Onset Of Convective Instability Research Articles

Stability of natural convection in a vertical layer of Navier-Stokes-Voigt fluid

The stability of natural convection in a vertical layer of viscoelastic fluid confining between rigid-isothermal side walls held at different temperatures is investigated numerically using the Chebyshev collocation method. The viscoelastic behavior is modelled by means of a constitutive equation encompassing the Navier-Stokes-Voigt fluid or the Kelvin-Voigt fluid of order zero. The onset of convective instability is examined by linearizing the governing equations for the perturbations and an appropriate extension of Squire's theorem is given making a case to consider only two-dimensional perturbation stability equations. The numerical solution of the stability eigenvalue problem leads to the determination of the neutral stability condition. The dependence of the Kelvin-Voigt parameter on the critical stability parameters and also on the Prandtl number at which the point of transition from stationary to travelling-wave mode occurs is thoroughly analysed. The Kelvin-Voigt parameter shows an important role on the travelling-wave mode instability where it inducts both stabilizing and destabilizing effects on the base flow depending on the values of Prandtl number, while its impact on the stationary mode is found to be very weak. Furthermore, the streamlines and isotherms of the perturbation modes presented herein demonstrate the development of complex dynamics at the critical state. The results of a Newtonian fluid are obtained as a particular case.

Bi-disperse convection under the action of an internal heat source

The aim of this paper is the investigation on hydrodynamic stability of an incompressible fluid saturating a horizontal layer of bi-disperse porous medium, heated from below and internally, in order to analyse the effect of the internal heat source on the onset of convective instabilities. The principle of exchange of stabilities is proved, so convection can arise only via stationary motions. Via linear instability and nonlinear stability analyses of the conduction solution, the linear threshold and the nonlinear threshold in the L2−norm for the onset of convective motions are determined and compared.

Forchheimer–Bénard Instability of the Non-Newtonian Fluid

The present paper examines the effect of vertical throughflow on the onset of convective instability in a horizontal porous layer filled with a non-Newtonian power-law fluid (PL). The permeable boundary layers are exposed to two different uniform constant temperature conditions, The Oberbeck-Boussinesq hypothesis is considered with the Darcy-Forchheimer model. A fourth-order eigenvalue problem is stemmed from the performance of the linear stability analysis, and the critical values are obtained using the shooting method combined with the Runge-Kutta method. The non-Newtonian Darcy-Rayleigh number (R), the Péclet number (Pe), the Forchheimer number (G), and the power-law index (n) are the parameters whose value play a crucial role in the onset of instability. The finding shows more stabilizing effects arise in pseudoplastic fluid than dilatant one at Peclet number Pe << 1 where the inverse behaviour takes place at large Peclet number even with the existence of the drag number or without it.

When does the onset of convection in an inclined porous layer become subcritical?

We consider the onset of convective instability in an inclined porous layer heated from below. Linearised stability theory tells us that there always exists a band of wavenumbers within which small-amplitude disturbances will grow, but this is true only when the inclination of the layer is less than 31.49032∘. At higher inclinations such disturbances always decay. However, it is also widely known that nonlinear convection may be computed for larger inclinations. This paper provides an initial explanation for how these two facts may be reconciled. It is generally assumed that the onset of convection in an inclined layer is supercritical, and, while this is certainly true when the layer is horizontal, there is no reason to assume that it remains so for other inclinations. The present paper, then, is a combined weakly-nonlinear and numerical investigation of the effect of inclination on the manner of onset. The weakly nonlinear analysis shows that the transition from a supercritical onset to a subcritical one takes place when the inclination is 24.247627∘, and this is confirmed using a detailed and focussed set of nonlinear numerical simulations.

Effects of thermodiffusion on Darcy–Brinkman convection in a rotating binary nanofluid saturating a porous layer

