Onset Of Thermosolutal Convection Research Articles

Magnetic and chemical reaction effects on double‐diffusive convection with the effect of heat and salt flux boundary conditions

AbstractThe magnetic and chemical reaction effects on the onset of thermosolutal convection in a penetrative internally heated fluid are investigated for the case when the plane layer has a constant temperature and salt concentration on the upper surface and a specified heat flux and salt flux on the bottom surface. The linear instability and nonlinear stability boundaries are analyzed using the Chebyshev collocation method to calculate the corresponding eigenvalue systems according to the boundary conditions. Two distinct internal heat source functions are examined: the first kind is a constant, and the second kind is to heat the bottom half of the surface while cooling the rest. The Rayleigh number is graphically represented with the various parameter effects. The outcomes indicate that utilizing magnetic, chemical reactions, and internal heat sources has a significant influence on defining the thermosolutal convection stabilization and destabilization thresholds. For all types of internal heat sources, it is found that the magnetic field has a stabilizing effect. It is also noted that the effects of chemical reaction, heat flux, and salt flux play important factors in the stability region.

Stability Analysis of Double Diffusive Convection in Local Thermal Non-equilibrium Porous Medium with Internal Heat Source and Reaction Effects

Abstract The internal heat source and reaction effects on the onset of thermosolutal convection in a local thermal non-equilibrium porous medium are examined, where the temperature of the fluid and the solid skeleton may differ. The linear instability and nonlinear stability theories of Darcy–Brinkman type with fixed boundary condition are carried out where the layer is heated and salted from below. The D 2 {D^{2}} Chebyshev tau technique is used to calculate the associated system of equations subject to the boundary conditions for both theories. Three different types of internal heat source function are considered, the first type increases across the layer, while the second decreases, and the third type heats and cools in a nonuniform way. The effect of different parameters on the Rayleigh number is depicted graphically. Moreover, the results detect that utilizing the internal heat source, reaction, and non-equilibrium have pronounced effects in determining the convection stability and instability thresholds.

Onset of thermosolutal convection in rotating horizontal nanofluid layers

Onset of thermosolutal convection in rotating horizontal nanofluid layers

Onset of Thermosolutal Convection in Couple-Stress Fluid in a Porous Media in the Presence of Magnetic Field

Onset of Thermosolutal Convection in Couple-Stress Fluid in a Porous Media in the Presence of Magnetic Field

Influence of Chemical Reaction on Stability of Thermo-Solutal Convection of Couple-Stress Fluid in a Horizontal Porous Layer

We consider the onset of thermo-solutal convection in a couple-stress fluid-saturated anisotropic porous medium, where the chemical equilibrium on the bounding surfaces and the solubility of the dissolved components depend on temperature. The entire study has been spilt into two parts: (i) linear stability analysis (ii) weakly non-linear stability analysis. Stationary case of linear stability analysis is discussed for two modes of bounding surfaces (a) realistic bounding surfaces i.e. Rigid-Rigid and Rigid-Free (R/R and R/F), (b) non-realistic bounding surfaces i.e. Free-Free (F/F). Howsoever, investigation of oscillatory state and weakly non-linear stability are restricted to F/F case. Galerkin method is used to solve the eigenvalue problem for R/R and R/F cases, whereas, exact solutions are obtained for F/F case.A comparative study among flow stability for above different cases is made as function of ratio of viscosities ( i.e., couple-stress viscosity to fluid viscosity which is defined as couple-stress parameter, \((C)\)) and effective chemical reaction (i.e. chemical reaction parameter, \((\chi )\)). It has been found that increasing viscosity of the couple-stress fluid, in terms of increasing \(C\), increases flow stability in all three cases, but among all cases its stabilization effect for R/R is maximum. However, in the absence of couple-stress parameter the maximum stability of flow is observed for F/F. Apart from this, the chemical reaction stabilizes the flow for all the three cases. Furthermore, stability analysis for F/F case indicates that couple-stress parameter stabilizes the system in all modes (stationary, oscillatory and finite amplitude) of convection.Damkohler number \((\chi )\) is found to delay the stationary convection, however, it speeds up the onset of oscillatory convection. The non-linear theory based on truncated representation of Fourier series method predicts the occurrence of sub-critical instability in the form of finite amplitude motion. The effect of \(C\) and \(\chi \) on heat and mass transfer is also examined.

