Weak Nonlinear Oscillatory Convection in a Viscoelastic Fluid-Saturated Porous Medium Under Gravity Modulation

A study of thermal instability driven by buoyancy force is carried out in an initially quiescent infinitely extended horizontal porous medium saturated with viscoelastic fluid. Modified Darcy’s law is used to explain characteristics of fluid motion. The time periodic gravity field has been considered, and its effect on the system has been investigated. A weak nonlinear stability analysis has been performed for the oscillatory mode of convection, and heat transport in terms of the Nusselt number, which is governed by the complex non-autonomous Ginzburg–Landau equation, is calculated. The influence of relaxation and retardation times of viscoelastic fluid on heat transfer has been discussed. Further, the study establishes that the heat transport can be controlled effectively by a mechanism that is external to the system. Finally, it is found that overstability advances the onset of convection, and hence increases heat transfer.     

Thermal convection of a viscoelastic fluid in an open-top porous layer heated from below

Buoyancy-driven convection of a viscoelastic fluid saturated in an open-top porous square box is studied based on a modified Darcy's law. The results are compared with those for a Newtonian fluid under the same boundary conditions and those for the viscoelastic fluid under a closed-top boundary. In particular, the critical Darcy–Rayleigh number Ra for onset of convection is determined first by using the linear stability theory. Then the effects of the relaxation time and the retardation time of the viscoelastic fluid on the heat transfer rate and the flow pattern are investigated numerically. The results reveal some interesting properties of thermal convection for the viscoelastic fluid. The relaxation time makes the fluid easier to destabilize while the retardation time tends to stabilize the fluid motion in the porous medium, and larger heat transfer rate can be achieved with larger value of the relaxation time and decreased retardation time. Furthermore, larger relaxation time facilitates earlier bifurcation of the flow pattern as Ra increases, but bifurcation can be postponed with increased retardation time. For larger ratio of relaxation time over retardation time, the flow pattern is more complicated and the frequency of flow oscillation also increases. Finally, large ratio of relaxation time over retardation time can make the open-top boundary impermeable due to the viscoelastic effect on the fluid.     

An analytical study of linear and non-linear double diffusive convection with Soret and Dufour effects in couple stress fluid

The onset of double diffusive convection in a two component couple stress fluid layer with Soret and Dufour effects has been studied using both linear and non-linear stability analysis. The linear theory depends on normal mode technique and non-linear analysis depends on a minimal representation of double Fourier series. The effect of couple stress parameter, the Soret and Dufour parameters, and the Prandtl number on the stationary and oscillatory convection are presented graphically. The Dufour parameter enhances the stability of the couple stress fluid system in case of both stationary and oscillatory mode. The effect of positive Soret parameter is to destabilize the system in case of stationary mode while it stabilizes the system in case of oscillatory mode. The negative Soret parameter enhances the stability in both stationary and oscillatory mode. The couple stress parameter enhances the stability of the system in both stationary and oscillatory modes. The Dufour parameter increases the heat transfer while the couple stress parameter has reverse effect. The Soret parameter has negligible influence on heat transfer. Both Dufour and Soret parameters increases the mass transfer while the couple stress parameter has dual effect depending on the value of the Rayleigh number.     

Magneto-convection in an anisotropic porous layer with Soret effect

Thermal instability in an electrically conducting two component Boussinesq fluid-saturated-porous medium has been investigated, in the presence of Soret coefficient. The porous medium is confined between two horizontal surfaces, and subjected to a constant vertical magnetic field. Flow in the porous medium is characterized by generalized Darcy model, which includes the time derivative term. Performing linear and non-linear stability analysis, the effect of magnetic field on the stability of flow through porous medium has been investigated. The normal mode method is used in linear stability analysis, while a weak non-linear analysis based on a minimal representation of double Fourier series method is used in non-linear analysis. The critical Rayleigh number, wave number for stationary and oscillatory modes, and frequency of oscillations are obtained analytically using linear theory. Effects of various parameters on stationary, oscillatory and finite amplitude convection, rate of heat and mass transfer have been obtained analytically and presented graphically.     