In this study, the onset of convection in a rotating horizontal porous layer saturated by a binary nanofluid was theoretically investigated under the influence of thermodiffusion and nanoparticles. To study the onset of convection, the Darcy–Brinkman model was employed for the porous medium, and the thermal Rayleigh number was derived analytically using linear stability analysis. Water–ammonia-based silver, copper, and alumina, three binary nanofluids, were considered, and the effect of nanoparticles on the stability of the system was analysed using addition factor analysis. The Brinkman model for viscosity and Bruggeman model under the mean field approach for thermal conductivity were used to study the impact of nanoparticles. Heat transport was examined by calculating the heat transfer coefficient. The effects of nondimensional parameters involved in binary nanofluids on the onset of convective instabilities are depicted pictorially. The dependence of the size of the convection cells on nondimensional parameters is also discussed. The Darcy–Taylor number decreases the size of convection cells, whereas the Darcy number increases the size of convection cells. The size of the convection cells is independent of the porosity of the medium, Lewis number, volume fraction of nanoparticles, Soret effect of nanoparticles, and the solute in binary nanofluids. Furthermore, the Darcy–Taylor number and porosity have stabilising effects, whereas the Darcy number, volume fraction of nanoparticles, and Lewis number have destabilising effects on the system. Soret effects of nanoparticles and solutes exhibit dual consequences on stability. They stabilise the binary nanofluid layer if δ4 < –1 and destabilise when δ4 > –1. Their effect is immaterial if δ4 = –1.

The effect of local thermal non-equilibrium on the onset of thermal instability for a metallic foam

Mixed convection in metallic foams modeled with Darcy's law under local thermal non-equilibrium conditions is investigated, where the solid phase thermal conductivity is assumed to be infinitely larger than its fluid phase counterpart. A linear and modal stability analysis was employed to evaluate the convective and absolute instability thresholds as well as their respective cell patterns. This analysis indicates that local thermal non-equilibrium always has a stabilizing effect and the spanwise uniform mode is always the most unstable. At the onset of convective instability, however, the number of equally unstable cell patterns increases with both the aspect ratio and the local thermal non-equilibrium strength.

Onset of convective instability in an inclined porous medium

The diffusion of a solute from a concentrated source into a horizontal, stationary, fluid-saturated porous medium can lead to a convective motion when a gravitationally unstable density stratification evolves. In an inclined porous medium, the convective flow becomes intricate as it originates from a combination of diffusion and lateral flow, which is dominant near the source of the solute. Here, we investigate the role of inclination on the onset of convective instability by linear stability analyses of Darcy's law and mass conservation for the flow and the concentration field. We find that the onset time increases with the angle of inclination (θ) until it reaches a cutoff angle beyond which the system remains stable. The cutoff angle increases with the Rayleigh number, Ra. The evolving wavenumber at the onset exhibits a lateral velocity that depends non-monotonically on θ and linearly on Ra. Instabilities are observed in gravitationally stable configurations (θ≥90°) solely due to the nonuniform base flow generating a velocity shear commonly associated with Kelvin–Helmholtz instability. These results quantify the role of medium tilt on convective instabilities, which is of great importance to geological CO2 sequestration.

Onset of transient convection in a porous medium with an embedded low-permeability layer

Deep saline aquifers used for CO2 sequestration are commonly made of sedimentary formations consisting of several layers of distinguishable permeability. In this work, the effect of a non-monotonic, vertically varying permeability profile on the onset of convective instability is studied theoretically using linear stability analyses. The onset time depends on the interaction between the permeability profile and the location of the concentration perturbation peak beyond which the concentration of CO2 decays. A thin low-permeability layer can either accelerate or delay the onset time of the convective instability depending on the nature of the permeability variation – whether the permeability transition is smooth or layered, the Rayleigh number (Ra), and the location of the permeability change (a^) relative to the perturbation peak (a^c*), which scales as a^c*≈14Ra−1 for homogeneous systems. However, the low permeable layer has no effect on the onset time when it is near the lower boundary of a medium with sufficiently large Ra (a^c*≪a^). This nontrivial dependence highlights the implication of ignoring geological features of a small spatial extent, indicating the importance of a detailed characterization of CO2 storage sites.

Dynamics of Disturbances Caused by Local Sources of Heat Release in the Earth’s Damp Atmosphere

The stability of a layer of moist air above a heated surface of land or ocean is studied under the sudden influence of internal sources of heat release, which depend on the pressure and density. A dispersion equation is obtained for small 3D perturbations in a rotating atmosphere. It contains the previously known waves (acoustic, internal waves, and Brunt–​​Vaisala waves) and makes it possible to obtain a modified criterion for the onset of convective instability. This dispersion equation also makes it possible to study the effect of internal heat release on the dynamics of perturbation in these waves, including the conditions for the occurrence of instability generated by an atmospheric electric discharge. Taking into account new considered effects can lead to the identification of new instabilities and the modification of the previously known ones. The proposed system of equations allows numerical calculations of vortex flows in moist air in the near-surface layer and a more detailed study of the mechanisms of the beginning of hurricanes and tornadoes.