Onset of Vibrational Convection in a Binary Fluid Saturated Non-Darcy Porous Layer Heated from Above

A linear stability analysis is used to investigate the influence of mechanical vibration on the onset of thermosolutal convection in a horizontal porous layer heated and salted from above. Vibrations are considered with arbitrary amplitude and frequency. The Brinkman extended Darcy model is used to describe the flow and the Oberbeck-Boussinesq approximation is employed. Continued fraction method and Floquet theory are used to determine the convective instability threshold. It is found that the solutal Rayleigh number has the stabilizing effect. The existence of a closed disconnected loop of synchronous mode is predicted in the marginal curve for moderate values of solutal Rayleigh number and vibration amplitude.

Onset of thermosolutal convection in a shallow porous layer heated and salted from below and subject to a horizontal heat flux balanced by a Soret mass flux

In the present paper, a thorough analytical and numerical investigation is carried out to examine the double-diffusive convective instability within a horizontal porous layer heated and salted from below. A situation is considered where a lateral perturbing heat flux applied to the system is balanced by the horizontally induced Soret mass flux. The parameters governing this problem are the thermal Rayleigh number, R T; the Lewis Number, Le; the buoyancy ratio, N; the Soret parameter, M; the ratio of the horizontal to vertical heat flux, a; and the aspect ratio, A r; of the porous layer. The present investigation is focused on the situation where MN = 1 , which describes an equilibrium state between the induced Soret mass flux and the imposed heat flux. For this situation, a rest state solution is possible. The analytical solution, derived on the basis of the parallel flow approximation, is validated numerically using a finite difference method by solving the full governing equations. In the M ∗– Le plane ( M ∗ = 1 / M ) , five distinct regions, describing different flow behaviors, are delineated and their location depends on the lateral heat flux parameter a. It is also demonstrated that supercritical and/or subcritical bifurcations are possible for specific ranges of M ∗ and Le. The effect of the lateral heating and the Soret parameter on the critical Rayleigh number, corresponding to the onset of parallel flow convection, is examined. The parameter a affects the flow and the heat transfer considerably, but its effect on the mass transfer is negligible.

The effect of temperature-dependent solubility on the onset of thermosolutal convection in a horizontal porous layer

We consider the onset of thermosolutal (double-diffusive) convection of a binary fluid in a horizontal porous layer subject to fixed temperatures and chemical equilibrium on the bounding surfaces, in the case when the solubility of the dissolved component depends on temperature. We use a linear stability analysis to investigate how the dissolution or precipitation of this component affects the onset of convection and the selection of an unstable wavenumber; we extend this analysis using a Galerkin method to predict the structure of the initial bifurcation and compare our analytical results with numerical integration of the full nonlinear equations. We find that the reactive term may be stabilizing or destabilizing, with subtle effects particularly when the thermal gradient is destabilizing but the solutal gradient is stabilizing. The preferred spatial wavelength of convective cells at onset may also be substantially increased or reduced, and strongly reactive systems tend to prefer direct to subcritical bifurcation. These results have implications for geothermal-reservoir management and ore prospecting.

A Linear Stability Study of the Gradient Zone of a Solar Pond

The linear stability of a plane layer with horizontal temperature and concentration stratification corresponding to gradient zone of a solar pond is investigated. The problem is described by Navier-Stokes equations with Boussinesq-Oberbeck approximation. Two source terms are introduced in the energy equations: the absorption of solar energy characterized by the extinction radiative coefficient μe and by the parameter f defined as the ratio of extracted heat flux to absorbed heat flux in the lower convective zone. The influence of the parameters μe and f on the onset of thermosolutal convection in the case of confined and infinite layers is analyzed. It is found that convection starts in an oscillatory state, independently of the RaS value. Different convection solutions were found for marginal stability and steady state.

