Critical Thermal Rayleigh Number Research Articles

Exploration of Coriolis Force on the Linear Stability of Couple Stress Fluid Flow Induced by Double Diffusive Convection

Abstract In this paper, the effect of Coriolis force is explored on convective instability of a doubly diffusive incompressible couple stress fluid layer with gravity acting downward. A linear stability analysis is used to obtain the conditions for the onset of stationary and oscillatory convection in the closed form. Being a multiparameter instability problem, results for some isolated cases have been presented to illustrate interesting corners of parameter space. It is found that the neutral curve for oscillatory onset forms a closed-loop which is separate from the neutral curve for stationary onset indicating the requirement of three critical thermal Rayleigh numbers to specify the linear instability criteria instead of the usual single value. Besides, the simultaneous presence of rotation and the addition of heavy solute to the bottom of the layer exhibit an intriguing possibility of destabilizing the system under certain conditions, in contrast to their stabilizing effect when they are present in isolation. The implication of couple stresses on each of the aforementioned anomalies is clearly brought out. The spatial wavelength of convective cells at the onset is also discussed.

Double-diffusive convection in a magnetic nanofluid layer with cross diffusion effects

The present paper investigates the onset of double-diffusive convection in a layer of magnetic nanofluid with the Soret and Dufour effects. The impact of three important slip mechanisms, viz. Brownian motion, thermophoresis, and magnetophoresis are included in the model that is used for the magnetic nanofluids (MNFs). We performed a linear stability analysis to investigate the problem and derived the results for water-based and ester-based magnetic nanofluids. The results are presented simultaneously for both the gravity as well as the microgravity environment for Rigid–Rigid boundaries. A numerical technique is employed to examine the nature of the stability, and it is found that the stability of the considered problem is stationary. It is also observed that the effect of increase in the values of Dufour parameter $$N_{TC}$$ and solutal Rayleigh number Rs is to delay, while increase in the values of Soret parameter $$N_{CT}$$ , concentration Rayleigh number $$R_n$$ , nonlinearity of fluid magnetization $$M_3$$ , Lewis number Le, and thermo-solutal Lewis number $$Le_s$$ is to advance the onset of double-diffusive magnetic nanofluid convection in both the gravity and microgravity environment. In the gravity environment, value of the critical thermal Rayleigh number $$Ra_c$$ first decreases as Langevin parameter $$\alpha _L$$ increases from 1 to 2, and then it starts increasing with the further increase in the value of $$\alpha _L$$ . This behavior is found to be just opposite to that observed for the critical magnetic Rayleigh number $$Ng_c$$ in the case of microgravity environment. Moreover, the values of $$Ra_c$$ and $$Ng_c$$ are found be higher in case of the ester-based MNFs compared with the water-based MNFs.

Numerical study of nanofluid thermo-bioconvection containing gravitactic microorganisms in porous media: Effect of vertical throughflow

An analytical investigation of the onset of nanofluid thermo-bioconvection in a fluid saturated by porous media containing gravitactic and nanoparticles microorganisms subjected to a vertical throughflow is presented. The heat conservation equation is formulated by introducing the convective term of nanoparticle flux. The fluid is stimulated with modified Brinkman model, normal mode analysis and six-term Galerkin methods are used to solve the governing equations. The combined effects of vertical throughflow, nanoparticles, gravitactic microorganisms, and porosity have been taken into account. The effects of bioconvection Rayleigh number, bioconvection Péclet number, nanoparticle Rayleigh number, Péclet number, bioconvection Lewis number, and porosity on critical thermal Rayleigh number have been examined. The analysis leads that critical wave number is the function of bioconvection parameters, nanofluid parameters and throughflow parameters. It is also found that vertical throughflow disturbs the formation of bioconvection pattern which are necessary for the development of bioconvection.

Influence of a uniform transverse magnetic field on the thermo-hydrodynamic stability in water-based nanofluids with metallic nanoparticles using the generalized Buongiorno’s mathematical model

