Onset Of Stationary Convection Research Articles

Effect of basic Temperature Gradients on Electrothermal Convection in a Rotating Layer of Viscoelastic (Walter’s B) Fluid Saturated Porous Medium Heated from Below

Combined Effect of Thermal Modulation and Rotation on the Onset of Convection in Walter's B Viscoelastic Fluid Saturated Porous Medium has been investigated. The problem is numerically solved using the Galerkin method after applying linear stability analysis and normal modes. It is possible to derive the dispersion relation while accounting for the effects of wave number, electric Rayleigh number, Darcy number, Taylor number, and thermal electric Rayleigh number. Graphically, the effects of the electric Rayleigh number, Darcy number, Wave number, and Taylor number on the onset of stationary convection have been studied. Under the boundary conditions considered, The kinematic viscoelasticity accounting for rheology of the nanofluid has no effect on the stationary convection for Walters’ (model B’) nanofluids and behaves like an ordinary Newtonian nanofluid.

Thermal Convective Instabilities and Chaos in a Rotating Hybrid Nanofluid Layer with Cattaneo–Christov Heat Flux Model

The linear and nonlinear dynamics of thermal convection of a rotating hybrid nanofluid layer heated from below with the Cattaneo–Christov heat flux model are studied in this paper. Starting from the flow equations of a hybrid nanofluid and exploiting the free boundary conditions, the analytical expressions of the stationary and oscillatory Rayleigh numbers of the base fluid are determined as a function of the dimensionless parameters of the heat transfer fluid and the thermophysical properties of the hybrid nanofluid. The effects of hybrid nanoparticles and Taylor number on the onset of stationary convection in the base fluid are investigated graphically. Then, a numerical study of the transition from natural convection to chaotic behaviour of the hybrid nanofluid is made using the truncated Galerkin approximation. This approximation allowed us to find a novel six-dimensional nonlinear system depending on the parameters of the base fluid and the thermophysical properties of the hybrid nanofluid that can be reduced to five, four, or three dimensions when we tend some parameters to zero. The different results showed that the addition of hybrid nanoparticles (alumina-copper) to a thermal fluid (water) subjected to the rotation force in the presence or absence of the thermal relaxation time allows control of the chaotic convection in the base fluid. On the other hand, the increase of the rescaled Taylor number and the Cattaneo number widens the domain of chaos in the hybrid nanofluid with the increase of the rescaled Rayleigh number of the base fluid.

Effect of salt fingers on heat transfer for different non‐uniform concentration profiles in a micropolar liquid

AbstractThis paper discusses the theoretical aspects of the effect of salt fingers on heat transfer for different non‐uniform concentration profiles in a micropolar liquid layer kept between two parallel plates of infinite extent separated by a thin layer, heated and soluted from above. The onset of salt finger convection (convection due to the salt finger process) is studied through the linear stability analysis theory. The system of partial differential equations is solved numerically using the normal mode analysis method and the required solution is found by applying the Galerkin method. The effect of heat transfer and concentration of micropolar liquid is obtained for (i) free–free, (ii) rigid–rigid, and (iii) rigid–free isothermal, permeable with no‐spin boundary conditions. The effect of different micropolar parameters (i.e., coupling parameter, micropolar heat conduction parameter, couple stress parameter, and inertia parameter) has been analyzed on the onset of stationary convection, and the results are depicted graphically. It is shown that different non‐uniform concentration gradients, diffusivity ratio, coupling parameter, and solutal Rayleigh number influence the heat transfer in the system. The phase of concentration flow for different boundary conditions is compared and analyzed.

Convective instabilities in binary mixture 3He-4He in porous media

The present study attempts to simulate and analyze the Rayleigh-Bénard convection of superfluid mixture (3He-4He) kept in the sparsely packed porous medium with stress-free boundary conditions. Theoretically the linear and nonlinear analysis are carried out near the onset of stationary convection. The nonlinear governing equations describing the motion with the Darcy model, temperature and concentration fields are expanded as the sequence of non-homogeneous linear equations. These equations are solved by employing the Fourier analysis of perturbations in terms of the non-dimensional expansion parameter ε until O(ε8) as proposed by Kuo (1961). The flow field and heat transfer characteristics are analyzed for different control parameters arising in the system such as the Rayleigh number (R), the separation ratio (ψ), which is the coupling between the temperature and concentration fields, and the Lewis number (Le). Apart from streamlines and isotherms, the novel flow visualization technique for the heat flow patterns in terms of heatlines is derived and plotted. Also, the kinetic energy, potential energy, and minimum entropy generation are analyzed. The Nusselt number is found to be get enhanced as the values of R and ψ are increased whereas it gets inhibited as Le increases.

