Critical Thermal Rayleigh Number Research Articles

Linear and nonlinear stability of double diffusive convection in a micropolar fluid saturated porous layer with magnetic field and throughflow effects

The linear and nonlinear stability of double-diffusive convection in a porous layer saturated with micropolar fluid is examined. A transverse magnetic field is applied to the flow together with vertical throughflow. The normal mode technique is employed for linear stability analysis, whereas the energy method is used for nonlinear stability analysis. The resulting eigenvalue problems corresponding to linear and nonlinear stability theories are solved numerically by employing the bvp4c routine in MATLAB 2022(b). The critical thermal Rayleigh numbers for both linear and nonlinear analyses are computed for the different values of the governing parameters and presented graphically. A comparison is made between linear and nonlinear stability results. It is observed that the flow is more stable whenever a magnetic field is added to the flow, although the subcritical instability region also slightly increases. Increasing the Darcy number, Lewis number, coupling number, and absolute value of the throughflow parameter destabilizes the flow. On the other hand, raising the porosity of the medium and micropolar parameters stabilizes the flow. Furthermore, there is no subcritical gap in the absence of the throughflow effect, which is a good agreement between the linear and nonlinear thresholds.

Instability of Jeffrey Fluid Throughflow in a Porous Layer Induced by Heat Source and Soret Effect

Abstract In this study, we investigated the instability of thermosolutal convection of Jeffrey fluid in a porous layer with internal heating and the Soret effect. The layer is bounded by two fixed permeable parallel plates which are assumed to be isothermal and isosolutal. An existing initial flow in the vertical direction is passing the layer at a constant speed. The flow fields are adequately presented by PDEs and transformed into dimensionless forms. A small perturbation to the basic flow profiles with linear stability analysis results the problem in an eigenvalue problem. The Runge-Kutta method is used to derive the numerical value of the critical thermal Rayleigh number. The convective instability for asymptotic cases for Le = 1 and Pe = 0 are also examined as special cases. The analysis reveals that for a non-positive Soret parameter, the flow is stable for all Lewis numbers and independent of the heat source. But with a positive Soret parameter in the absence of a heat source, the fluid flow is stable for Le = 3 while the influence of a heat source destabilizes the flow for Le > 2. In high and low shear flows with increasing solutal gradient, the solutal Rayleigh number shows a highly destabilizing nature for all Le. Moreover, smaller relaxation and higher retardation time are the most unstable characteristics of the heat source system. In convective longitudinal rolls, the uni-cellular streamline patterns tend to become bi-cellular by the influence of positive Soret parameters and energy sources.

Double-diffusive thermosolutal Hadley–Prats flow: stability analysis

ABSTRACT A mathematical model is formulated to study the onset of double-diffusion fluid flow through an infinite permeable channel with internal heat source and viscous dissipation effects under the influence of convection conditions. Numerically, the dimensionless emerging eigenvalue problem is tackled using the fourth-order Runge–Kutta scheme, specifically applied to longitudinal roll disturbances. Critical values of wave number and vertical thermal Rayleigh number are determined. A higher value of Gebhart number is observed to correlate significantly with the destabilizing phenomena in Hadley–Prats flow. Concentration-based internal heat generation also strongly modifies the critical thermal Rayleigh number. Also, it is found that the horizontal mass flow and viscous dissipation exert a substantial influence on the onset of instability in the flow regime. Linear instability analysis indicates that greater values of Lewis number in the porous medium stabilize the convection process for both values of mass diffusion parameter ( C z ) . Enhancing C z from negative to positive values diminishes the critical thermal Rayleigh ( R z ) value and consequently induces instability in the porous medium. Increased concentration-based internal heat generation generates the destabilization process in the flow region.

