetic and insulated to temperature perturbations, while at upper stress-free boundary a general convective-radiative exchange condition on perturbed temperature is imposed. The resulting eigenvalue problem is solved numerically using the Galerkin method. It is found that increasing in the dimensionless heat source strength Ns, magnetic number M1 Darcy number Da and the non-linearity of magnetization parameter M3 is to hasten, while increase in the ratio of viscosities Λ, Biot number Bi and magnetic susceptibility χ is to delay the onset of ferroconvection. Further, increase in Bi, Da−1 and Ns and decrease in Λ, M1 and M3 is to diminish the dimension of convection cells.

The effects of local thermal nonequilibrium and MFD viscosity on the onset of Brinkman ferroconvection

The simultaneous effect of local thermal nonequilibrium (LTNE) and magnetic field dependent (MFD) viscosity on thermal convective instability in a horizontal ferrofluid saturated Brinkman porous layer in the presence of a uniform vertical magnetic field is studied analytically. The results indicate that the onset of Brinkman ferroconvection is delayed with increasing MFD viscosity parameter but the critical wave number is found to be independent of this parameter. When compared to the simultaneous presence of buoyancy and magnetic forces, it is observed that the onset of Brinkman ferroconvection is delayed more when the magnetic forces alone are present. Asymptotic solutions for both small and large values of scaled inter-phase heat transfer coefficient H t are compared with those computed numerically and good agreement is found between them. Besides, the influence of magnetic and LTNE parameters on the stability characteristics of the system is also discussed. The available results in the literature are recovered as particular cases from the present study.

Effects of MFD viscosity and LTNE on the onset of ferromagnetic convection in a porous medium

The combined effect of magnetic field dependent (MFD) viscosity and a local thermal non-equilibrium (LTNE) on the criterion for the onset of ferromagnetic convection in a ferrofluid saturated horizontal porous layer heated from below in the presence of a uniform vertical magnetic field is studied analytically using linear stability theory. A modified Darcy equation is used to describe the flow in the porous medium and a two-field model for temperature each representing the solid as well as fluid phases separately is used for energy equation. It is demonstrated that the principle of exchange of stability is valid. The results indicate that the onset of ferromagnetic convection is delayed with an increase in the MFD viscosity parameter but shows no influence on the critical wave number. Moreover, the system is found to be more stable when the magnetic forces alone are present. Asymptotic solutions for both small and large values of scaled interphase heat transfer coefficient H t are presented and compared with those computed numerically. An excellent agreement is obtained between the asymptotic and the numerical results. Besides, the influence of magnetic and LTNE parameters on the stability characteristics of the system is also discussed.

Effect of internal heat generation on the onset of Brinkman–Benard convection in a ferrofluid saturated porous layer

The effect of internal heat generation on the criterion for the onset of convection in a horizontal ferrofluid saturated porous layer in the presence of a uniform magnetic field is studied using the Brinkman–Lapwood extended Darcy flow model with fluid viscosity different from effective viscosity. The lower boundary is taken to be rigid – isothermal and the upper surface is free and subject to the general thermal condition. The resulting eigenvalue problem is solved numerically using the Galerkin method with the Rayleigh number as the eigenvalue. The stability of the system is found to be dependent on the dimensionless internal heat source strength N s , Darcy number Da, viscosity ratio Λ , Biot number Bi, magnetic Rayleigh number, R m and the nonlinearity of fluid magnetization parameter M 3. The effect of increase in the value of R m , N s and M 3 is to hasten, while increase in the value of Bi, Λ and Da −1 is to delay the onset of thermomagnetic convection in a ferrofluid saturated porous layer. Further, increase in the value of Bi, Da −1 and N s as well as decrease in Λ , R m and M 3 is to diminish the dimension of convection cells.

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.

