Stability Analysis Of Double-diffusive Convection Research Articles

The Nonlinear Stability Analysis of Double-Diffusive Convection with Viscous Dissipation Effect

The Nonlinear Stability Analysis of Double-Diffusive Convection with Viscous Dissipation Effect

INFLUENCE OF MAGNETIC FIELD ON THE STABILITY OF DOUBLE DIFFUSIVE NANOFLUID CONVECTION IN A VERTICAL HOMOGENEOUS POROUS CHANNEL

The stability analysis of double-diffusive convection of nanofluid flow in a vertical porous channel is investigated in the presence of the transverse magnetic effect. A uniform magnetic field perpendicular to the flow direction is applied. Nanofluid is modeled using a model that combines Brownian motion and thermophoresis. The boundaries of the channel are assumed to be nonpermeable, isothermal, and isosolutal. A normal mode analysis is used to obtain the eigenvalue problem for the perturbed state. The Chebyshev spectral collocation technique is used to solve the resulting eigenvalue problem. The critical Rayleigh number and corresponding critical wavenumber are presented graphically for the effect of various parameters at least stable mode on the planes (Ha, Rac) and (Ha, αc), respectively. It is observed that transverse magnetic parameters tend to have a major effect on convective instability.

Linear and Weak Nonlinear Double Diffusive Convection in a Viscoelastic Fluid Saturated Anisotropic Porous Medium with Internal Heat Source

This paper deals linear and weak nonlinear stability analysis of double-diffusive convection in an anisotropic porous layer with internal heat source saturated by viscoelastic fluid. For linear stability analysis we use normal mode technique and obtained the expression for oscillatory thermal Rayleigh number which is used to plot neutral stability curve for oscillatory case. For nonlinear analysis truncated representation of Fourier series upto two terms is used. The system of time dependent nonlinear equation is solved numerically and plot the curve for heat transfer and mass transfer with respect to time for different parameters. Effect of thermal anisotropy parameter, mechanical anisotropy parameter, relaxation parameter, retardation parameter, internal heat source parameter, solute Rayleigh number, diffusivity ratio, Darcy-Prandtl number on the onset of convection, heat and mass transfers have been discussed. We also draw the stream lines, isotherms, isohalines at different times.

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.

Stability analysis of double-diffusive convection of Rivlin-Ericksen elastico-viscous nanofluid saturating a porous medium: a revised model

In the present paper we study the stability analysis of double-diffusive convection in a horizontal layer of an elastico-viscous nanofluid in a porous medium. The Rivlin-Ericksen fluid model is employed to describe the rheological behavior of the nanofluid. The dispersion relation describing the effect of various parameters is obtained by applying normal modes method and by using linear stability analysis. In this model we assume that the nanoparticle concentration flux is zero on the boundaries and we can give the values of temperature and concentration at the boundaries. The assumed boundary conditions neutralize the possibility of oscillatory convection and only stationary convection occurs. For stationary convection; it is observed that the Rivlin-Ericksen elastico-viscous nanofluid fluid behaves like an ordinary Newtonian nanofluid. The solutal-Rayleigh Number, thermo-nanofluid Lewis number, thermo-solutal Lewis number, Soret parameter and Dufour parameter have stabilizing effect on the stationary convection. A very good agreement is found between the present paper and earlier published results.

Stability analysis of soret-driven double-diffusive convection of Maxwell fluid in a porous medium

Stability analysis of double-diffusive convection for viscoelastic fluid with Soret effect in a porous medium is investigated using a modified-Maxwell–Darcy model. We use the linear stability analysis to investigate how the Soret parameter and the relaxation time of viscoelastic fluid effect the onset of convection and the selection of an unstable wavenumber. It is found that the Soret effect is to destabilize the system for oscillatory convection. The relaxation time also enhances the instability of the system. The effects of Soret coefficient and relaxation time on the heat transfer rate in a porous medium are studied using the nonlinear stability analysis, the variation of the Nusselt number with respect to the Rayleigh number is derived for stationary and oscillatory convection modes. Some previous results can be reduced as the special cases of the present paper.

Linear Stability Analysis of Double-Diffusive Convection in Porous Media, with Application to Geological Storage of CO2

Onset of double-diffusive buoyancy-driven flow resulted from vertical temperature and concentration gradients in a horizontal layer of a saturated and homogenous porous medium is investigated using amplification factor theory. After injection of CO2 into a deep saline aquifer, the density of the brine saturated with CO2 increases slightly. This increase in density induces natural convection. The effect of geothermal gradient is also considered in this work as a second incentive for convection and the double-diffusion convection was studied. Linear stability analysis is used to predict the inception of instabilities and initial wavelength of the convective instabilities. The analysis presented is applied to acid gas injection (as an analogue for CO2 storage) into saline aquifers in the Alberta basin. It is found that the geothermal gradient does not have significant effect on the onset of convection for these aquifers. It is shown that the geothermal effects on the onset of natural convection are negligible as compared to the solutal effects induced by dissolution and diffusion of CO2 in deep saline aquifers. Therefore, the linear stability analysis and the long-term numerical simulation of CO2 sequestration into such saline aquifers may be assumed to be isothermal in terms of natural convection occurrence.