Linear and nonlinear stability analyses of micropolar fluid flow in horizontal porous layers

The linear and nonlinear stability analyses of micropolar fluid flow in a horizontal porous layer heated from below in the presence of throughflow is numerically investigated. The Brinkman model is considered to govern the micropolar fluid flow within the porous region. The main purpose of the present study is to investigate the behavior of the subcritical region for micropolar fluid parameters in the presence of throughflow. The energy approach is used to analyze nonlinear stability, whereas the normal mode scheme is used to investigate linear stability. The obtained eigenvalue problems related to linear and nonlinear stability analyses are solved numerically using the bvp4c routine in MATLAB. Finally, the critical thermal Rayleigh number is determined for the given values of the governing parameters. It is observed that the subcritical area decreases as the Darcy number (Da), micropolar parameter (m), and absolute value of throughflow parameter (|Pe|) decrease. Furthermore, there is no subcritical gap in the absence of the throughflow effect for micropolar fluid flow, which is a good agreement for the linear and nonlinear thresholds.

Nonlinear magnetoconvection of a Maxwell fluid in a porous layer with chemical reaction

ABSTRACT In this paper, the linear and nonlinear instability of magnetoconvection in a Darcy-Benard setup saturated by a Maxwell fluid with chemical reactions is studied. The governing non-dimension equations are solved using the normal modes, and we obtain the expressions for steady and oscillatory thermal Rayleigh numbers. The effects of different physical parameters such as the Damkohler number (), Hartmann number (), Solute Rayleigh number (), relaxation parameter (), Magnetic Prandtl number (), Lewis number () on stationary and oscillatory critical thermal Rayleigh numbers are presented and described. Enhancing the values of the Solute Rayleigh number and Lewis number makes the system unstable. Also, the Hartmann number and Damkohler number have a contrasting effect on stationary and oscillatory convection. Enhancing the value of the relaxation parameter makes the system more stable. In order to study heat transport by convection, the well-known equation, the Landau-Ginzburg equation, is derived.

Onset of triple diffusive thermosolutal convection in a composite system

AbstractIn a composite system possessing rigid‐rigid boundaries, the significance of thermal diffusion on the onset of triple diffusive convection is analyzed. The Darcy–Brinkman–Rayleigh–Benard model is employed to model the porous media. The regular perturbation methodology has been used to solve the governing equations of the composite system with the Boussinesq approximation. The critical thermal Rayleigh number, which determines the stability of the system, is estimated analytically. A graphical assessment of the impact of multiple physical attributes on the stability of the system is made. It is observed that the Soret parameters and solute Rayleigh numbers have a stabilizing effect, whereas the Darcy number exhibits a destabilizing effect upon the onset of triple diffusive convection in the composite system.

EFFECT OF VISCOUS DISSIPATION AND INTERNAL HEAT SOURCE ON MONO-DIFFUSIVE THERMOCONVECTIVE STABILITY IN A HORIZONTAL POROUS MEDIUM LAYER

A mathematical model is developed for studying the onset of mono-diffusive convective fluid flow in a horizontal porous layer with temperature gradient, internal heat generation, and viscous dissipation effects. Darcy's model is used for the porous medium, which is considered to be isotropic and homogenous. A linear instability analysis is conducted, and transverse or longitudinal roll disturbances are examined. The dimensionless emerging eigenvalue problem is solved numerically with the Runge-Kutta and shooting methods for both cases of disturbances, i.e,. longitudinal and transverse rolls. Critical wave number and critical vertical thermal Rayleigh number <i>R<sub>z</sub></i> are identified. For higher values of Gebhart number, Ge, a significant destabilizing effect of Hadley-Prats flow is computed. Internal heat generation also strongly modifies the critical vertical Rayleigh number. Extensive interpretation of the solutions related to the onset of convection is provided. The study is relevant to geophysical flows and materials processing systems.

INSTABILITY INVESTIGATION OF THERMO-BIOCONVECTION OF OXYTACTIC MICROORGANISM IN JEFFREY NANOLIQUID WITH EFFECTS OF INTERNAL HEAT SOURCE

The intent of this article is to investigate the influence of internal heat source on the stability of a suspension containing oxytactic microorganisms in a shallow horizontal porous fluid layer saturated by Jeffrey nanoliquid. The Jeffrey-Buongiorno model governs the nanofluid bioconvection flow. Normal mode analysis is utilized, and the principle of exchange of stability is invoked due to the absence of opposing agencies. The stability criteria is defined in terms of critical thermal Rayleigh number as a function of various flow governing parameters by using the weighted residual Galerkin method. It is perceived that the increment of uniform heat supply and the presence of oxytactic microorganism cells as well as the nanoparticles enhances heat transfer and constitutes an unstable system that hastens bioconvection. The nanoparticle Lewis number is found to have a dual impact on the system stability that relies on the nanoparticle Rayleigh number and exhibits destabilizing nature for top-heavy nanoparticle concentration. It is also observed that the Jeffrey parameter produces nonoscillatory instability in the system.