Intricacies of coupled molecular diffusion on double diffusive viscoelastic porous convection

The effect of cross-diffusion on the onset of convective instability in a horizontal porous layer saturated with double diffusive viscoelastic fluid is investigated. A modified Darcy–Oldroyd-B model is used to describe the viscoelastic fluid flow in a porous medium. A linear instability analysis has been performed to obtain the condition for the onset of stationary and oscillatory convection and the numerical results for NaCl–MgCl2 system are presented. Besides, the eigenvalues are computed numerically using a Galerkin method and the results are found to compare well with those obtained analytically. The simultaneous effect of viscoelasticity and the cross diffusion has led to significant changes in the convective pattern. The cross-diffusion terms induce stabilizing or destabilizing influence on the system depending on the magnitude as well as stabilizing/destabilizing impact of one of the stratifying components. The range of strain retardation parameter beyond which oscillatory convection ceases to occur is found to increase with increasing stress relaxation parameter irrespective of cross diffusion effects. The cross over boundary between stationary and oscillatory convection is demarcated by identifying a codimension-two point in the viscoelastic parameters plane. The results of Maxwell fluid are highlighted as a particular case.

Assessment of Critical Convection and Associated Rotation States in Models of Sun-like Stars Including a Stable Layer

Abstract Recent numerical simulations of rotating stellar convection have suggested the possible existence of retrograde (slow equator, fast poles) or so-called antisolar differential rotation states in slowly rotating stars possessing a large Rossby number. We aim to understand whether such rotational states exist from the onset of convective instability or are the outcome of complex nonlinear interactions in the turbulent convective envelope. To this end, we have made a systematic linear analysis of the critical state of convection in a series of 15 numerical simulations published in Brun et al. We have assessed their degree of supercriticality and most-unstable mode properties, and computed the second-order mean zonal flow response. We find that none of the linear critical cases show a retrograde state at the onset of convection even when their nonlinear counterparts do. We also find that the presence of a stably stratified layer coupled to the convectively unstable upper layer leads to interesting gravity-wave excitation and angular momentum transport. We conclude that retrograde states of differential rotation are probably the outcome of complex mode–mode interactions in the turbulent convection layer and are, as a consequence, likely to exist in real stars.

Two- and Three-Dimensional Absolute Instabilities in a Porous Medium with Inclined Temperature Gradient and Vertical Throughflow

A linear stability analysis for the onset of mixed convection in a saturated porous medium through an absolute instability of both two- and three-dimensional disturbances is performed. Relevant control parameters associated with the inclined temperature gradient and the vertical throughflow are the vertical and horizontal Rayleigh numbers, $$R_{\mathrm{v}}$$ and $$R_{\mathrm{h}}$$, and the vertical Peclet number, $$Q_{\mathrm{v}}$$, respectively. This work extends previous studies on the very same problem in two fronts. For two-dimensional disturbances, the present results do not agree with the literature for a few of the parametric conditions reported. This is caused by the collision of the convectively unstable downstream propagating branch with multiple upstream propagating branches, which generates several saddle points and, hence, makes the identification of the correct pinching point more difficult. In other words, literature results are all saddle points but not always pinching points. For three-dimensional disturbances, this issue is not present and the current results agree with the literature. On the other hand, due to the inherent difficulties associated with a three-dimensional absolute instability analysis, literature results have only been able to report the group velocities at the onset of convective instability. When their real parts are zero, transition occurs directly from stable to absolutely unstable. Otherwise, transition occurs from stable to convectively unstable first and nothing can be said about the onset of absolute instability. In this work, a novel technique recently developed by the authors allowed the identification of the onset of absolute instability under all parametric conditions investigated in the literature, extending earlier results. Doing so confirmed the dichotomy already observed in these earlier studies, i.e., the onset of absolute instability for two- and three-dimensional longitudinal modes indeed differs.