Weakly nonlinear thermohaline rotating convection in a sparsely packed porous medium

This paper discusses the analytical study of linear and weakly nonlinear stability of thermohaline convection in a sparsely packed porous due to rotation. In the linear stability analysis, we used the normal mode technique to find the critical Rayleigh number and it is a function of q,Ta,Λ,R2, and L. Marginal stability curves were traced corresponding to thermal Rayleigh number and studied bifurcation points. In the weakly nonlinear analysis, a nonlinear two-dimensional Landau Ginzburg equation was derived at the onset of stationary convection, studied the secondary instabilities and heat transport due to convection. One dimensional coupled Landau Ginzburg equations are derived at the onset of oscillatory convection and analyzed the stability regions of steady-state, standing waves, and travelling waves.

Rayleigh-Benard convection in a binary fluid-saturated anisotropic porous layer with variable viscosity effect

Rayleigh-Benard convection due to buoyancy that occurred in a horizontal binary fluid layer saturated anisotropic porous media is investigated numerically. The system is heated from below and cooled from above. The temperature-dependent viscosity effect was applied to the double-diffusive binary fluid and the critical Rayleigh number for free-free, rigid-free, and rigid-rigid representing the lower-upper boundary were obtained by using the single-term Galerkin expansion procedure. Both boundaries are conducted to temperature. The effect of temperature-dependent viscosity, mechanical anisotropy, thermal anisotropy, Soret, and Dufour parameters on the onset of stationary convection are discussed and shown graphically.

Effect of anisotropic permeability on double‐diffusive bidisperse porous medium

AbstractThe model of thermosolutal convection in a fluid‐saturated bidisperse porous medium of Darcy type is studied in this paper. The permeability is allowed to be horizontally isotropic for both the macro‐ and microphases. The linear instability and nonlinear stability are analyzed by taking the Soret effect into account. Furthermore, the effect of anisotropy parameter, Soret coefficient, and other physical parameters on the stability of the system are investigated. It is shown that the linear instability boundaries and the energy stability boundaries do not coincide when the layer is heated and salted from below, where a region of potential subcritical instability occurs. The results reveal that the horizontal to vertical permeability ratio plays a crucial role in the stability of the system. It is also observed that for large values of the salt Rayleigh number, the onset of thermal convection is more likely to be via oscillatory convection rather than stationary convection. Furthermore, the onset of stationary convection is significantly influenced by the presence of the Soret coefficient.

Electrohydrodynamic Instability of a Rotating Walters’ (model B’) Fluid in a Porous Medium: Brinkman model

Abstract In this study, the instability of Walters’ (model B’) viscoelastic fluid in a Darcy-Brinkman-Boussinesq system heated from below saturating a porous medium in electrohydrodynamics is considered. By applying the linear stability analysis and normal modes, the dispersion relation accounting for the effect of Prandtl number, electric Rayleigh number, Darcy number, Brinkman-Darcy number, Taylor number and kinematic viscoelasticity parameter is derived. The effects of electric Rayleigh number, Darcy number, Brinkman-Darcy number and Taylor number on the onset of stationary convection have been investigated both analytically and graphically.

Finite amplitude cellular convection under the influence of a vertical magnetic field

At the onset of stationary convection, the effect of a vertical magnetic field on the heat transfer of Rayleigh-Benard convection for an electrically conducting fluid is studied. The nonlinear governing equations describing the motion, temperature and magnetic fields are expanded as the sequence of non-homogeneous linear equations, which depend on the solutions of the linear stability problem. Infinite number of steady state with finite amplitude solutions are obtained for the stress-free boundary conditions. The perturbation method proposed by Kuo (1961) is used for the first time to highlight the heat transfer features of magnetoconvection. An explicit expression at the onset of convection in terms of parameters of the system is obtained. The dependence of heat transfer rate on Rayleigh number (R), Chandrasekhar number, thermal and magnetic Prandtl numbers is extensively examined until sixth order using an expansion of R as proposed by Kuo (1961). The results show that the magnetic field dampens the heat flow for stationary convection, i.e., the onset of convection shifts to higher values of R as the vertical magnetic field increases. Under the uniform magnetic field, heat flow gets enhanced as the thermal Prandtl number increases, whereas heat flow diminishes for the increase in magnetic Prandtl number. The results of flow field and heat transfer characteristics are depicted in the form of streamlines and isotherms, respectively. The presence of magnetic field changes the flow structure of streamlines from unicellular to multicellular patterns. This is due to the magnetic susceptibility of colder fluid flow towards the magnetic field. The flow field is analyzed with respect to the topological invariant relation. To trace the path of convective heat transport, the concept of Heatfunction has been employed. This methodology explains the comprehensive interpretation of energy distribution in terms of heatlines.