Analysis of Magneto-Thermo-Bioconvection of Nanofluid Containing Gyrotactic Microorganisms Through Porous Media

The present work aims to study the combined effect of external magnetic field and heating from below on the bioconvection of nanofluid containing gyrotactic microorganisms. Modified Maxwell’s equations and generalized Buongiorno’s model are used to frame the flow constitutive equations of nanofluid saturated in the Darcy-Brinkman porous medium. The present investigation has been done in the framework of linear stability. The resulting eigenvalue problem along with boundary conditions has been solved using Chebyshev-Gauss-Lobato Spectral Method. Appropriate transformations are used to convert Neumann boundary conditions into Dirichlet’s one. The influence of various pertinent parameters on the critical thermal Rayleigh number has been shown through graphs and tables. It has been observed that magnetic field stablizes the system whereas fast moving microorganisms hasten the onset of convection.

MULTI-DIFFUSIVE CONVECTION IN A ROTATING POROUS LAYER UNDER THE EFFECTS OF SUSPENDED PARTICLES AND GRAVITY FIELD: A BRINKMAN MODEL

The onset of multi-diffusive convection problem is analysed theoretically to include the effects of suspended particles and rotation through a porous medium. In the present paper, Brinkman model is considered for the porous medium. The variations in fluid density are due to the variation in stratifying components having different thermal and solute diffusivities. Linear stability analysis procedure along with normal mode method is employed to obtain a dispersion relation in terms of thermal and solute Rayleigh number. Further, the case of stationary convection (when the growth rate vanishes) is also discussed and a dispersion relationship between thermal and solute Rayleigh numbers is obtained to study the effect of various embedded parameters. The critical thermal and solute Rayleigh numbers can be obtained with the help of critical dimensionless wave number for varying values of physical parameters.

The onset of Darcy–Brinkman–Rayleigh–Benard convection in a composite system with thermal diffusion

AbstractThe effect of thermal diffusion on the onset of double‐diffusive convection is analyzed by considering a composite system of the fluid‐saturated porous layer. The porous layer of the composite system is modeled employing the Darcy–Brinkman–Rayleigh–Benard model. The boundaries of the composite system are assumed to be rigid‐rigid under normal gravitational force. The governing equations satisfying the Darcy–Brinkman–Rayleigh–Benard model are solved using the regular perturbation method. A mathematical expression is derived for the critical thermal Rayleigh number by solving the coupled equation. The impact of various physical parameters on the onset of convection is graphically represented under the influence of the Soret effect and the stability of the system is analyzed.

Effect of TD viscosity and Coriolis force on thermohaline convection of ferrofluid in Darcy porous matrix: A Galerkin numerical technique

The effect of TD viscosity (Temperature Dependent viscosity) and Coriolis force in horizontal ferromagnetic fluid layer with Darcy porous matrix is investigated theoretically. This fluid layer is heated from below and it is salted from above. The non-linear behavior is omitted by use of a linear stability theory. The normal mode technique is taken to account to investigate the convective system with free-free boundaries. To investigate the entire system the exact solution is obtained. The stationary instability is obtained. But, oscillatory instability can not occur in the study. The critical thermal magnetic Rayleigh number N S C is calculated for sufficient large values of M 1. In the analysis, we have attempted to analyze T a , V and D a on the system in many situations.