Linear stability of horizontal throughflow in a Brinkman porous medium with mixed thermal boundary conditions

The onset of convective instability in a horizontal throughflow due to external heating and viscous dissipation is investigated. The flow is considered along a porous layer with high permeability, bounded by two impermeable surfaces subjected to external heating. Hence, the third kind of thermal boundary conditions are used at the two boundaries. The Brinkman extended Darcy's law is adopted for the equations governing flow through the layer. The base flow is assumed to be inclined to the horizontal axis at an angle γ. An average mass flow along the horizontal direction is considered, the magnitude of which is given by Pe (Péclet number). The disturbances in the flow are assumed in the form of two-dimensional oblique rolls inclined to the base flow at an angle γ(0≤γ≤π2). The four limiting cases with different combinations of B0,B1→0,∞, are also discussed. The effect of viscous dissipation and external heating at the two boundaries, on the instability of the flow is discussed. The instability remains unaffected by the direction of the throughflow. However, the instability is affected by the magnitude of the throughflow. Viscous dissipation has a stabilizing effect on the flow, as long as the external heating at the bottom boundary is higher than that at the upper boundary. The longitudinal rolls (at γ=π2) are the preferred mode of instabilities.

Convective overshoot and macroscopic diffusion in pure-hydrogen-atmosphere white dwarfs

Abstract We present a theoretical description of macroscopic diffusion caused by convective overshoot in pure-hydrogen DA white dwarfs using 3D, closed-bottom, radiation hydrodynamics co5bold simulations. We rely on a new grid of deep 3D white dwarf models in the temperature range $11\, 400 \le T_{\mathrm{eff}} \le 18\, 000$ K where tracer particles and a tracer density are used to derive macroscopic diffusion coefficients driven by convective overshoot. These diffusion coefficients are compared to microscopic diffusion coefficients from 1D structures. We find that the mass of the fully mixed region is likely to increase by up to 2.5 orders of magnitude while inferred accretion rates increase by a more moderate order of magnitude. We present evidence that an increase in settling time of up to 2 orders of magnitude is to be expected, which is of significance for time-variability studies of polluted white dwarfs. Our grid also provides the most robust constraint on the onset of convective instabilities in DA white dwarfs to be in the effective temperature range from 18 000 to 18 250 K.

The effect of pulsating throughflow on the onset of magneto convection in a layer of nanofluid confined within a Hele-Shaw cell

In this article, the joint effect of pulsating throughflow and magnetic field on the onset of convective instability in a nanofluid layer, bounded in a Hele-Shaw cell is presented within the context of linear stability theory and frozen profile approach. The model utilized for nanofluid combines the impacts of Brownian motion and thermophoresis, while for Hele-Shaw cell, Hele-Shaw model is considered. The Galerkin technique is utilized to solve the eigenvalue problem. The outcome of the important parameters on the stability framework is examined analytically. It is observed that the pulsating throughflow and magnetic field have both stabilizing effects. The impact of increasing the Hele-Shaw number [Formula: see text], the modified diffusive ratio [Formula: see text] and the nanoparticle Rayleigh number [Formula: see text] is to quicken the convective motion, while the Lewis number [Formula: see text] has dual impact on the stability framework in the existence of pulsating throughflow. It is also established that the oscillatory mode of convective motion is possible only when the value of the magnetic Prandtl number [Formula: see text] is not greater than unity.

Modeling Craton Destruction by Hydration‐Induced Weakening of the Upper Mantle

AbstractGrowing evidence shows that lithospheric mantle beneath cratons may contain a certain amount of water that originated from dehydration of subducted slabs or mantle metasomatism. As water can significantly reduce the viscosity of nominally anhydrous minerals such as olivine, hydration‐induced rheological weakening is a possible mechanism for the lithospheric thinning of cratons. Using 2‐D thermomechanical numerical models, we investigated the influence of water on dislocation and diffusion creep of olivine during the evolution of cratonic lithosphere. Modeling results indicate that dislocation creep of wet olivine alone is insufficient to trigger dramatic lithospheric thinning within a timescale of tens of millions of years, even with an extremely high water content. However, if diffusion creep is incorporated, significant convective instability will occur at the base of the lithosphere and drive lithospheric mantle dripping, which results in intense lithospheric thinning. We performed semianalytical models to better understand the influence of various parameters on the onset of convective instability. The convective instability promoted by hydration weakening drives lithospheric mantle dripping beneath cratons and thus provides a possible mechanism for cratonic thinning.