Effect of Carreau-Yasuda rheological parameters on subcritical Lapwood convection in horizontal porous cavity saturated by shear-thinning fluid

The present study is focused on Lapwood convection in isotropic porous media saturated with non-Newtonian shear thinning fluid. The non-Newtonian rheological behavior of the fluid is modeled using the general viscosity model of Carreau–Yasuda. The convection configuration consists of a shallow porous cavity with a finite aspect ratio and subject to a vertical constant heat flux, whereas the vertical walls are maintained impermeable and adiabatic. An approximate analytical solution is developed on the basis of the parallel flow assumption, and numerical solutions are obtained by solving the full governing equations. The Darcy model with the Boussinesq approximation and energy transport equations are solved numerically using a finite difference method. The results are obtained in terms of the Nusselt number and the flow fields as functions of the governing parameters. A good agreement is obtained between the analytical approximation and the numerical solution of the full governing equations. The effects of the rheological parameters of the Carreau-Yasuda fluid and Rayleigh number on the onset of subcritical convection thresholds are demonstrated. Regardless of the aspect ratio of the enclosure and thermal boundary condition type, the subcritical convective flows are seen to occur below the onset of stationary convection. Correlations are proposed to estimate the subcritical Rayleigh number for the onset of finite amplitude convection as a function of the fluid rheological parameters. Linear stability of the convective motion, predicted by the parallel flow approximation, is studied, and the onset of Hopf bifurcation, from steady convective flow to oscillatory behavior, is found to depend strongly on the rheological parameters. In general, Hopf bifurcation is triggered earlier as the fluid becomes more and more shear-thinning.

INVESTIGATION ON COUPLED CONVECTION WITH INTERNAL HEATING IN MICROPOLAR FLUID

The effect of uniform distribution of internal heat generation on the linear stability analysis of the Benard-Marangoni convection (coupled driven convection) in micropolar fluid is investigated theoretically. The upper free surface is assumed to be non-deformable and the lower boundary is taken to be rigid and isothermal with fixed temperature and span-vanishing boundaries. The eigenvalue problem is solved numerically using the Galerkin method. The influence of the internal heat generation in micropolar fluid with various parameters on the onset of stationary convection has been analyzed and also comparison has been made with the Newtonian fluid. We found that the effect of internal heating is to destabilize the micropolar fluid system.

On the onset of triple-diffusive convection in a layer of nanofluid

On the onset of triple-diffusive convection in a horizontal layer of nanofluid heated from below and salted from above and below is studied both analytically and numerically. The effects of thermophoresis and Brownian diffusion parameters are also introduced through Buongiorno model in the governing equations. By using linear stability analysis based on perturbation theory and applying normal modes analysis method, the dispersion relation accounting for the effect of various parameters is derived. The influences of solute-Rayleigh number, analogous solute-Rayleigh number, thermo-nanofluid Lewis number, modified diffusivity ratio and nanoparticle Rayleigh number on the stability of stationary convection are presented analytically and graphically. The solute Rayleigh number and analogous solute Rayleigh number have stabilizing effects on the onset of stationary convection for both top-heavy and bottom-heavy arrangements. The thermo-nanofluid Lewis number and diffusivity ratio have stabilizing effects on the onset of stationary convection while nanoparticle Rayleigh number has destabilizing effect on the onset of stationary convection. The necessary conditions for the existence of oscillatory modes are also obtained. A very good agreement is found between the results of present paper and earlier published results.

Triple-diffusive Convection in a Micropolar Rotating Ferrofluid

This paper deals with the theoretical investigation of the triple-diffusive convection in a micropolar ferrofluid layer heated and soluted below subjected to a transverse uniform magnetic field in the presence of uniform vertical rotation. For a flat fluid layer contained between two free boundaries, an exact solution is obtained. A linear stabilityanalysis theory and normal mode analysis method have been carried out to study the onset convection. The influence of various parameters like rotation, solute gradients, and micropolar parameters (i.e. coupling parameter, spindiffusion parameter and micropolar heat conduction parameter) has been analyzed on the onset of stationary convection. The critical magnetic thermal Rayleigh number for the onset of instability is also determined numerically for sufficiently large value of buoyancy magnetization parameter M1(ratio of the magnetic to gravitational forces). The principle of exchange of stabilities is found to hold true for the micropolar fluid heated from below in the absence of micropolar viscous effect, microinertia, solute gradient and rotation. The oscillatory modes are introduced due to the presence of the micropolar viscous effect, microinertia, solute gradient and rotation, which were non-existent in their absence.