Implication of cross-diffusion on the stability of double diffusive convection in an imposed magnetic field

The effects of cross-diffusion on linear and weak nonlinear stability of double diffusive convection in an electrically conducting horizontal fluid layer with an imposed vertical magnetic field are investigated. The criterion for the onset of stationary and oscillatory convection is obtained analytically by performing the linear instability analysis. Several noteworthy departures from those of doubly diffusive fluid systems are unveiled under certain parametric conditions. It is shown that (i) disconnected closed convex oscillatory neutral curve separated from the stationary neutral curve exists requiring three critical thermal Rayleigh numbers to completely specify the linear instability criteria instead of a usual single critical value, (ii) an electrically conducting fluid layer in the presence of magnetic field can be destabilized by stable solute concentration gradient, and (iii) a doubly diffusive conducting fluid layer can be destabilized in the presence of magnetic field. It is demonstrated that small variations in the off-diagonal elements enforce discrepancies in the instability criteria. A weak nonlinear stationary stability analysis has been performed using a perturbation method and a cubic Landau equation is derived and the stability of bifurcating equilibrium solution is discussed. It is found that subcritical bifurcation occurs depending on the choices of governing parameters.

Convective heat and mass transports and chaos in two-component systems: comparison of results of physically realistic boundary conditions with those of artificial ones

Linear and weakly nonlinear stability analyses of double-diffusive convection in two-component liquids with either potassium chloride (KCl) or sodium chloride (NaCl) aqueous solution, and heat being present is investigated in the paper for free, and rigid, isothermal, iso-solutal boundaries. Using the thermophysical values of the aqueous solutions, we have shown that the stationary convection is the preferred mode at onset and that sub-critical motion is possible. We found that the critical thermal Rayleigh number for water + NaCl + heat is higher compared to that of water + KCl+ heat. The study shows that for water + KCl + heat, the transition from convective motion to chaotic motion occurs at $$r_{\mathrm{H}}=27.2$$ for free boundaries and at 48.5 for rigid boundaries. Here, $$r_{\mathrm{H}}$$ denotes the Hopf thermal Rayleigh number. Further, the existence of windows of mildly chaotic points and fully periodic intervals are reported using Lyapunov exponents and bifurcation diagrams. Chaotic motions in both the aqueous solutions are nearly identical. The percentage increase in heat transport in the double-diffusive system involving NaCl is nearly 1% more than that of KCl in the case of free boundaries, whereas in the case of realistic boundaries it is nearly 1.6%. The comparison of the Nusselt and the Sherwood numbers between water + KCl and water + NaCl leads us to the conclusion that the aqueous solution with lower Lewis number transports maximum heat in the case of free boundaries and opposite is seen in the case of rigid boundaries due to the boundary effect. The many qualitative similarities between the results of artificial and realistic boundaries are highlighted.

Double‐diffusive convection in a rotating viscoelastic fluid layer

AbstractThe linear instability and a weakly nonlinear stability of a rotating double‐diffusive convection in a viscoelastic fluid layer are investigated. An Oldroyd‐B constitutive equation is used to describe the rheological behavior of the viscoelastic fluid. Several remarkable departures from those of single‐diffusive and double‐diffusive viscoelastic fluid systems are explored by performing the linear instability analysis. Under certain conditions, it is observed that (i) a non‐rotating double diffusive viscoelastic fluid layer becomes destabilized by rotation, (ii) a rotating viscoelastic fluid layer gets destabilized by the addition of heavy solute to the bottom of the layer, (iii) disconnected closed convex oscillatory neutral curve from the stationary neutral curve exists, and (iv) three critical thermal Rayleigh numbers are required to specify the linear instability criteria, as opposed to a single critical value. The nature of linear instability characteristics for Oldroyd‐B, Maxwell and Newtonian fluids is found to be contradictory for the identical values of governing parameters. By performing a weakly nonlinear stability analysis, a cubic Landau equation for the amplitude corresponding to stationary convection is derived and the stability of bifurcating equilibrium solution is discussed. The viscoelastic parameters influence the stability of stationary bifurcation despite their effect is not felt on the stationary onset. The influence of various parameters on heat and mass transfer is also elucidated.

