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.     

Rotating Brinkman–Lapwood Convection with Modulation

The purpose of this article is to analyze, theoretically, the effect of modulation on rotating Brinkman–Lapwood convection, i.e., buoyancy-driven convection in a sparse porous medium subjected to rotation. Darcy–Brinkman momentum equation with Coriolis term has been used to describe the flow. The system is considered rotating about an axis with non-uniform rotation speed. In particular, we assume that the rotation speed is varying sinusoidally with time. A linear stability analysis has been performed to find the critical Rayleigh number in modulated case. The effect of modulated rotation speed is found to have a stabilizing effect on the onset of convection for different values of modulation frequency and the other physical parameters involved.     

Fluid convection in a rotating porous layer under modulated temperature on the boundaries

The linear stability of thermal convection in a rotating horizontal layer of fluid-saturated porous medium, confined between two rigid boundaries, is studied for temperature modulation, using Brinkman’s model. In addition to a steady temperature difference between the walls of the porous layer, a time-dependent periodic perturbation is applied to the wall temperatures. Only infinitesimal disturbances are considered. The combined effect of rotation, permeability and modulation of walls’ temperature on the stability of flow through porous medium has been investigated using Galerkin method and Floquet theory. The critical Rayleigh number is calculated as function of amplitude and frequency of modulation, Taylor number, porous parameter and Prandtl number. It is found that both, rotation and permeability are having stabilizing influence on the onset of thermal instability. Further it is also found that it is possible to advance or delay the onset of convection by proper tuning of the frequency of modulation of the walls’ temperature.     

The Effect of Rotation on the Onset of Double Diffusive Convection in a Sparsely Packed Anisotropic Porous Layer

The effect of rotation on the onset of double diffusive convection in a sparsely packed anisotropic porous layer, which is heated and salted from below, is investigated analytically using the linear and nonlinear theories. The Brinkman model that includes the Coriolis term is employed for the momentum equation. The critical Rayleigh number, wavenumber for stationary and oscillatory modes and a dispersion relation are obtained analytically using linear theory. The effect of anisotropy parameters, Taylor number, Darcy number, solute Rayleigh number, Lewis number, Darcy–Prandtl number, and normalized porosity on the stationary, oscillatory and finite amplitude convection is shown graphically. It is found that contrary to its usual influence on the onset of convection in the absence of rotation, the mechanical anisotropy parameter show contrasting effect on the onset criterion at moderate and high rotation rates. The nonlinear theory based on the truncated representation of Fourier series method is used to find the heat and mass transfers. The effect of various parameters on heat and mass transfer is shown graphically. Some of the convection systems previously reported in the literature is shown to be special cases of the system presented in this study.     

Effect of rotation on the convective stability of large-scale disturbances in a turbulent fluid

Linear equations of turbulent convection in the presence of rotation are derived within the framework of a phenomenological approach. It is shown that the amplification of large-scale disturbances is possible only for the essentially inhomogeneous problem when the dimensions of the large-scale disturbance exceed the characteristic scale of density variation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 3–9, July–August, 1989.     

Local nonsimilarity method for the two-phase boundary layer in mixed convection laminar film condensation

The two-phase boundary layer in laminar film condensation was solved by Koh for the free convection regime and forced convection regime using the similarity method. But the problem on mixed convection does not admit similarity solutions. The current work develops a local nonsimilarity method for the full spectrum of mixed convection, with a generic boundary layer formulation reduced to two specific cases mathematically identical to Koh’s formulations on the two limiting cases for either free or forced convection. Other solution methods for mixed convection in the literature are compared and critically evaluated to ensure a high level of accuracy in the current method. The current solution is used to extend Fujii’s correlation for mixed convection to the region where the energy convection effect is significant but has been hitherto neglected. The modified Fujii correlation provides a practical engineering tool for evaluating laminar film condensation with a mixed convection boundary layer.     

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.     

Effects of a.c. Electric Field and Rotation on Bénard–Marangoni Convection

The coupled buoyancy and thermocapillary instability, the Bénard–Marangoniproblem, in an electrically conducting fluid layer whose upper surface is deformed and subject to a temperature gradient is studied. Both influences of an a.c. electric field and rotation are investigated. Special attention is directed at the occurrence of convection both in the form of stationary motion and oscillatory convection. The linear stability problem is solved for different values of the relevant dimensionless numbers, namely the a.c. electric Rayleigh number, the Taylor, Rayleigh, Biot, Crispation and Prandtl numbers. For steady convection, it is found that by increasing the angular velocity, one reinforces the stability of the fluid layer whatever the values of the surface deformation and the applied a.c. electric field. We have also determined the regions of oscillatory instability and discussed the competition between both stationary and oscillatory convections. This revised version was published online in July 2006 with corrections to the Cover Date.     