Linear and Nonlinear Double-Diffusive Convection in a Fluid-Saturated Porous Layer with Cross-Diffusion Effects

The double-diffusive convection in a horizontal fluid-saturated porous layer, which is heated and salted from below in the presence of Soret and Dufour effects, is studied analytically using both linear and nonlinear stability analyses. The linear analysis is based on the usual normal mode technique, while the nonlinear analysis is based on truncated representation of Fourier series. The generalized Darcy model that includes the time derivative is employed for the momentum equation. The critical Rayleigh number, wavenumber for stationary and oscillatory modes, and frequency of oscillations are obtained analytically using linear theory. The effects of solute Rayleigh number, Lewis number, normalized porosity parameter, Vadasz number, Soret and Dufour parameters on the stationary, oscillatory convection, and heat and mass transfers are shown graphically. The Vadasz number has dual effect on the threshold of the oscillatory convection. Some known results are recovered as special cases of the present problem.     

Thermal Instability in a Porous Medium Layer Saturated with a Viscoelastic Nanofluid

The onset of convection in a horizontal layer of a porous medium saturated with a viscoelastic nanofluid was studied in this article. The modified Darcy model was applied to simulate the momentum equation in porous media. An Oldroyd-B type constitutive equation was used to describe the rheological behavior of viscoelastic nanofluids. The model used for the viscoelastic nanofluid incorporates the effects of Brownian motion and thermophoresis. The onset criterion for stationary and oscillatory convection was analytically derived. The effects of the concentration Rayleigh number, Prandtl number, Lewis number, capacity ratio, relaxation, and retardation parameters on the stability of the system were investigated. Oscillatory instability is possible in both bottom- and top-heavy nanoparticle distributions. Results indicated that there is competition among the processes of thermophoresis, Brownian diffusion, and viscoelasticity that causes the convection to set in through oscillatory rather than stationary modes. Regimes of stationary and oscillatory convection for various parameters were derived and are discussed in detail.     

Linear and Non-linear Double Diffusive Convection in a Fluid-Saturated Anisotropic Porous Layer with Cross-Diffusion Effects

The double diffusive convection in a horizontal anisotropic porous layer saturated with a Boussinesq binary fluid, which is heated and salted from below in the presence of Soret and DuFour effects is studied analytically using both linear and non-linear stability analyses. The linear analysis is based on the usual normal mode technique, while the non-linear analysis is based on a minimal representation of double Fourier series. The generalized Darcy model including the time derivative term is employed for the momentum equation. The critical Rayleigh number, wavenumbers for stationary and oscillatory modes, and frequency of oscillations are obtained analytically using linear theory. The effects of anisotropy parameter, solute Rayleigh number, and Soret and DuFour parameters on the stationary, oscillatory convection, and heat and mass transfer are shown graphically. Some known results are recovered as special cases of the present problem.     

Cross Diffusion Convection in a Newtonian Fluid-Saturated Rotating Porous Medium

Effect of rotation on linear and nonlinear instability of cross-diffusive convection in an anisotropic porous medium saturated with Newtonian fluid has been investigated. Normal mode technique has been used for linear stability analysis, however nonlinear analysis is done using spectral method, involving only two terms. The Darcy model with Coriolis terms, has been employed in the momentum equation. Nonlinear analysis is used to find the thermal and concentration Nusselt numbers. The effects of various parameters, including Soret and Dufour parameters, on stationary and oscillatory convection, have been obtained, and shown graphically.     

Coriolis effect on thermal convective instability of viscoelastic fluids in a rotating porous cylindrical annulus

Linear stability analysis of thermal convection is studied for a viscoelastic fluid in a rotating porous cylindrical annulus. The modified Darcy–Jeffrey model with the addition of the Coriolis term in a rotating frame of reference is applied to characterize the non-Newtonian rheology in porous media. We investigate how the interaction among the Coriolis force, viscoelasticity, and bounded sidewalls affects the preferred mode at the onset of convection. The results show that for a slowly rotating case, the oscillatory mode is always preferred for any considered cylindrical radii. However, for a moderately rotating case, the oscillatory preferred mode only arises intermittently as the outer cylindrical radius gradually increases. This result is quite different from the case for viscoelastic fluids in a rotating porous layer or in a porous cylinder without rotation. Further, we discover that for a pair of given cylindrical radii when the Taylor number exceeds a critical value depending on the viscoelastic parameters, the oscillatory convection does not occur. We also examine how the variations of the Taylor number and the viscoelastic parameters affect the patterns of temperature disturbance at the onset of convection.     