An analytic study of the development of bioconvection in a suspension of randomly swimming gyrotactic microorganisms and nanoparticles

By formulating a mathematical model for randomly swimming gyrotactic microorganisms, the development of bioconvection in a suspension containing nanoparticles is studied. The modelled system of nonlinear partial differential equations are solved by using linear stability analysis, normal mode analysis and Galerkin weighted residual method to obtain secular equation containing parameters characterising the onset of bioconvection. The effect of thermophoresis and Brownian motion in the suspension of random swimming gyrotactic microorganisms has been analysed. The numerical results showing dependence of critical thermal Rayleigh number on bioconvection Rayleigh number, nanoparticle Rayleigh number, and nanoparticle Lewis number are also presented graphically. The analysis reveals that thermophoresis, nanoparticle mass diffusivity and volumetric fraction enhance the development of bioconvection. We also found that the impact of the thermal Rayleigh number on the bioconvection Rayleigh number is more significant than the nanoparticle Rayleigh number.

The Effect of Magnetic Field on the Stability of Double-Diffusive Convection in a Porous Layer with Horizontal Mass Throughflow

The onset of double-diffusive convection in an electrically conducting fluid-saturated porous layer is studied. The convective flow in the porous medium is induced by horizontal temperature and concentration gradients with net horizontal mass throughflow. The effect of magnetic field on the instability of convection is taken into consideration. The stability of the steady-state solution is investigated in two different approaches, namely the linear instability analysis and nonlinear stability analysis. The nonlinear stability analysis is performed by constructing the energy functional. The eigenvalue problems which are derived from the stability analyses are numerically integrated using the shooting and Runge–Kutta methods. The variation in the critical thermal Rayleigh number against each flow governing parameter is shown graphically. It is observed that Hartman number $$Ha^{2}$$ delays the onset of convection to commence and helps to reduce the region of subcritical instabilities. When the solute is concentrated at lower boundary of the porous layer, the onset of convection is in the form of stationary modes, but it switches to oscillatory mode of convection when the solute is concentrated at upper boundary. Interestingly, Hartman number $$Ha^{2}$$ plays an important role in delaying this transition from stationary mode to oscillatory mode of convection.

Ferroconvection in an Anisotropic Porous Medium with Variable Gravity

A linear stability assessment was performed to study the impact of internal heating and variable gravity in an anisotropic porous medium of a ferrofluid layer system on the onset of Benard-Marangoni convection. The system is heated from below with both the lower and upper limits are considered as completely insulated to the disturbance of the temperature. The eigenvalue problem is solved by using regular perturbation technique to obtain the critical Marangoni number and also the critical thermal Rayleigh number. It is noted that the increase of value anisotropic permeability, Darcy number and also magnetic number will enhance the convection of the system while the increasing values of anisotropic thermal diffusivity will help to stabilize the system.

Thermal radiation and surface roughness effects on the thermo-magneto-hydrodynamic stability of alumina–copper oxide hybrid nanofluids utilizing the generalized Buongiorno’s nanofluid model

Sequel to the fact that hybrid nanofluidic systems (e.g. scalable micro-/nanofluidic device) exhibit greater thermal resistance with increasing nanoparticle concentration, little is known on the significance of thermal radiation, surface roughness and linear stability of water conveying alumina and copper oxide nanoparticles. This study presents the effects of thermal radiation and surface roughness on the complex dynamics of water conveying alumina and copper oxide nanoparticles, in the case where the thermophysical properties of the resulting mixture vary meaningfully with the volume fraction of solid nanomaterials, as well as with the Brownian motion and thermophoresis microscopic phenomena. Based on the linear stability theory and normal mode analysis method, the basic partial differential equations governing the transport phenomenon were non-dimensionalized to obtain the simplified stability equations. The optimum values of the critical thermal Rayleigh number depicting the onset of thermo-magneto-hydrodynamic instabilities were obtained using the power series method and the Chock–Schechter numerical integration. The increase in the strength of Lorentz forces, thermal radiation and surface roughness has a stronger stabilizing impact on the appearance of convection cells. On the contrary, the stability diminishes with the increasing values of the volumetric fraction and diameter of nanomaterials. The partial substitution of the alumina nanoparticles by the copper oxide nanomaterials in the mixture stabilizes importantly the hybrid nanofluidic medium.

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.