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.

Naissance de régimes de convection thermosolutale dans une cellule rectangulaire poreuse

The onset of thermosolutal convection in a rectangular cell filled with a porous medium saturated by a binary fluid is studied. The vertical walls are maintained at different, but uniform, temperatures and concentrations. The horizontal walls are impermeable and adiabatic. When the ratio of resulting solutal and thermal buoyancy forces is equal to − 1, we obtain an equilibrium solution corresponding to a purely diffusive regime. We demonstrate that this regime is linearly stable up to a critical thermal Rayleigh number, Ra c1 , depending on the aspect ratio of the cell and the Lewis number. These analytical results are in good agreement with numerical direct simulations which allow us to describe the double-diffusive convective regimes taking place when the equilibrium regime loses its stability.

Gravitational modulation of thermosolutal convection during directional solidification

During directional solidification of a binary alloy at constant velocity, thermosolutal convection may occur due to the temperature and solute gradients associated with the solidification process. For vertical growth in an ideal furnace (lacking horizontal gradients) a quiescent state is possible. The effect of a time-periodic vertical gravitational acceleration (or equivalently vibration) on the onset of thermosolutal convection is calculated based on linear stability using Floquet theory. Numerical calculations for the onset of instability have been carried out for a semiconductor alloy with Schmidt number of 10 and Prandtl number of 0.1 with primary emphasis on large modulation frequencies in a microgravity environment for which the background gravitational acceleration is negligible. The numerical results demonstrate that there is a significant difference in stability depending on whether a heavier or lighter solute is rejected. For large modulation frequencies, the stability behavior can be described by either the method of averaging or an asymptotic resonant mode analysis.

The effect of gravity modulation on thermosolutal convection in an infinite layer of fluid

The effect of time-periodic vertical gravity modulation on the onset of thermosolutal convection in an infinite horizontal layer with stress-free boundaries is investigated using Floquet theory for the linear stability analysis. Situations for which the fluid layer is stably stratified in either the fingering or diffusive regimes of double-diffusive convection are considered. Results are presented both with and without steady background acceleration. Modulation may stabilize an unstable base solution or destabilize a stable base solution. In addition to synchronous and subharmonic response to the modulation frequency, instability in the double-diffusive system can occur via a complex conjugate mode. In the diffusive regime, where oscillatory onset occurs in the unmodulated system, regions of resonant instability occur and exhibit strong coupling with the unmodulated oscillatory frequency. The response to modulation of the fundamental instability of the unmodulated system is described both analytically and numerically; in the double-diffusive system this mode persists under subcritical conditions as a high-frequency lobe.

Onset of Thermosolutal Convection in a Liquid Layer Having Deformable Free Surface – I. Stationary Convection

Abstract The effect of surface deformation on the onset of thermosolutal convection in a horizontal thin layer heated from below and salted from above is examined, using linear stability theory. It is found that two critical Crispation numbers Cr1 and Cr2 exist. For Cr ≦ Cr1 the instability mechanism remains unefTected by the deformation of the free surface, whereas for Cr1 < Cr < Cr2 the critical Marangoni number (Mc) decreases, showing instability due to deformation. If Cr = Cr2, Mc is obtained for two values of the wave number. When Cr > Cr2 , Mc decreases as the wave number tends to zero. Further, the effect of the Marangoni number, Biot number, Bond number etc. on the stability characteristics of the problem is discussed.

Onset of Thermosolutal Convection in a Liquid Layer Having Deformable Free Surface – II. Overstability

Abstract The onset of thermosolutal convection driven by a temperature and concentration dependent surface tension is investigated for a thin layer of fluid having a deformable free surface. It is shown that there exist two Crispation numbers (Cr) for oscillatory modes of instability. It is further shown that Cr and the frequency of the oscillatory mode are strongly coupled for large values of Cr. It is found that Cr destabilizes the system for both cases, salted from below and salted from above