The onset of nanofluid convection in the presence of an externally applied magnetic field is investigated numerically based on the non-homogeneous Buongiorno’s mathematical model. In this study, we use the latest experimental correlations and powerful analytical models for expressing the thermo-physical properties of some electrically conducting nanofluids, such as copper-water, sliver-water and gold-water nanofluids, in which the Brownian motion and thermophoresis effects on slip flow in nanofluids are taken into account in this model (i.e., two-phase transport model). In this paper, we assume that the nanofluid has Newtonian behavior, confined horizontally between two infinite impermeable boundaries and heated from below, in such a way that the nanoparticles tend to concentrate near the upper wall. Considering the basic state of the nanofluidic system, the linear stability theory has been successfully applied to obtain the principal stability equations, which are solved numerically for an imposed volumetric fraction of nanoparticles and no-slip impermeable conditions at the isothermal walls bounding the nanofluid layer. The linear boundary-value problem obtained in this investigation is converted into a pure initial-value problem, so that we can solve it numerically by the fourth-fifth-order Runge-Kutta-Fehlberg method. The generalized Buongiorno’s mathematical model proposed in this study allows performing a highly accurate computational analysis. In addition, the obtained results show that the stability of the studied nanofluidic system depends on several parameters, namely, the magnetic Chandrasekhar number Q , the reference value for the volumetric fraction of nanoparticles $ \phi_0$ and the size of nanoparticles $ d_p$ . In this analysis, the thermo-hydrodynamic stability of the studied nanofluid is controlled through the critical thermal Rayleigh number $ R_{ac}$ , which characterizes the onset of convection cells, whose size is $ L_c=2\pi/a_c$ . Furthermore, the effects of various pertinent parameters on the critical stability parameters $ R_{ac}$ and $ a_c$ are discussed in more detail through graphical and tabular illustrations, for three types of nanofluids including copper-water, sliver-water, and gold-water.

Double-diffusive Hadley–Prats flow in a horizontal porous layer with a concentration based internal heat source

Double-diffusive Hadley–Prats flow with a concentration based heat source is investigated through linear and non-linear stability analyses. The resultant eigenvalue problems for both theories are solved numerically using Shooting and fourth order Runga–Kutta methods, with the critical thermal Rayleigh number being evaluated with respect to various flow governing parameters such as the magnitudes of the heat source and mass flow. It is observed, in the linear case, that an increase in the horizontal thermal Rayleigh number is stabilising for both positive and negative values of the solutal Rayleigh number. In the non-linear case, a destabilising effect is identified at higher mass flow rates. An increase in both the heat source and mass flow results in destabilisation.

Numerical Investigations of Electro-Thermal Convection in Dielectric Nanofluid Layer

The numerical investigations of a dielectric nanofluid under the simultaneous action of an Alternating Current electric field and a uniform adverse temperature gradient has been studied using the linear theory and normal mode analysis. The eigen-value problem is solved analytically and numerically using the Galerkin technique. Effect of various non-dimensional parameters on thermal Rayleigh number Ra for the stationary convection has been determined. The boundaries are considered to be stress-free. Thermal Rayleigh number has been computed for various values of electric Rayleigh number for the onset of instability using the software-MATHEMATICA Version 5.2. Numerical results are presented graphically. The thermal Rayleigh number decreases linearly as the concentration Rayleigh number Rn increases. Moreover for stability motion, the value of thermal Rayleigh number Ra occurs only for negative values of Rea and it is positive. The value of critical thermal Rayleigh number Ra decreases with decreasing the value of Rea. It is observed that the role of electric field is not of much sinificance the convection in the fluid layer in case of stability motions. In case of stability, the thermal Rayleigh number is independent of Prandtl number and hence Prandtl number has no role to play.

Nonlinear Stability of Double-diffusive Convection in a Porous Layer with Throughflow and Concentration based Internal Heat Source

In this study, nonlinear stability analysis of double-diffusive convection in a horizontal fluid saturated porous layer has been investigated. Concentration based internal heat source and vertical throughflow effects are considered during investigation. Energy method has been implemented to develop the nonlinear stability analysis. Runge–Kutta and shooting methods have been used to solve the eigenvalue problem. Critical thermal Rayleigh number is obtained for assigned values of governing parameters. Results of linear and nonlinear theories have been compared. It is observed that for downward throughflow, when Peclet number Pe is high, the effect of concentration based internal heat source $$\gamma $$ is insignificant.

Linear stability analysis for ferromagnetic fluids in the presence of magnetic field, compressibility, internal heat source and rotation through a porous medium

The effects of magnetic field and heat source strength on thermal convection of a compressible rotating ferromagnetic fluid through a porous medium are investigated theoretically using linear stability theory. A normal mode analysis method is employed to find solutions for the fluid layer confined between parallel planes with free boundaries. The cases of stationary and oscillatory instabilities are discussed. For the stationary state, compressibility, medium porosity and temperature gradient due to heat source have destabilizing effects, whereas rotation and ratio of magnetic permeability delay the onset of convection. The magnetic field and medium permeability have both stabilizing and destabilizing effects under certain conditions. The variations in the stationary critical thermal Rayleigh number and neutral instability curves in (Ra1,x)-plane for various values of physical parameters are shown graphically to depict the stability characteristics. The sufficient conditions for the non-existence of overstability are obtained and the principle of exchange of stabilities holds true in the absence of magnetic field and rotation under certain conditions.

