Stability Of Double Diffusive Convection Research Articles

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.

Conditional and unconditional stability for double diffusive convection when the viscosity has a maximum

We here analyse two models of double-diffusive convection in fluid layer when viscosity depends on temperature quadratically. However, to a linearized instability analysis, conditional and global (unconditional) nonlinear stability theories are applied. For the first model, we establish an unconditional nonlinear energy stability. Moreover, in the second model the standard energy method does not yield unconditional stability so a conditional energy analysis is employed to achieve nonlinear results. In addition, the nonlinear stability bounds is found to be independent of the salt field and a presentation of the region of possible subcritical instabilities is given.

Stability of Double-Diffusive Convection in a Porous Medium with Temperature-Dependent Viscosity: Brinkman–Forchheimer Model

In this article, the problem of double-diffusive convection in a porous layer when the viscosity depends on the temperature and using Brinkman–Forchheimer model has been introduced by using the linear and nonlinear energy theories. For linear theory, the critical Rayleigh numbers have been derived and then numerically calculated. However, for nonlinear theory, the critical threshold was derived in three different ways and the results were compared with the results of the linear analysis.

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.

Global Exponential Nonlinear Stability for Double Diffusive Convection in Porous Medium

Nonlinear stability of the motionless double-diffusive solution of the problem of an infinite horizontal fluid layer saturated porous medium is studied. The layer is heated and salted from below. By introducing two balance fields and through defining new energy functionals it is proved that for CLe ≥ R, Le ≤ 1 the motionless double-diffusive solution is always stable and for CLe < R, Le < 1 the solution is globally exponentially and nonlinearly stable whenever R < 4π2+LeC, where Le, C and R are the Lewis number, Rayleigh number for solute and heat, respectively. Moreover, the nonlinear stability proved here is global and exponential, and the stabilizing effect of the concentration is also proved.

Study of heat and mass transport in Couple-Stress liquid under G-jitter effect

Abstract In the present paper, we deal with the effect of G-jitter (time periodic gravity modulation) on the stability of double diffusive convection in couple stress liquid by method of non-linear analysis. The infinitesimal disturbances are expanded in terms of power series of amplitude of modulation. Couple stress term has been employed in momentum equation. For the stationary convection, the Ginzburg–Landau equation has been used to investigate the effect of modulation on heat and mass transfer. The effect of Prandtl number, couple stress parameter, Lewis number and solute Rayleigh number has also been investigated.

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.

Soret effect on the double diffusive convection instability due to viscous dissipation in a horizontal porous channel

The effect of the Soret parameter on the convective stability of double diffusive convection solely because of the viscous dissipation in a horizontal porous channel is studied. The lower boundary is adiabatic whereas the upper boundary is considered to be isothermal. The convective stability of the present system is governed by the solutal Rayleigh number (R) and is influenced by the viscous dissipation parameter (ξ), Lewis number (Le) and Soret parameter (Sr). For non positive values of the Soret parameter, the longitudinal rolls happen to be the most unstable ones when ξ takes small values. With positive values of the Soret parameter, the transverse rolls are seen to be the most unstable for relatively smaller values of the viscous dissipation parameter. As the viscous dissipation effect becomes stronger, the longitudinal rolls become the most unstable ones even for positive Soret parameter and the transverse rolls become more unstable for non-positive Soret parameter. It is observed that the Soret parameter has significant effect on convective instability and this is discussed. It has also been noticed that viscous dissipation shows a dual effect in presence of the Soret effect. For fixed values of the viscous dissipation parameter and Lewis number, negative values for the Soret number advances the onset of convection. Though positive values of the Soret parameter stabilizes the flow with smaller values of the Lewis number, but it destabilizes the flow when Lewis number and viscous dissipation parameter take larger values.

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.

Soret Dufour Driven Thermosolutal Instability of Darcy-Maxwell Fluid

Linear stability of double diffusive convection of Darcy-Maxwell fluid with Soret and Dufour effects is investigated. The effects of the Soret and Dufour numbers, Lewis number, relaxation time and solutal Darcy Rayleigh number on the stationary and oscillatory convection are presented graphically. The Dufour number enhances the stability of Darcy-Maxwell fluid for stationary convection while it has a stabilizing character for overstability. The Soret number is to destabilize the system in case of both stationary and oscillatory modes. In the limiting case some previous results have been recovered.

Stability of double-diffusive convection induced by selective absorption of radiation in a fluid layer

Linear and nonlinear stability analyses were performed on a fluid layer with a concentration-based internal heat source. Clear bimodal behaviour in the neutral curve (with stationary and oscillatory modes) is observed in the region of the onset of oscillatory convection, which is a previously unobserved phenomenon in radiation-induced convection. The numerical results for the linear instability analysis suggest a critical value γc of γ, a measure for the strength of the internal heat source, for which oscillatory convection is inhibited when γ > γc. Linear instability analyses on the effect of varying the ratio of the salt concentrations at the upper and lower boundaries conclude that the ratio has a significant effect on the stability boundary. A nonlinear analysis using an energy approach confirms that the linear theory describes the stability boundary most accurately when γ is such that the linear theory predicts the onset of mostly stationary convection. Nevertheless, the agreement between the linear and nonlinear stability thresholds deteriorates for larger values of the solute Rayleigh number for any value of γ.

