Linear and non-linear double diffusive convection in a rotating porous layer using a thermal non-equilibrium model

Double diffusive convection in a fluid-saturated rotating porous layer heated from below and cooled from above is studied when the fluid and solid phases are not in local thermal equilibrium, using both linear and non-linear stability analyses. The Darcy model that includes the time derivative and Coriolis terms is employed as momentum equation. A two-field model that represents the fluid and solid phase temperature fields separately is used for energy equation. The onset criterion for stationary, oscillatory and finite amplitude convection is derived analytically. It is found that small inter-phase heat transfer coefficient has significant effect on the stability of the system. There is a competition between the processes of thermal and solute diffusions that causes the convection to set in through either oscillatory or finite amplitude mode rather than stationary. The effect of solute Rayleigh number, porosity modified conductivity ratio, Lewis number, diffusivity ratio, Vadasz number and Taylor number on the stability of the system is investigated. The non-linear theory based on the truncated representation of Fourier series method predicts the occurrence of subcritical instability in the form of finite amplitude motions. The effect of thermal non-equilibrium on heat and mass transfer is also brought out.     

Combined effect of thermal modulation and rotation on the onset of stationary convection in a porous layer

The stability of a fluid-saturated horizontal rotating porous layer subjected to time-periodic temperature modulation is investigated when the condition for the principle of exchange of stabilities is valid. The linear stability analysis is used to study the effect of infinitesimal disturbances. A regular perturbation method based on small amplitude of applied temperature field is used to compute the critical values of Darcy–Rayleigh number and wavenumber. The shift in critical Darcy–Rayleigh number is calculated as a function of frequency of modulation, Taylor number, and Darcy–Prandtl number. It is established that the convection can be advanced by the low frequency in-phase and lower-wall temperature modulation, where as delayed by the out-of-phase modulation. The effect of Taylor number and Darcy–Prandtl number on the stability of the system is also discussed. We found that by proper tuning of modulation frequency, Taylor number, and Darcy–Prandtl number it is possible to advance or delay the onset of convection.     

Thermal instability in a rotating anisotropic porous layer saturated by a viscoelastic fluid

Linear and non-linear thermal instability in a rotating anisotropic porous medium, saturated with viscoelastic fluid, has been investigated for free-free surfaces. The linear theory is being related to the normal mode method and non-linear analysis is based on minimal representation of the truncated Fourier series analysis containing only two terms. The extended Darcy model, which includes the time derivative and Coriolis terms has been employed in the momentum equation. The criteria for both stationary and oscillatory convection is derived analytically. The rotation inhibits the onset of convection in both stationary and oscillatory modes. A weak non-linear theory based on the truncated representation of Fourier series method is used to find the thermal Nusselt number. The transient behaviour of the Nusselt number is also investigated by solving the finite amplitude equations using a numerical method. The results obtained during the analysis have been presented graphically.     

A two-phase,two-component model for natural convection in a porous medium

A numerical study of natural convection in a two-phase, two-component flow in a porous medium heated from below is presented. Interphase mass and energy transfer, latent heat and bouyancy effects are major physical features. This study extends earlier studies of natural convection based on single-phase, saturated porous medium models. The appearance of two-phase heat pipe zones in the flow has a marked effect on the fluid and heat flows as well as on the performance of the numerical methods. The numerical techniques for handling phase change, Jacobian construction and time step selection are discussed.     

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.     

Bifurcation analysis for thermal convection in a rotating porous layer

The linear and weakly nonlinear thermal convection in a rotating porous layer is investigated by constructing a simplified model involving a system of fifth-order nonlinear ordinary differential equations. The flow in the porous medium is described by Lap wood-Brinkman-extended Darcy model with fluid viscosity different from effective viscosity. Conditions for the occurrence of possible bifurcations are obtained. It is established that Hopf bifurcation is possible only at a lower value of the Rayleigh number than that of simple bifurcation. In contrast to the non-rotating case, it is found that the ratio of viscosities as well as the Darcy number plays a dual role on the steady onset and some important observations are made on the stability characteristics of the system. The results obtained from weakly nonlinear theory reveal that, the steady bifurcating solution may be either sub-critical or supercritical depending on the choice of physical parameters. Heat transfer is calculated in terms of Nusselt number.     