Numerical and analytical investigations of thermosolutal instability in rotating Rivlin-Ericksen fluid in porous medium with Hall current

Numerical and analytical investigations of the thermosolutal instability in a viscoelastic Rivlin-Ericksen fluid are carried out in the presence of a uniform vertical magnetic field to include the Hall current with a uniform angular velocity in a porous medium. For stationary convection, the stable solute gradient parameter and the rotation have stabilizing effects on the system, whereas the magnetic field and the medium permeability have stabilizing or destabilizing effects on the system under certain conditions. The Hall current in the presence of rotation has stabilizing effects for sufficiently large Taylor numbers, whereas in the absence of rotation, the Hall current always has destabilizing effects. These effects have also been shown graphically. The viscoelastic effects disappear for stationary convection. The stable solute parameter, the rotation, the medium permeability, the magnetic field parameter, the Hall current, and the vis-coelasticity introduce oscillatory modes into the system, which are non-existent in their absence. The sufficient conditions for the non-existence of overstability are also obtained.     

Effect of rotation on the formation of longitudinal vortices in mixed convection flow over a flat plate

This paper presents a numerical study of the effect of rotation on the formation of longitudinal vortices in mixed convection flow over a flat plate. The criterion on the position of marking the onset of longitudinal vortices is defined in this paper. The onset position characterized by the Goertler number G _δ depends on the Grashof number, the rotation number Ro, the Prandtl number Pr and the wave number. The results show that negative rotation stabilizes the boundary layer flow on the surface. On the contrary, positive rotation destabilizes the flow. The numerical data are compared with the experimental results.     

The Onset of Darcy–Brinkman Electroconvection in a Dielectric Fluid Saturated Porous Layer

The combined effect of a vertical AC electric field and the boundaries on the onset of Darcy–Brinkman convection in a dielectric fluid saturated porous layer heated either from below or above is investigated using linear stability theory. The isothermal bounding surfaces of the porous layer are considered to be either rigid or free. It is established that the principle of exchange of stability is valid irrespective of the nature of velocity boundary conditions. The eigenvalue problem is solved exactly for free–free (F/F) boundaries and numerically using the Galerkin technique for rigid–rigid (R/R) and lower-rigid and upper-free (F/R) boundaries. It is observed that all the boundaries exhibit qualitatively similar results. The presence of electric field is emphasized on the stability of the system and it is shown that increasing the AC electric Rayleigh number R _ea is to facilitate the transfer of heat more effectively and to hasten the onset of Darcy–Brinkman convection. Whereas, increase in the ratio of viscosities Λ and the inverse Darcy number Da ⁻¹ is to delay the onset of Darcy–Brinkman electroconvection. Besides, increasing R _ea and Da ⁻¹ as well as decreasing Λ are to reduce the size of convection cells.     

Laminar natural convection of pure and saline water along a vertical isothermal cylinder

 In most studies concerning laminar natural convection along a vertical isothermal cylinder a linear relationship between fluid density and temperature has been used and kinematic viscosity and thermal diffusivity have been considered constant calculated at ambient temperature. However, it is known that the density–temperature relationship for water is non-linear at low temperatures and kinematic viscosity and thermal diffusivity are functions of temperature. In this study the problem of laminar natural convection of pure and saline water along a vertical isothermal cylinder has been investigated in the temperature range between 20 and 0 ^∘C taking into account the temperature dependence of ν, α and ρ. The results are obtained with the numerical solution of the boundary layer equations. The variation of ν, α and ρ with temperature has a strong influence on free convection characteristics. Received on 17 May 1999     

Convection in a rotating medium above a thermally inhomogeneous horizontal surface

The linear stationary problem of convection in a medium rotating about a vertical axis above a thermally inhomogeneous horizontal surface is theoretically investigated. Attention is mainly focused on the case of a homogeneous medium, but certain stratification effects and especially the convection characteristics in binary mixtures (for example, in saline sea water) are also considered. When the rotation is rapid (large Taylor numbers) the convective cells are strongly elongated in the vertical direction, though they also contain a thin Ekman boundary layer. The importance of the boundary conditions on the horizontal surface (in parallel with the no-slip conditions, more general conditions that may follow from the quadratic turbulent friction model are considered) is shown. In the case of binary mixtures, the differential diffusion and rotation effects may together result in the appearance of “induced salt fingers”, the deep penetration of convection into an arbitrarily stably stratified medium. The convective motions may then have a considerable effect on the background vertical temperature and admixture distributions. Attention is drawn to an original manifestation of the analogy between the rotation and stratification effects: in a non-rotating, stably stratified medium, near a thermally inhomogeneous vertical surface, the convection also penetrates deep into the medium, but in the horizontal direction, so that, when the coordinate system is rotated through 90°, the solution coincides with the case of a rotating non-stratified fluid considered here.     

The effect of vertical vibrations on the onset of convection in a planar rotating layer

The effect of vertical vibrations on the convection in a rotating planar fluid layer heated from below was studied. In this case a modulation parameter, the acceleration due to gravity, appears in the problem. The modulation of the parameter may have a significant effect on the onset of convective instability. Parameter modulation in nonrotating layers has been investigated in earlier work [1–3]. The presence of rotation significantly increases the complexity of the mathematical problem, introducing an additional dependence of the solution on the Taylor number Ta and the Prandtl number Pr. Furthermore, an oscillatory convection regime can occur at the stability limit in rotating fluids with Pr < 1. Parameter modulation in the rotating fluid may not only lead to a change in the stability limit and critical wavelength but also to a change in the eigenfrequency of the oscillatory convection. Rauscher and Kelly [4] examined the effect of parameter modulation on the convective stability of a rotating fluid only for the particular case of a sinusoidal variation in the temperature gradient with a small amplitude for Pr = 1, i.e., the effect of modulation was studied on only a steady convection regime.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 12–22, July–August, 1984.     