Effects of Vertical Throughflow and Variable Gravity on Hadley–Prats Flow in a Porous Medium

The stability of convection in a horizontal porous layer which is saturated with fluid and induced by horizontal temperature gradients subjected to horizontal mass flow is investigated by means of linear and nonlinear stability analysis. The effects of variable gravity field and vertical throughflow are also considered in this analysis. The nonlinear stability analysis part has been developed via energy functional. Shooting and Runge–Kutta methods have been used to solve eigenvalue problem in both cases. Critical vertical thermal Rayleigh numbers for both linear and nonlinear analyses \(R_\mathrm{L}\) and \(R_\mathrm{E}\) are evaluated for different values of horizontal Rayleigh number \(R_x\), horizontal Peclet number Pe, vertical Peclet number \(Q_\mathrm{v}\) and variable gravity parameter \(\eta \). Comparison is made between linear and nonlinear stability results. It has been observed that linear stability results are overpredicting the onset of convection compared with nonlinear theory, and hence subcritical instabilities would arise before one gets the onset of linear stability threshold.

Rayleigh-Benard Instability in a Horizontal Porous Layer Affected by Rotation

This study examines the Benard convection of an infinite horizontal porous layer permeated by an incompressible thermally conducting viscous fluid in the presence of Coriolis forces. The porous layer is controlled by the Brinkman model. Analytical and numerical solutions are obtained for the cases of stationary convection and overstability. The critical thermal Rayleigh numbers are obtained for different values of the permeability of porous medium, Chandrasekhar number and Taylor number for different boundary conditions. The related eigenvalue problem is solved using the Chebyshev polynomial Tau method.

Linear Stability of Double Diffusive Convection of Hadley-prats Flow with Viscous Dissipation in a Porous Media

Linear stability analysis of double diffusive convection in a horizontal fluid saturated porous layer has been carried out. Effects of viscous dissipation, horizontal mass flow are taken into account. Combined effects of horizontal mass flow, viscous dissipation may cause instability in the fluid system. To carry out linear stability analysis, disturbances are the form of longitudinal rolls, transverse rolls have been considered. In order to solve eigenvalue problem numerically, Runge-Kutta and shooting methods have been employed for the case of longitudinal rolls. Chebyshev-Tau method has been used for the case of transverse rolls. Critical wave number aC , critical vertical thermal Rayleigh number RaC are evaluated for assigned values of flow governing parameters. For the onset of convection, physical explanation is given.

Tropical atmospheric circulations with humidity effects.

The main objective of this article is to study the effect of the moisture on the planetary scale atmospheric circulation over the tropics. The modelling we adopt is the Boussinesq equations coupled with a diffusive equation of humidity, and the humidity-dependent heat source is modelled by a linear approximation of the humidity. The rigorous mathematical analysis is carried out using the dynamic transition theory. In particular, we obtain mixed transitions, also known as random transitions, as described in Ma & Wang (2010 Discrete Contin. Dyn. Syst.26, 1399-1417. (doi:10.3934/dcds.2010.26.1399); 2011 Adv. Atmos. Sci.28, 612-622. (doi:10.1007/s00376-010-9089-0)). The analysis also indicates the need to include turbulent friction terms in the model to obtain correct convection scales for the large-scale tropical atmospheric circulations, leading in particular to the right critical temperature gradient and the length scale for the Walker circulation. In short, the analysis shows that the effect of moisture lowers the magnitude of the critical thermal Rayleigh number and does not change the essential characteristics of dynamical behaviour of the system.