On the onset of stationary convection in double-diffusive binary viscoelastic fluid saturated anisotropic porous layer

Linear stability of double diffusive convection in a binary viscoelastic fluid saturated anisotropic porous layer has been studied analytically. A sufficient condition for the occurrence of stationary convection has been derived in terms of the parameters of the system alone. It is further proved the above result is uniformly valid for any combination of the bounding surfaces.

Effect of variable gravity on thermal instability of rotating nanofluid in porous medium

Effect of variable gravity on the onset of thermal convection in a horizontal layer of rotating nanofluid in a porous medium is investigated. For the porous medium Darcy model is used. A linear stability analysis based upon normal mode technique is used to find solution of the fluid layer confined between two free-free boundaries. Rayleigh numbers on the onset of stationary convection and oscillatory convection have been derived by using Galerkin method. Graphs have been plotted to study the effect of Taylor number (rotation), Lewis number and variable gravity parameter on the stationary convection.

Mean flow effects in magneto-convection

In the present study the theoretical and numerical results of Rayleigh-Bérnard convection in an electrically conducing fluid in the presence of a vertical external magnetic field is reported. The effect of mean flow at the onset of stationary convection and close to the bifurcation is analyzed. The coefficients of the corresponding amplitude equations are analytically calculated and numerical simulations are performed. Finally, the secondary instabilities, such as Eckhaus, zig-zag and skew-varicose at the onset of stationary convection, associated to the roll solution are studied. It is observed that most of the critical modes appear close to the zig-zag instability.

Triple- Diffusive Convection in a Micropolar Ferrofluid in the Presence of Rotation

This paper deals with the theoretical investigation of the triple-diffusive convection in a micropolar ferrofluid layer heated and soluted below subjected to a transverse uniform magnetic field in the presence of uniform vertical rotation. For a flat fluid layer contained between two free boundaries, an exact solution is obtained. A linear stability analysis theory and normal mode analysis method have been employed to study the onset convection. The influence of various parameters like rotation, solute gradients, and micropolar parameters (i.e., the coupling parameter, spin diffusion parameter and micropolar heat conduction parameter) on the onset of stationary convection has been analyzed. The critical magnetic thermal Rayleigh number for the onset of instability is also determined numerically for sufficiently large value of the buoyancy magnetization parameter M1 (ratio of the magnetic to gravitational forces). The principle of exchange of stabilities is found to hold true for the micropolar fluid heated from below in the absence of micropolar viscous effect, microinertia, solute gradient and rotation. The oscillatory modes are introduced due to the presence of the micropolar viscous effect, microinertia , solute gradient and rotation, which were non-existent in their absence. In this paper, an attempt is also made to obtain the sufficient conditions for the non-existence of overstability.

Küppers-Lortz instability in rotating Rayleigh-Bénard convection in a porous medium

The Darcy-Lapwood-Brinkman model with the Boussinesq approximation is used to study Kuppers-Lortz (KL) instability in the nonlinear regime of rotating Rayleigh-Benard convection in a sparsely packed porous medium near the onset of stationary convection. The threshold Taylor numbers and critical angles for the onset of KL instability are obtained for different values of Λ, M for finite Prandtl numbers (1.5≤Pr≤100). Heat transfer is studied from Nusselt number at the onset of stationary convection.

Effect of Internal Heat Generation on Benard-Marangoni Convection in Micropolar Fluid with Feedback Control

The effect of uniform distribution of internal heat generation on the linear stability analysis of the Benard-Marangoni convection in an Eringen's micropolar fluids with feedback control is investigated theoretically. The upper free surface is assumed to be non-deformable and the lower boundary is taken to be rigid and isothermal with fixed temperature and span-vanishing boundaries. The eigenvalue is solved numerically using the Galerkin method. The influence of the internal heat generation; Q and feedback control; K in micropolar fluids with various parameters on the onset of stationary convection has been analysed.

A Mathematical Theorem on the Onset of Stationary Convection in Couple-Stress Fluid

The thermal instability of a couple-stress fluid heated from below is investigated. Following the linearized stability theory and normal mode analysis, the paper mathematically establishes that the onset of instability at marginal state, cannot manifest itself as stationary convection, if the thermal Rayleigh number R and the couple-stress parameter F,