Linear and nonlinear stability of double diffusive convection in a couple stress fluid–saturated porous layer

The linear and nonlinear stability of double diffusive convection in a layer of couple stress fluid–saturated porous medium is theoretically investigated in this work. Applying the linear stability theory, the criterion for the onset of steady and oscillatory convection is obtained. Emphasizing the presence of couple stresses, it is shown that their effect is to delay the onset of convection and oscillatory convection always occurs at a lower value of the Rayleigh number at which steady convection sets in. The nonlinear stability analysis is carried out by constructing a system of nonlinear autonomous ordinary differential equations using a truncated representation of Fourier series method and also employing modified perturbation theory with the help of self-adjoint operator technique. The results obtained from these two methods are found to complement each other. Besides, heat and mass transport are calculated in terms of Nusselt numbers. In addition, the transient behavior of Nusselt numbers is analyzed by solving the nonlinear system of ordinary differential equations numerically using the Runge–Kutta–Gill method. Streamlines, isotherms, and isohalines are also displayed.

A new approach to nonlinear [formula omitted]-stability of double diffusive convection in porous media: Necessary and sufficient conditions for global stability via a linearization principle

A new approach to nonlinear L 2 -stability for double diffusive convection in porous media is given. An auxiliary system Σ of PDEs and two functionals V, W are introduced. Denoting by L and N the linear and nonlinear operators involved in Σ, it is shown that Σ-solutions are linearly linked to the dynamic perturbations, and that V and W depend directly on L-eigenvalues, while (along Σ) d V d t and d W d t not only depend directly on L-eigenvalues but also are independent of N. The nonlinear L 2 -stability (instability) of the rest state is reduced to the stability (instability) of the zero solution of a linear system of ODEs. Necessary and sufficient conditions for general, global L 2 -stability (i.e. absence of regions of subcritical instabilities for any Rayleigh number) are obtained, and these are extended to cover the presence of a uniform rotation about the vertical axis.

Structural stability for the Brinkman equations of flow in double diffusive convection

This paper continues the investigation of structural stability for the Brinkman equations modeling the double diffusive convection for flow in a porous medium. It supplements earlier results of Straughan and Hutter [B. Straughan, K. Hutter, A priori bounds and structural stability for double diffusive convection incorporating the Soret effect, Proc. R. Soc. Lond. Ser. A 455 (1999) 767–777].

On the stability of double diffusive convection in a porous layer with throughflow

The effect of throughflow on the stability of double diffusive convection in a porous layer is investigated for different types of hydrodynamic boundary conditions. The lower and upper boundaries are assumed to be insulating to temperature and concentration perturbations. The resulting eigenvalue problem is solved by the Galerkin technique. The curvature of the basic temperature as well as solute concentration gradients significantly affects the stability of the system. It is observed that, for a suitable choice of parametric values, Hopf bifurcation occurs always prior to direct bifurcation, and the throughflow alters the nature of bifurcation. In contrast to the single component system, it is found that throughflow is (a) destabilizing even if the lower and upper boundaries are of the same type, and (b) stabilizing as well as destabilizing, irrespective of its direction, when the boundaries are of different types.

Non-darcian effects on double diffusive convection in a sparsely packed porous medium

The linear and non-linear stability of double diffusive convection in a sparsely packed porous layer is studied using the Brinkman model. In the case of linear theory conditions for both simple and Hopf bifurcations are obtained. It is found that Hopf bifurcation always occurs at a lower value of the Rayleigh number than one obtained for simple bifurcation and noted that an increase in the value of viscosity ratio is to delay the onset of convection. Non-linear theory is studied in terms of a simplified model, which is exact to second order in the amplitude of the motion, and also using modified perturbation theory with the help of self-adjoint operator technique. It is observed that steady solutions may be either subcritical or supercritical depending on the choice of physical parameters. Nusselt numbers are calculated for various values of physical parameters and representative streamlines, isotherms and isohalines are presented.

A weak nonlinear stability analysis of double diffusive convection with cross-diffusion in a fluid-saturated porous medium

The effect of “Cross Diffusion” on the linear and nonlinear stability of double diffusive convection in a fluid-saturated porous medium has been studied analytically. In the case of linear theory, the normal mode technique has been used and the condition for the maintenance of “finger” and “diffusive” instabilities have been obtained. It has been found that fingers can form by taking cross diffusion terms of appropriate sign and magnitude even though both components make stabilizing contributions to the net vertical density gradient. It has also been shown that “finger” and “diffusive” instabilities can never occur simultaneously. The nonlinear theory is based on the truncated representation of Fourier series and it has been found that the finite amplitude convection may occur when both initial property gradients are stabilizing. Further, the region of finite amplitude instability always encloses the region of infinitesimal oscillatory instability. The effects of permeability and cross-diffusion terms on the heat and mass transports have also been clearly brought out.

Numerical Study of the Onset of Double-Diffusive Cellular Convection Due to a Uniform Lateral Heat Flux

When a fluid with a vertical solute gradient of (−dS/dy)0 is heated laterally, roll cells start to form at the boundary, developing into a series of convective layers. Numerical experiments were performed to investigate the onset of the abovementioned double-diffusive convection under the application of a uniform lateral heat flux. The paper reports the results and discussion of the following aspects of the stability of double-diffusive convection; (i) the relationship between the critical value, (Ra/Rs)c, above which convection cells form along the vertical wall and the nondimensional slot width, (d/L), (ii) the effect of the Lewis number on (Ra/Rs)c. It was also confirmed that values of (Ra/Rs)c as well as H/L (the nondimensional vertical size of incipient cells) obtained in this numerical experiment for wide slot widths (d/L&gt;∼30), agreed well with those obtained previously by physical experiments.