Lock-exchange gravity currents over rough bottoms

In nature, density driven currents often flow over or within a bottom roughness: a sea breeze encountering tall buildings, a shallow flow encountering aquatic vegetation, or a dense oceanic current flowing over a rough bottom. Laboratory experiments investigating the mechanisms by which bottom roughness enhances or inhibits entrainment and dilution in a lock-exchange dense gravity current have been conducted. The bottom roughness has been idealized by an array of vertical, rigid cylinders. Both spacing (sparse vs. dense configuration) and height of the roughness elements compared with the height of the current have been varied. Two-dimensional density fields have been obtained. Experimental results suggest that enhancement of the entrainment/dilution of the current can occur due to two different mechanisms. For a sparse configuration, the dense current propagates between the cylinders and the entrainment is enhanced by the vortices generated in the wake of the cylindrical obstacles. For a dense configuration, the dense current rides on top of the cylinders and the dilution is enhanced by the onset of convective instability between the dense current above the cylinders and the ambient lighter water between the cylinders. For low values of the ratio of the cylinder to lock height \(\lambda \) the dense current behavior approaches that of a current over a smooth bottom, while the largest deviations from the smooth bottom case are observed for large values of \(\lambda \).

Magnetic field effect on the onset of Soret-driven convection of a nanofluid confined within a Hele-Shaw cell

The effect of a magnetic field on the early stages of Soret-driven convection of a nanoparticle suspension with large negative separation ratio χ confined within a Hele-Shaw cell, heated from above, was analyzed. Taking the Lorentz force into account, new stability equations were formulated in a similar (τ, ζ)-domain as well as in a global (τ, z)-domain by introducing the Hele-Shaw Rayleigh number based on the Soret flux (RsH) and the Hele-Shaw Hartmann number (HaH). With and without the quasi-steadiness assumptions, the resulting stability equations were solved analytically by expanding the disturbances as a series of orthogonal functions, and also the numerical shooting method was used. The critical time of the onset of convection and the corresponding wave number were obtained as a function of RsH and HaH. It was found that the magnetic field plays a critical role in the onset of convective instability. The onset time increases with increasing HaH and decreasing RsH. The linear stability limits are independent of the solution methods, if the trial functions for the disturbance quantities are properly chosen. Based on the results of the linear stability analysis, a non-linear analysis was conducted using direct numerical simulations. The non-linear analysis revealed that the convective motion can be apparent far after the linear stability limit.

Effect of Phase Thermal Modulation Without Stationary Temperature Gradient on the Threshold of Convection

The convective instability of a horizontal fluid layer subject to a time varying gradient of temperature is investigated. The stationary component of the temperature gradient is considered equal to zero and the oscillating components imposed on the horizontal boundaries are in phase and with the same amplitude. The aim of the present paper is to examine the effect of this type of modulation on the onset of convective instability. We show that unlike the case where the equilibrium configuration is stable in the absence of modulation, we have instability when the temperature at the horizontal boundaries is modulated in phase. Also, we observe that in the limit of low and high dimensionless frequency of modulation, ω < 0.5 and ω > 140, the basic state tends to a stable equilibrium configuration and for an intermediate dimensionless frequency, the system is potentially unstable. The results obtained from analytical asymptotic study for low and high dimensionless frequency are in good agreement with the numerical ones.

Linear instability of the vertical throughflow in a horizontal porous layer saturated by a power-law fluid

The effects of the vertical throughflow of a non-Newtonian power-law fluid on the onset of convective instability in a horizontal porous layer are investigated. The extended Darcy’s model of momentum diffusion is employed together with the Oberbeck–Boussinesq approximation. A stationary basic solution for the vertical throughflow is determined analytically. The basic velocity and temperature fields turn out to be independent of the non-Newtonian rheology. A linear stability analysis is carried out, leading to a fourth-order eigenvalue problem. A numerical solution of the eigenvalue problem is employed to obtain the neutral stability curves and the critical Rayleigh number for the onset of instability. The governing parameters of the transition to instability are the Péclet number associated with the throughflow, and the power-law index of the fluid. These parameters influence the position of the neutral stability curve and also the critical Rayleigh number. The asymptotic cases of absolute pseudoplasticity, absolute dilatancy, and small Péclet number are discussed in detail. The latter case leads to a simple analytical solution that approximates fairly well the numerical data when the Péclet number is smaller than unity.