Development Of Convective Instability Research Articles

Convective instability in a two-layer system of reacting fluids with concentration-dependent diffusion

The development of convective instability in a two-layer system of miscible fluids placed in a narrow vertical gap has been studied theoretically and experimentally. The upper and lower layers are formed with aqueous solutions of acid and base, respectively. When the layers are brought into contact, the frontal neutralization reaction begins. We have found experimentally a new type of convective instability, which is characterized by the spatial localization and the periodicity of the structure observed for the first time in the miscible systems. We have tested a number of different acid–base systems and have found a similar patterning there. In our opinion, it may indicate that the discovered effect is of a general nature and should be taken into account in reaction–diffusion–convection problems as another tool with which the reaction can govern the movement of the reacting fluids. We have shown that, at least in one case (aqueous solutions of nitric acid and sodium hydroxide), a new type of instability called as the concentration-dependent diffusion convection is responsible for the onset of the fluid flow. It arises when the diffusion coefficients of species are different and depend on their concentrations. This type of instability can be attributed to a variety of double-diffusion convection. A mathematical model of the new phenomenon has been developed using the system of reaction–diffusion–convection equations written in the Hele–Shaw approximation. It is shown that the instability can be reproduced in the numerical experiment if only one takes into account the concentration dependence of the diffusion coefficients of the reagents. The dynamics of the base state, its linear stability and nonlinear development of the instability are presented. It is also shown that by varying the concentration of acid in the upper layer one can achieve the occurrence of chemo-convective solitary cell in the bulk of an almost immobile fluid. Good agreement between the experimental data and the results of numerical simulations is observed.

Effects of hydrodynamic dispersion on the stability of buoyancy-driven porous media convection in the presence of first order chemical reaction

Carbon dioxide (text {CO}_2) sequestration is one of several long-term solutions suggested to decrease the amount of greenhouse gases in the atmosphere. Among different methods of carbon dioxide sequestration, the dissolution of text {CO}_2 in deep saline aquifers is considered one of the most effective. A significant number of studies are currently being carried out to provide a good understanding of the physical mechanisms involved in this type of storage. The present work focuses on the hydrodynamic part of the problem: setting a model for carbon dioxide-loaded flows in an idealised two-dimensional geometry. It considers the impact of hydrodynamic dispersion in porous media on the development of convective instabilities. Particular attention is paid to the mathematical form of the dispersion tensor widely used in porous media studies, and a new type of bifurcation is investigated. We show that the analysis of bifurcations from the no-flow steady-state solution is a continuous but non-smooth problem, which is a key feature of the analysis. Although the problem is non-smooth, it is also shown that the basic behaviours of linear stability analysis are observed in its solution.

Transitional regimes of penetrative convection in a plain layer

Penetrative convection in a plain layer of water is considered for the interval of temperatures containing the point of density maximum. When the unstable and stable layers are equal in the static conductive state, the development of convective instability is investigated from the formation of steady structures to chaotic motion. Scales of the periodicity cell for regimes with a strong nonlinear effect were chosen with particular attention. Specifics are shown for steady structures with small-scale vortices near upper boundaries and for periodic motions with synchronized oscillations of the lower parts of vertically elongated profiles of temperature that move symmetrically. With the increase of supercriticality, the motion loses reflection symmetry, becomes doubly periodic, and finally becomes quasi-periodic before transition to chaotic motion. Domains of hysteresis are investigated for which motions with different structure and heat fluxes coexist, and it is illustrated by the dependence of the Nusselt number on supercriticallity.

Effect of walls on the development of convective instability in a vertical channel of limited height

Convective instability in a plane vertical channel is considered. Under the simplest boundary conditions, a characteristic equation is obtained and critical conditions for Rayleigh convection in the form of two-dimensional flows in the plane of the channel are determined. It is discussed whether or not such instability may occur in evaporation processes for which a transition to an intense evaporation mode related to the development of Rayleigh convection was previously observed.