Stability analysis of double-diffusive convection of Maxwell fluid in a porous medium heated from below

Linear stability analysis of Maxwell fluid in the Bénard problem for a double-diffusive mixture in a porous medium is studied based on the Darcy–Maxwell model. The critical Rayleigh number and the corresponding wave number for the exchange of stability are obtained. On the other hand, the effect of the relaxation time of Maxwell fluid on the critical Rayleigh number is discussed. In limiting cases, some results published previously are recovered from our results.

Stability analysis of double-diffusive convection in superposed fluid and porous layers using a one-equation model

Linear stability analysis has been carried out to predict the onset of double-diffusive convection in superposed fluid and porous layers using a one-equation model. The eigenvalue problem is solved numerically by a finite difference scheme. Results have been obtained for the thermal convection and salt-finger cases. Comparing with the results obtained for the same problems by Chen and Chen [F. Chen, C.F. Chen, J. Heat Transfer 110 (1988) 403–409] using a two-equation model, we find that these two methods give the same general characteristics of the marginal stability curves, however, there are differences in the critical conditions and the flow streamlines at onset. Carefully conducted experiments are needed to determine which model gives the more realistic results.

On numerical stability analysis of double-diffusive convection in confined enclosures

The onset of thermosolutal convection and finite-amplitude flows, due to vertical gradients of heat and solute, in a horizontal rectangular enclosure are investigated analytically and numerically. Dirichlet or Neumann boundary conditions for temperature and solute concentration are applied to the two horizontal walls of the enclosure, while the two vertical ones are assumed impermeable and insulated. The cases of stress-free and non-slip horizontal boundaries are considered. The governing equations are solved numerically using a finite element method. To study the linear stability of the quiescent state and of the fully developed flows, a reliable numerical technique is implemented on the basis of Galerkin and finite element methods. The thresholds for finite-amplitude, oscillatory and monotonic convection instabilities are determined explicitly in terms of the governing parameters. In the diffusive mode (solute is stabilizing) it is demonstrated that overstability and subcritical convection may set in at a Rayleigh number well below the threshold of monotonic instability, when the thermal to solutal diffusivity ratio is greater than unity. In an infinite layer with rigid boundaries, the wavelength at the onset of overstability was found to be a function of the governing parameters. Analytical solutions, for finite-amplitude convection, are derived on the basis of a weak nonlinear perturbation theory for general cases and on the basis of the parallel flow approximation for a shallow enclosure subject to Neumann boundary conditions. The stability of the parallel flow solution is studied and the threshold for Hopf bifurcation is determined. For a relatively large aspect ratio enclosure, the numerical solution indicates horizontally travelling waves developing near the threshold of the oscillatory convection. Multiple confined steady and unsteady states are found to coexist. Finally, note that all the numerical solutions presented in this paper were found to be stable.

Eigenvalues of Block Matrices Arising from Problems in Fluid Mechanics

Block matrices with a special structure arise from mixed finite element discretizations of incompressible flow problems. This paper is concerned with an analysis of the eigenvalue problem for such matrices and the derivation of two shifted eigenvalue problems that are more suited to numerical solution by iterative algorithms like simultaneous iteration and Arnoldi's method. The application of the shifted eigenvalue problems to the determination of the eigenvalue of smallest real part is discussed and a numerical example arising from a stability analysis of double-diffusive convection is described.

Double-diffusive convection caused by coupled molecular diffusion

Double-diffusive convection is studied for the case where a large coupled diffusion (or cross-diffusion) effect is present. The Soret effect is a familiar example of this cross-diffusion where the flux of the solute depends not only on its own spatial gradient but also on the in situ temperature gradient. The linear stability analysis of double-diffusive convection has been extended to include the two cross-diffusion flux terms and it has been shown that, with a sufficiently large coupled diffusion effect, fingers can form even when the concentrations of both components make the fluid's density gradient statically stable. The conditions under which the diffusive instability can occur are compared with those for the formation of fingers and it is shown that these two types of double-diffusive convection cannot occur together in any particular set of linear property gradients. We then consider finite-amplitude, steady, infinitely long fingers and show that a sufficiently large cross-diffusion effect can again allow fingers to exist when the concentrations of both solutes increase with depth. It is also shown that the diffusion of properties from an initially sharp interface may set up vertical gradients that are favourable for the formation of fingers.