Effect of the Buoyancy Ratio on Oscillatory Double-Diffusive Convection in Binary Mixture

In this work, a numerical study of double-diffusive convection in binary mixture has been presented. A square cavity filled with a binary mixture and exposed to opposition solute and thermal gradients has been considered. The following flow parameters were considered: Prandtl number Pr = 10, Lewis number Le = 10, and buoyancy ratio varies 0 ≤ N ≤ 2. The finite volume method with SIMPLER algorithm was used to solving numerically the mathematical model. Our computer code is validated and shows a good agreement with literature available results. The obtained results show a strongest dependence of the thermal structure and solute effect with the buoyancy ratio. The oscillatory double-diffusive flow appeared from periodic time-evolution where the phenomena retuned in each period time. A critical thermal Rayleigh number RaTcr and corresponding dominated frequency for the onset of oscillatory double-diffusive convection were determined for each buoyancy ratio N, and the results show a strongest dependence between the buoyancy ratio and critical Rayleigh number. Also, the dominance of solute force increases the intensity of the flow better than the case of the dominance of thermal force.

Effect of Soret and Temperature Dependent Viscosity on Thermohaline Convection in a Ferrofluid Saturating a Porous Medium

Soret driven ferrothermoconvective instability in multi-component fluids has a wide range of applications in heat and mass transfer. This paper deals with the theoretical investigation of the effect of temperature dependent viscosity on a Soret driven ferrothermohaline convection heated from below and salted from above subjected to a transverse uniform magnetic field in the presence of a porous medium. The Brinkman model is used in the study. It is found that the stationary mode of instability is preferred. For a horizontal fluid layer contained between two free boundaries an exact solution is examined using the normal mode technique for a linear stability analysis. The effect of salinity has been included in magnetization and density of the fluid. The critical thermal magnetic Rayleigh number for the onset of instability is obtained numerically for sufficiently large values of the buoyancy magnetization parameter M1 using the method of numerical Galerkin technique. It is found that magnetization and permeability of the porous medium destabilize the system. The effect of temperature dependent viscosity stabilizes the system on the onset of convection.

Hall Effect on Bénard Convection of Compressible Viscoelastic Fluid through Porous Medium

An investigation made on the effect of Hall currents on thermal instability of a compressible Walter’s B′ elasticoviscous fluid through porous medium is considered. The analysis is carried out within the framework of linear stability theory and normal mode technique. For the case of stationary convection, Hall currents and compressibility have postponed the onset of convection through porous medium. Moreover, medium permeability hasten postpone the onset of convection, and magnetic field has duel character on the onset of convection. The critical Rayleigh numbers and the wave numbers of the associated disturbances for the onset of instability as stationary convection have been obtained and the behavior of various parameters on critical thermal Rayleigh numbers has been depicted graphically. The magnetic field, Hall currents found to introduce oscillatory modes, in the absence of these effects the principle of exchange of stabilities is valid.

Hall Effect on Thermal Instability of Viscoelastic Dusty Fluid in Porous Medium

The effect of Hall currents and suspended dusty particles on the hydromagnetic stability of a compressible, electrically conducting Rivlin-Ericksen elastico viscous fluid in a porous medium is considered. Following the linearized stability theory and normal mode analysis the dispersion relation is obtained. For the case of stationary convection, Hall currents and suspended particles are found to have destabilizing effects whereas compressibility and magnetic field have stabilizing effects on the system. The medium permeability, however, has stabilizing and destabilizing effects on thermal instability in contrast to its destabilizing effect in the absence of the magnetic field. The critical Rayleigh numbers and the wave numbers of the associated disturbances for the onset of instability as stationary convection are obtained and the behavior of various parameters on critical thermal Rayleigh numbers are depicted graphically. The magnetic field, Hall currents and viscoelasticity parameter are found to introduce oscillatory modes in the systems, which did not exist in the absence of these parameters

Numerical study of the onset of thermosolutal convection in rotating spherical shells

The influence of an externally enforced compositional gradient on the onset of convection of a mixture of two components in a rotating fluid spherical shell is studied for Ekman numbers E = 10−3 and E = 10−6, Prandtl numbers σ = 0.1, 0.001, Lewis numbers τ = 0.01, 0.1, 0.8, and radius ratio η = 0.35. The Boussinesq approximation of the governing equations is derived by taking the denser component of the mixture for the equation of the concentration. Differential and internal heating, an external compositional gradient, and the Soret and Dufour effects are included in the model. By neglecting these two last effects, and by considering only differential heating, it is found that the critical thermal Rayleigh number \documentclass[12pt]{minimal}\begin{document}$R_{e}^c$\end{document}Rec depends strongly on the direction of the compositional gradient. The results are compared with those obtained previously for pure fluids of the same σ. The influence of the mixture becomes significant when the compositional Rayleigh number Rc is at least of the same order of magnitude as the known \documentclass[12pt]{minimal}\begin{document}$R_{e}^c$\end{document}Rec computed without mixture. For positive and sufficiently large compositional gradients, \documentclass[12pt]{minimal}\begin{document}$R_e^c$\end{document}Rec decreases and changes sign, indicating that the compositional convection becomes the main source of instability. Then the critical wave number mc decreases, and the drifting waves slow down drastically giving rise to an almost stationary pattern of convection. Negative gradients delay the onset of convection and determine a substantial increase of mc and ωc for Rc sufficiently high. Potential laws are obtained numerically from the dependence of \documentclass[12pt]{minimal}\begin{document}$R_{e}^c$\end{document}Rec and of the critical frequency ωc on Rc, for the moderate and small Ekman numbers explored.