Onset of Darcy–Brinkman convection in a binary viscoelastic fluid saturated porous layer

The onset of Darcy–Brinkman double-diffusive convection in a binary viscoelastic fluid-saturated porous layer is studied using both linear and weakly nonlinear stability analyses. The Oldroyd-B model is employed to describe the rheological behavior of the fluid. An extended form of Darcy–Oldroyd law incorporating the Brinkman’s correction and time derivative is used to describe the fluid flow and the Oberbeck–Boussinesq approximation is invoked. The onset criterion for stationary and oscillatory convection is derived analytically. The effects of rheological parameters, Darcy number, normalized porosity, Lewis number, solute Rayleigh number, and Darcy–Prandtl number on the stability of the system is investigated. The results indicated that there is a competition among the processes of thermal, solute diffusions and viscoelasticity that causes the convection to set in through the oscillatory modes rather than the stationary. The Darcy–Prandtl number has a dual effect on the threshold of oscillatory convection. The nonlinear theory based on the method of truncated representation of Fourier series is used to find the transient heat and mass transfer. Some existing results are reproduced as the particular cases of present study.     

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.     

Stability of Thermal Convection in a Fluid-Porous System Saturated with an Oldroyd-B Fluid Heated from Below

A linear stability analysis is conducted for thermal convection in a two-layer system composed of a fluid layer overlying a porous medium saturated with an Oldroyd-B fluid heated from below. It is found that the convection pattern in the system is controlled by the porous medium when the ratio of the depth of the fluid layer to that of the porous medium is small. However, the fluid layer takes a predominant role if the depth ratio exceeds a critical value. Compared with stationary convection, the switching point from a porous-dominated mode to a fluid-dominated mode for oscillatory convection is located at a lower depth ratio. The effects of different parameters on stationary convection and oscillatory convection are also investigated in detail.     

The convective instability of Maxwell fluid-saturated porous layer using a thermal non-equilibrium model

A linear stability analysis is carried out to study viscoelastic fluid convection in a horizontal porous layer heated from below and cooled from above when the solid and fluid phases are not in a local thermal equilibrium. The modified Darcy–Brinkman–Maxwell model is used for the momentum equation and two-field model is used for the energy equation each representing the solid and fluid phases separately. The conditions for the onset of stationary and oscillatory convection are obtained analytically. Linear stability analysis suggests that, there is a competition between the processes of viscoelasticity and thermal diffusion that causes the first convective instability to be oscillatory rather than stationary. Elasticity is found to destabilize the system. Besides, the effects of Darcy number, thermal non-equilibrium and the Darcy–Prandtl number on the stability of the system are analyzed in detail.     

The onset of convection in a nanofluid saturated porous layer using Darcy model with cross diffusion

Linear and nonlinear stability analysis for the onset of convection in a horizontal layer of a porous medium saturated by a nanofluid is studied. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. The modified Darcy equation that includes the time derivative term is used to model the momentum equation. In conjunction with the Brownian motion, the nanoparticle fraction becomes stratified, hence the viscosity and the conductivity are stratified. The nanofluid is assumed to be diluted and this enables the porous medium to be treated as a weakly heterogeneous medium with variation, in the vertical direction, of conductivity and viscosity. The critical Rayleigh number, wave number for stationary and oscillatory mode and frequency of oscillations are obtained analytically using linear theory and the non-linear analysis is made with minimal representation of the truncated Fourier series analysis involving only two terms. The effect of various parameters on the stationary and oscillatory convection is shown pictorially. We also study the effect of time on transient Nusselt number and Sherwood number which is found to be oscillatory when time is small. However, when time becomes very large both the transient Nusselt value and Sherwood value approaches to their steady state values.     