Moderate Time Scale Finite Amplitude Analysis of Large Stefan Number Convection in Rotating Mushy Layers

We investigate the steady state convection amplitude for solutal convection occurring during the solidification of a rotating mushy layer in a binary alloy system for a new Darcy equation formulation. We adopt a large far field temperature and assume that the initial composition is very close to the eutectic composition. The linear stability analysis showed that rotation stabilised solutal convection. The results of the weak non-linear analysis of stationary convection indicates the presence of Hopf bifurcation, associated with the oscillatory mode, developing at Ta = 3.     

A New Class of Equations for Rotationally Constrained Flows

The incompressible Navier–Stokes equation is considered in the limit of rapid rotation (small Ekman number). The analysis is limited to horizontal scales small enough so that both horizontal and vertical velocities are comparable, but the horizontal velocity components are still in geostrophic balance. Asymptotic analysis leads to a pair of nonlinear equations for the vertical velocity and vertical vorticity coupled by vertical stretching. Statistically stationary states are maintained against viscous dissipation by boundary forcing or energy injection at larger scales. For thermal forcing direct numerical simulation of the reduced equations reveals the presence of intense vortical structures spanning the layer depth, in excellent agreement with simulations of the Boussinesq equations for rotating convection by Julien et al. (1996). Received 30 May 1997 and accepted 4 January 1998     

Mixed convection from a cylinder oscillating vertically in a quiescent fluid

 The problem of heat convection from a vertically oscillating cylinder in a quiescent fluid is investigated. The governing equations of motion and energy are solved numerically in a non-inertial frame of references to determine the flow field and heat transfer characteristics under different conditions. The main dominating parameters are Keulegan–Carpenter number, KC, frequency parameter, β, Grashof number, Gr and Prandtl number, Pr. The ranges considered for these parameters are KC ≤ 10, β≤40 and Gr ≤ 10⁵ while Prandtl number is kept constant. The study revealed that the effect of amplitude and frequency of oscillation on heat transfer is strongly influenced by the Grashof number range. In the forced convection regime (Gr = 0), the increase of KC creates extensive vortex motion at all cylinder positions that leads to a significant increase in heat transfer. A similar trend, but with a lesser extent, is also observed for the increase of β. However, at high Grashof numbers, the effect of oscillation on heat convection is only significant at large values of KC. Received on 5 June 2000 / Published online: 29 November 2001     

On Mixed Oscillatory and Steady Modes of Nonlinear Convection in Mushy Layers

We consider the problem of mixed oscillatory and steady modes of nonlinear compositional convection in horizontal mushy layers during the solidification of binary alloys. Under a near-eutectic approximation and the limit of large far-field temperature, we determine a number of two- and three-dimensional weakly nonlinear mixed solutions, and the stability of these solutions with respect to arbitrary three-dimensional disturbances is then investigated. The present investigation is an extension of the problem of mixed oscillatory and steady modes of convection, which was investigated by Riahi (J Fluid Mech 517: 71–101, 2004), where some calculated results were inaccurate due to the presence of a singular point in the equation for the linear frequency. Here we resolve the problem and find some significant new results. In particular, over a wide range of the parameter values, we find that the properties of the preferred and stable solution in the form of particular subcritical mixed standing and steady hexagons appeared to be now in much better agreement with the available experimental results (Tai et al., Nature 359:406–408, 1992) than the one reported in Riahi (J Fluid Mech 517:71–101, 2004). We also determined a number of new types of preferred supercritical solutions, which can be preferred over particular values of the parameters and at relatively higher values of the amplitude of convection.     

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.     

Effect of Vertical Modulation on the Onset of Filtration Convection

The present paper examines the effect of vertical harmonic vibration on the onset of convection in an infinite horizontal layer of fluid saturating a porous medium. A constant temperature distribution is assigned on the rigid boundaries, so that there exists a vertical temperature gradient. The mathematical model is described by equations of filtration convection in the Darcy–Oberbeck–Boussinesq approximation. The linear stability analysis for the quasi-equilibrium solution is performed using Floquet theory. Employment of the method of continued fractions allows derivation of the dispersion equation for the Floquet exponent σ in an explicit form. The neutral curves of the Rayleigh number Ra versus horizontal wave number α for the synchronous and subharmonic resonant modes are constructed for different values of frequency Ω and amplitude A of vibration. Asymptotic formulas for these curves are derived for large values of Ω using the method of averaging, and, for small values of Ω, using the WKB method. It is shown that, at some finite frequencies of vibration, there exist regions of parametric instability. Investigations carried out in the paper demonstrate that, depending on the governing parameters of the problem, vertical vibration can significantly affect the stability of the system by increasing or decreasing its susceptibility to convection.