A nonlinear stability analysis of a rotating double-diffusive magnetized ferrofluid

A nonlinear stability result for a double-diffusive magnetized ferrofluid layer rotating about a vertical axis for stress-free boundaries is derived via generalized energy method. The mathematical emphasis is on how to control the nonlinear terms caused by magnetic body and inertia forces. The result is compared with the result obtained by linear instability theory. The critical magnetic thermal Rayleigh number given by energy theory is slightly less than those given by linear theory and thus indicates the existence of subcritical instability for ferrofluids. For non-ferrofluids, it is observed that the nonlinear critical stability thermal Rayleigh number coincides with that of linear critical stability thermal Rayleigh number. For lower values of magnetic parameters, this coincidence is immediately lost. The effect of magnetic parameter, M3, solute gradient, S1, and Taylor number, TA1, on subcritical instability region have been analyzed. We also demonstrate coupling between the buoyancy and magnetic forces in the nonlinear stability analysis.

Hot springs mediate spatial exchange of heat and mass in the enclosed sediment domain: A stability perspective

In many natural environments, such as in underwater hot springs and hydrothermal vents, thermal gradients are accompanied with changes in the concentration of chemical compounds transported to the seawater, causing the so-called double-diffusive, mixed convection. To study the physical scenarios in such systems, a vertical channel filled with a porous medium saturated with saline water is considered. The motion in the sediment-filled channel is induced by two buoyancy forces and an external pressure gradient, similar to the situation in a vent with an upward flow direction. The fluid flow has been modeled by an extended Darcy model, and the flow instability mechanisms have been studied numerically. The linear stability analysis is performed considering a wide range of Darcy number (Da = 10 −5 –10 −8). The instability boundary curve showed three distinct dynamic regimes: (i) Rayleigh–Taylor (R–T), (ii) log–log non-linear variation, and (iii) log–log linear variation. The domain of different regimes were sensitive to external pressure gradient as well as permeability. Similar to cross-diffusive natural convection in pure viscous fluids, a linear relationship between logarithmic absolute values of critical thermal Rayleigh number (∣Ra T ∣) and solute Rayleigh number (Ra C ) is found in the third regime. Based on the permeability, for any solute Rayleigh number (Ra C ), there existed a minimum value of Reynolds number (Re), below which R–T type of instability appeared. Above this minimum value, the instability was due to two buoyancy forces, known as buoyant instability. Simulations of secondary flow via energy analysis demonstrated the development of complex dynamics at the critical state in all three regimes characterized by transition of multi to uni-cellular structures and vice verse.

Thermal instability of compressible Walters’ (Model B′) fluid in the presence of Hall currents and suspended particles

Effect of Hall currents and suspended particles is considered on the hydromagnetic stability of a compressible, electrically conducting Walters? (Model B?) elastico-viscous fluid. After linearizing the relevant hydromagnetic equations, the perturbation equations are analyzed in terms of normal modes. A dispersion relation governing the effects of visco-elasticity, magnetic field, Hall currents and suspended particles is derived. It has been found that for stationary convection, the Walters? (Model B?) fluid behaves like an ordinary Newtonian fluid due to the vanishing of the visco-elastic parameter. The compressibility and magnetic field have a stabilizing effect on the system, as such their effect is to postpone the onset of thermal instability whereas Hall currents and suspended particles are found to hasten the onset of thermal instability for permissible range of values of various parameters. Also, the dispersion relation is analyzed numerically and the results shown graphically. The critical Rayleigh numbers and the wavenumbers of the associated disturbances for the onset of instability as stationary convection are obtained and the behavior of various parameters on critical thermal Rayleigh numbers has been depicted graphically. The visco-elasticity, suspended particles and Hall currents (hence magnetic field) introduce oscillatory modes in the system which were non-existent in their absence.