The onset of ferroconvection in a horizontal ferrofluid saturated porous layer heated from below and cooled from above with constant heat flux subject to MFD viscosity

The effect of magnetic field dependent (MFD) viscosity on the onset of ferroconvection in a ferrofluid saturated horizontal porous layer is investigated theoretically. The bounding surfaces of the porous medium are considered to be either rigid-ferromagnetic or stress free with constant heat flux conditions. The resulting eigenvalue problem is solved numerically using the Galerkin technique and also analytically by regular perturbation technique. It is found that increase in porous parameter, MFD viscosity parameter and decrease in the magnetic number is to delay the onset of ferroconvection, while the nonlinearity of fluid magnetization has no influence on the stability of the system.

Brinkman–Benard–Marangoni convection in a magnetized ferrofluid saturated porous layer

A classical linear stability theory is used to study the onset of Brinkman–Benard–Marangoni (BBM) convection in an initially quiescent magnetized ferrofluid saturated horizontal layer of a very coarse porous medium in the presence of a uniform vertical magnetic field. The lower rigid boundary is subject to a fixed heat flux, while the upper boundary is free and open to the ambient air on which a general thermal condition which encompasses fixed temperature and fixed heat flux as particular cases is invoked. The resulting eigenvalue problem is solved using the Galerkin technique and also by a regular perturbation technique when both boundaries are insulated to temperature perturbations. The differences as well as similarities with the corresponding problem in ordinary viscous fluids and also the influence of different forces when they are acting in isolation are particularly highlighted. It is found that increase in the Biot number, porous parameter and decrease in the magnetic number as well as nonlinearity of fluid magnetization has a stabilizing effect on the onset of BBM ferroconvection. Moreover, the nonlinearity of fluid magnetization is observed to have no consequence on the onset of convection in the case of fixed heat flux boundary conditions. The present study reproduces already established results in the literature as particular cases.

Penetrative ferroconvection in a porous layer

Penetrative convection in a horizontal ferrofluid-saturated porous layer in the presence of a uniform applied vertical magnetic field has been investigated via the internal heating model using the Brinkman-extended Darcy equation. The rigid-isothermal boundaries of the porous layer are considered to be either paramagnetic or ferromagnetic. The eigenvalue problem is solved numerically using the Galerkin method with either thermal or magnetic Rayleigh number as the eigenvalue. The stability of the system is significantly affected by the internal heating in the porous layer. It is noted that the paramagnetic boundaries with large magnetic susceptibility delay the onset of penetrative ferroconvection the most when compared to very low magnetic susceptibility as well as ferromagnetic boundaries. An increase in the value of magnetic Rayleigh number (R m ), heat source strength (N S ) and non-linearity of magnetization (M 3) is to hasten the onset of ferroconvection. In addition, the stability of the system when heated from above and also in the absence of thermal buoyancy has been discussed in detail.

Effect of Boundary Conditions on the Onset of Thermomagnetic Convection in a Ferrofluid Saturated Porous Medium

The onset of thermomagnetic convection in a ferrofluid saturated horizontal porous layer in the presence of a uniform vertical magnetic field is investigated for a variety of velocity and temperature boundary conditions. The Brinkman–Lapwood extended Darcy equation, with fluid viscosity different from effective viscosity, is used to describe the flow in the porous medium. The lower boundary of the porous layer is assumed to be rigid-ferromagnetic, while the upper boundary is considered to be either rigid-ferromagnetic or stress-free. The thermal conditions include fixed heat flux at the lower boundary, and a general convective-radiative exchange at the upper boundary, which encompasses fixed temperature and heat flux as particular cases. The resulting eigenvalue problem is solved using the Galerkin technique and also by using regular perturbation technique when both boundaries are insulated to temperature perturbations. It is found that the increase in the Biot number and the viscosity ratio, and the decrease in the magnetic as well as in the Darcy number is to delay the onset of ferroconvection. Besides, the nonlinearity of fluid magnetization has no effect on the onset of convection in the case of fixed heat flux boundary conditions.