Neutrino crown of a protoneutron star in the process of collapse: Physical formulation of the problem

A spherically symmetric gravitational collapse of the iron core of a star develops in the form of a hydrodynamic process, where the role of intrinsic neutrino radiation ever increases. The early stage of the collapse has a homological character within the interior of the core, but there is a delay in exterior layers. Hydrodynamic calculations reveal that, at the late stage of the collapse (it is the stage within which the majority of neutrinos are emitted), a structure is formed that consists of a neutron-star germ nontransparent to neutrinos and exterior layers accreting onto it, which are, on the contrary, transparent to neutrinos. They are separated by a semitransparent layer occurring between the front of the accretion shock wave and the germ surface forming a neutrinosphere. By using a typical quasistationary character of this layer, which is referred to as the neutrino crown of a protoneutron star, a stationary model is developed here that supplements hydrodynamic calculations of the collapse process, which are rather rough within this model. In particular, these calculations reveal the crucial significance of the semitransparent crown for a possible transition of the collapse into an explosion having a scale of a supernova explosion. If there is no such possibility, the same crown determines the important properties of a quiet collapse that are associated with the development of convective instability, etc., in it. The model formulated here, which is comparatively simple (in relation to hydrodynamic calculations) owing to an adequate physical formulation of problem, is intended for analyzing special features of the crown. This formulation of the problem demonstrates some new possibilities of neutrino hydrodynamics, which is an analog of the well-known radiative hydrodynamics involving photons instead of neutrinos.

Convective destabilization of a thickened continental lithosphere

Removal or delamination of the lithospheric mantle in a late stage of mountain building is a process often invoked to explain syn orogenic extension, high temperature metamorphism, magmatism and uplift. One mechanism that could explain the lithospheric root detachment is the development of convective instabilities within the peridotitic lithosphere due to its high density. This mechanism is studied by two-dimensional convective numerical simulations in the simple case of a strongly temperature dependent viscous rheology appropriate for upper mantle rocks. We neglect here the weakening effect of a brittle rheology and of a crustal layer, and therefore we did not model tectonic deformations. Depending on the upper mantle viscosity and activation energy, a 300 km thick root can be inferred to be either indefinitely stable or to thicken with time or to thin with time. When the lithosphere is initially thicker than its equilibrium thickness, the convective flow at the base and on the sides of the lithospheric root is strong enough to cancel downwards heat conduction and to progressively remove the root. This flow is due to the finite density perturbations induced by the topography of the isotherms on the base and at the sides of the root. We derive two general parameterizations of the convective removal duration as a function of the equilibrium thickness, the thickening factor, the root width, and the rheological temperature scale. Using these relationships, and assuming that the lithospheric equilibrium thickness is about 100 km, the removal duration of a 250 km thick root ranges from 55 to 750 Myr depending on the root width. It is too small to explain the long term stability of cratonic lithospheric root, but too long to explain any sudden change in the stress and strain states in mountain belts development.

Numerical simulations in astrophysics:Supernovae explosions, magnetorotational model and neutrino emission

Theories of stellar evolution and stellar explosion are based on results of numerical simulations and even qualitative results are not available to get analytically. Supernovae are the last stage in the evolution of massive stars, following the onset of instability, collapse and formation of a neutron star. Formation of a neutron star is accompanied by a huge amount of energy, approximately 20% of the rest mass energy of the star, but almost all this energy is released in the form of weakly interacting and hardly registrated neutrino. About 0.1% of the released neutrino energy would be enough for producing a supernovae explosion, but even transformation of such a small part of the neutrino energy into the kinetic energy of matter meets serious problems. Two variants are investigated for obtaining explosion. The first one is based on development of convective instability, and more effective heating of the outer layers by a neutrino flux.The second model is based on transformation of a rotational energy of a rapidly rotating neutron star with its envelope into the energy of explosion due to action of a magnetic field as a transformation mechanism. Calculations in this model in 1‐ and 2‐dimensions give a stable value of transformation of the rotational energy into the energy of explosion on the level of few percents. This occurrance to be enough for explanation of the energy release in supernova explosion. The last model gives a direct demonstration of nonlinear interaction between hydrodynamical and hydromagnetic systems. At first a field is amplified by differential rotation, then this enhanced field leads to transformation of the rotational energy into the energy of explosion.