多孔介质内黏弹性流体的热对流稳定性研究

基于修正的Darcy模型, 介绍了多孔介质内黏弹性流体热对流稳定性研究的现状和主要进展. 通过线性稳定性理论, 分析计算多孔介质几何形状(水平多孔介质层、多孔圆柱以及多孔方腔)、热边界条件(底部等温加热、底部等热流加热、底部对流换热以及顶部自由开口边界)、黏弹性流体的流动模型(Darcy-Jeffrey, Darcy-Brinkman-Oldroyd以及Darcy-Brinkman -Maxwell模型)、局部热非平衡效应以及旋转效应对黏弹性流体热对流失稳的临界Rayleigh数的影响. 利用弱非线性分析方法, 揭示失稳临界点附近热对流流动的分叉情况, 以及失稳临界点附近黏弹性流体换热Nusselt数的解析表达式. 采用数值模拟方法, 研究高Rayleigh数下黏弹性流体换热Nusselt数和流场的演化规律,分析各参数对黏弹性流体热对流失稳和对流换热速率的影响.主要结果:(1)流体的黏弹性能够促进振荡对流的发生;(2)旋转效应、流体与多孔介质间的传热能够抑制黏弹性流体的热对流失稳;(3)在临界Rayleigh数附近,静态对流分叉解是超临界稳定的, 而振荡对流分叉可能是超临界或者亚临界的,主要取决于流体的黏弹性参数、Prandtl数以及Darcy数;(4)随着Rayleigh数的增加,热对流的流场从单个涡胞逐渐演化为多个不规则单元涡胞, 最后发展为混沌状态.      

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.     

The onset of double diffusive convection in a viscoelastic fluid layer

The onset of double diffusive convection in a viscoelastic fluid layer is studied using a linear and a weak nonlinear stability analyses. The onset criterion for stationary, oscillatory and finite amplitude convection is derived analytically. There is a competition between the processes of thermal diffusion, solute diffusion and viscoelasticity that causes the convection to set in through oscillatory mode rather than stationary. The effect of Deborah number, retardation parameter, solutal Rayleigh number, Prandtl number, Lewis number on the stability of the system is investigated. It is shown that the critical frequency increases with Deborah number and solutal Rayleigh number while it decreases with retardation parameter and Lewis number. The nonlinear theory based on the truncated representation of Fourier series method is used to find the heat and mass transfers. The transient behaviour of the Nusselt number and Sherwood number is investigated by solving the finite amplitude equations using Runge-Kutta method. The effect of viscoelastic parameters on heat and mass transfer is brought out.     

Overstability Analysis of Thermo-bioconvection Saturating a Porous Medium in a Suspension of Gyrotactic Microorganisms

A linear stability analysis is performed to analyze bioconvection in a dilute suspension of gyrotactic microorganisms in horizontal shallow fluid layer cooling from below and saturated by a porous medium, in the rigid boundary case. It is established that due to cooling from below thermally stratified layer is stabilized, which opposes the development of bioconvection and the situations for oscillatory convection may take place. The stability criterion is obtained in terms of thermal Rayleigh number, bioconvection Rayleigh number, gyrotactic number, bioconvection Peclet number, measure of cell eccentricity, Prandtl number, and Lewis number. It is observed that the presence of porous medium results in decrease of the magnitude of critical bioconvection Rayleigh number in comparison with its non-existence; hence due to porous effect, the system becomes less stable.     

Effect of Thermal Modulation on the Onset of Convection in a Viscoelastic Fluid Saturated Porous Layer

The effect of thermal modulation on the onset of convection in a horizontal, anisotropic porous layer saturated by a viscoelastic fluid is investigated by a linear stability analysis. Darcy’s law with viscoelastic correction is used to describe the fluid motion. The perturbation method is used to find the critical Rayleigh number and the corresponding wavenumber for small amplitude thermal modulation. The stability of the system characterized by a correction Rayleigh number is calculated as a function of the thermal and mechanical anisotropy parameters, the viscoelastic parameters and the frequency of modulation. It is found that the onset of convection can be delayed or advanced by the factors represented by these parameters. The results of the problem have possible implications in mantle convection.     

Non-Linear Two Dimensional Double Diffusive Convection in a Rotating Porous Layer Saturated by a Viscoelastic Fluid

Double diffusive convection in a rotating anisotropic porous layer, saturated by a viscoelastic fluid, heated from below and cooled from above has been studied making linear and non-linear stability analyses. The fluid and solid phases are considered to be in equilibrium. In momentum equation, we have employed the Darcy equation which includes both time derivative and Coriolis terms. The linear theory based on normal mode method is considered to find the criteria for the onset of stationary and oscillatory convection. A weak non-linear analysis based on minimal representation of truncated Fourier series analysis containing only two terms has been used to find the Nusselt number and Sherwood number as functions of time. We have solved the finite amplitude equations using a numerical scheme. The results obtained, during the above analyses, have been presented graphically and the effects of various parameters on heat and mass transfer have been discussed. Finally, we have drawn the steady and unsteady streamlines, isotherms, and isohalines for various parameters.