Linear stability analysis of convection in two-layer system with an evaporating vapor-liquid interface

Classical theories have successfully provided an explanation for convection in a liquid layer heated from below without evaporation. However, these theories are inadequate to account for the convective instabilities in an evaporating liquid layer, especially in the case when it is cooled from below. In the present paper, we study the onset of Marangoni convection in a liquid layer being overlain by a vapor layer. A new two-sided model is put forward instead of the one-sided model in previous studies. Marangoni-Bénard instabilities in evaporating liquid thin layers are investigated with a linear instability analysis. We define a new evaporation Biot number, which is different from that in previous studies and discuss the influences of reference evaporating velocity and evaporation Biot number on the vapor-liquid system. At the end, we explain why the instability occurs even when an evaporating liquid layer is cooled from below. The project supported by the National Natural Science Foundation of China (10372105) and the Knowledge Innovation Program of Chinese Academy of Sciences (KJCX2-SW-L05 and KGCX-SW-409) The English text was polished by Keren Wang.     

Three dimensional unsteady convection and mass transfer flow through porous medium

The problem of three dimensional unsteady convection flow through a porous medium, with effect of mass transfer bounded by an infinite vertical porous plate is discussed, when the suction at the plate is transverse sinusoidal and the plate temperature oscillates in time about a constant mean. Assuming the free stream velocity to be uniform, approximate solutions are obtained for the flow field, the temperature field, the skin-friction and the rate of heat transfer. The dependence of solution on Pr (Prandtl number), Gr (Grashof number based on temperature), Gc (modified Grashof number based on concentration difference), Sc (Schimdt number), the frequency and the permeability parameter is also investigated.     

Stability of buoyancy-driven convection in an Oldroyd-B fluid-saturated anisotropic porous layer

The nonlinear stability of thermal convection in a layer of an Oldroyd-B fluid-saturated Darcy porous medium with anisotropic permeability and thermal diffusivity is investigated with the perturbation method. A modified Darcy-Oldroyd model is used to describe the flow in a layer of an anisotropic porous medium. The results of the linear instability theory are delineated. The thresholds for the stationary and oscillatory convection boundaries are established, and the crossover boundary between them is demarcated by identifying a codimension-two point in the viscoelastic parameter plane. The stability of the stationary and oscillatory bifurcating solutions is analyzed by deriving the cubic Landau equations. It shows that these solutions always bifurcate supercritically. The heat transfer is estimated in terms of the Nusselt number for the stationary and oscillatory modes. The result shows that, when the ratio of the thermal to mechanical anisotropy parameters increases, the heat transfer decreases.     

Natural convection in a horizontal porous layer with anisotropic thermal diffusivity

The present paper is concerned with free convection in a horizontal porous layer with anisotropic thermal diffusivity. It is assumed that the diffusivity has rotational symmetry, with a symmetry axis making an arbitrary angle against the vertical. The critical Rayleigh number and wave number at marginal stability are calculated and the steady motion occurring at convection onset is examined. It is found that there are two different types of convection cells, depending on whether the longitudinal diffusivity is larger than the transverse diffusivity or not. In the former case, the convection cells are rectangular with vertical lateral walls. In the latter case, however, the lateral cell walls are tilted as well as curved.     

Nonlinear evolution analysis and stability for convection in a mushy layer with permeable interface

We consider the problem of two- and three-dimensional nonlinear buoyant flows in horizontal mushy layers during the solidification of binary alloys. We study the nonlinear evolution of such flow based on a recently developed realistic model for the mushy layer with permeable interface. The evolution approach is based on a Landau type equation for the amplitude of the secondary nonlinear solution, which can be in the form of rolls, squares, rectangles or hexagons. Using both analytical and computational methods, we calculate the solutions to the evolution equation near the onset of motion for both subcritical and supercritical regimes and determine the stable solutions. We find, in particular, that for several investigated cases with different parameter regimes, secondary solution in the form of subcritical down-hexagons or supercritical up-hexagons can be stable. However, the preferred solution for smallest values of the Rayleigh number and the amplitude of motion is in the form of subcritical down-hexagons. This result appears to agree with the experimental observation on the form of the convective flow near the onset of motion.     

Linear and non-linear analyses of convection in a micropolar fluid occupying a porous medium

Linear and weakly non-linear analyses of convection in a micropolar fluid occupying a high-porosity medium are performed. The Brinkman–Eringen momentum equation is considered. The linear and non-linear analyses are, respectively, based on the normal mode technique and truncated representation of Fourier series. The linear theory for a two-phase system reiterates that the preferred mode of convection is stationary as in the case of a single-phase system. An autonomous system of differential equations representing cellular convection arising in the study is considered to analyse the critical points. The Nusselt number is obtained as a function of micropolar and porous medium parameters.     

Linear stability of a fluid channel with a porous layer in the center简

We perform a Poiseuille flow in a channel linear stability analysis of a inserted with one porous layer in the centre, and focus mainly on the effect of porous filling ratio. The spectral collocation technique is adopted to solve the coupled linear stability problem. We investigate the effect of permeability, σ, with fixed porous filling ratio ψ = 1/3 and then the effect of change in porous filling ratio. As shown in the paper, with increasing σ, almost each eigenvalue on the upper left branch has two subbranches at ψ = 1/3. The channel flow with one porous layer inserted at its middle （ψ = 1/3） is more stable than the structure of two porous layers at upper and bottom walls with the same parameters. By decreasing the filling ratio ψ, the modes on the upper left branch are almost in pairs and move in opposite directions, especially one of the two unstable modes moves back to a stable mode, while the other becomes more instable. It is concluded that there are at most two unstable modes with decreasing filling ratio ψ. By analyzing the relation between ψ and the maximum imaginary part of the streamwise phase speed, Cimax, we find that increasing Re has a destabilizing effect and the effect is more obvious for small Re, where ψ a remarkable drop in Cimax can be observed. The most unstable mode is more sensitive at small filling ratio ψ, and decreasing ψ can not always increase the linear stability. There is a maximum value of Cimax which appears at a small porous filling ratio when Re is larger than 2 000. And the value of filling ratio 0 corresponding to the maximum value of Cimax in the most unstable state is increased with in- creasing Re. There is a critical value of porous filling ratio （= 0.24） for Re = 500; the structure will become stable as ψ grows to surpass the threshold of 0.24; When porous filling ratio ψ increases from 0.4 to 0.6, there is hardly any changes in the values of Cimax. We have also observed that the critical Reynolds number is especially sensitive for small ψ where the fastest drop is observed, and there may be a wide range in which the porous filling ratio has less effect on the stability （ψ ranges from 0.2 to 0.6 at σ = 0.002）. At larger permeability, σ, the critical Reynolds number tends to converge no matter what the value of porous filling ratio is.     

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.     

The effect of anisotropic permeability on free convective boundary layer flow in porous media

The effect of an anisotropic permeability on thermal boundary layer flow in porous media is studied. The convective flow is induced by a vertical, uniformly heated surface embedded in a fluid-saturated medium. A leading-order boundary layer theory is presented. It is shown that the thickness of the resulting boundary layer flow is different from that obtained in an isotropic porous medium. In general, an anisotropic permeability induces a fluid drift in the spanwise direction, the strength of which depends on the precise nature of the anisotropy. Conditions are found which determine whether or not the boundary layer flow is three-dimensional.     

Integral transform analysis of natural convection in porous enclosures

A hybrid numerical-analytical solution for steady-state natural convection in a porous cavity is proposed, based on application of the ideas in the generalized integral transform technique. The integral transformation process reduces the original coupled partial differential equations, for temperature and stream function, into an infinite system of non-linear ordinary differential equations for the transformed potentials, which is adaptively truncated and numerically solved through well-established algorithms. The approach is applied to a vertical rectangular enclosure subjected to uniform internal heat generation. The convergence characteristics of the explicit inversion formulae are illustrated and critical comparisons with previously reported purely numerical solutions are performed.     

Horizontal thermal convection in a porous medium

Linearised instability and nonlinear stability bounds for thermal convection in a fluid-filled porous finite box are derived. A nonuniform temperature field in the steady state is generated by maintaining the vertical walls at different temperatures. The linearised instability threshold is shown to be well above the global stability boundary, which is strongly suggested by the lack of symmetry in the perturbed system. The numerical results are evaluated utilising a newly developed Legendre-polynomial-based spectral method.     

Two fluid flow and heat transfer in an inclined channel containing porous and fluid layer

Convective flow and heat transfer in an inclined channel bounded by two rigid plates held at constant different temperatures with one region filled with porous matrix saturated with a viscous fluid and another region with a clear viscous fluid different from the fluid in first region is studied analytically. The coupled nonlinear governing equations are solved using regular perturbation method. It is found that the presence of porous matrix in one of the region reduces the velocity and temperature. Results have been presented for a wide range of governing parameters such as Grashof number, porous parameter, angle of inclination, ratio of heights of the two layers and also the ratio of viscosities.