Numerical simulation of collapsing volcanic columns with particles of two sizes

A three‐phase thermofluid‐dynamic model was employed to simulate the behavior of collapsing volcanic columns and related pyroclastic flows. The model accounts for the mechanical and thermal nonequilibrium between a gas phase and two solid phases representative of particles of two different sizes. The gas phase has two components: hot water vapor leaving the vent and atmospheric air. Collisions between particles of the same size were accounted by a solids elasticity modulus, whereas a semiempirical correlation was employed to account for particle‐particle interactions between particles of different sizes. The gas phase turbulence was modeled by a turbulent subgrid scale model. The partial differential equations of conservation of mass, momentum, and energy were solved numerically, by a finite difference scheme, on an axisymmetric physical domain for different granulometric compositions at the vent. Simulations were limited to particles of few hundreds microns, and therefore to dilute flows, in order to mantain a reasonable computational load. Results show the formation of the initial vertical jet, column collapse, building of a pyroclastic fountain followed by the generation of a radially spreading pyroclastic flow, and the development of convective instabilities from the upper layer of the flow which lead to the formation of coignimbritic or phoenix clouds. The analysis of the spatial and temporal distributions of the two solid phases in the different parts of the domain shows nonequilibrium effects between them and allow us to quantify important emplacement processes as pyroclast sedimentation and ash dispersion. In particular, the importance of coupling effects between the two solid phases leads to relevant differences between the behavior of columns with one or two solid phases. A significant influence of the granulometric composition was observed on the pyroclastic flow runout, flow thickness, and particle distribution in the flow and phoenix cloud. The results from simulations appear to be qualitatively in agreement with simple laboratory experiments and field observations.

Convective instabilities in a variable viscosity fluid cooled from above

Whether in the mantle or in magma chambers, convective flows are characterized by large variations of viscosity. We study the influence of the viscosity structure on the development of convective instabilities in a viscous fluid which is cooled from above. The upper and lower boundaries of the fluid are stress-free. A viscosity dependence with depth of the form ν 0 + ν 1 exp(−γ.z) is assumed. After the temperature of the top boundary is lowered, velocity and temperature perturbations are followed numerically until convective breakdown occurs. Viscosity contrasts of up to 10 7 and Rayleigh numbers of up to 10 8 are studied. For intermediate viscosity contrasts (around 10 3), convective breakdown is characterized by the almost simultaneous appearance of two modes of instability. One involves the whole fluid layer, has a large horizontal wavelength (several times the layer depth) and exhibits plate-like behaviour. The other mode has a much smaller wavelength and develops below a rigid lid. The “whole layer” mode dominates for small viscosity contrasts but is suppressed by viscous dissipation at large viscosity contrasts. For the “rigid lid” mode, we emphasize that it is the form of the viscosity variation which determines the instability. For steep viscosity profiles, convective flow does not penetrate deeply in the viscous region and only weak convection develops. We propose a simple method to define the rigid lid thickness. We are thus able to compute the true depth extent and the effective driving temperature difference of convective flow. Because viscosity contrasts in the convecting region do not exceed 100, simple scaling arguments are sufficient to describe the instability. The critical wavelength is proportional to the thickness of the thermal boundary layer below the rigid lid. Convection occurs when a Rayleigh number defined locally exceeds a critical value of 160–200. Finally, we show that a local Rayleigh number can be computed at any depth in the fluid and that convection develops below depth z r (the rigid lid thickness) such that this number is maximum. The simple similarity laws are applied to the upper mantle beneath oceans and yield estimates of 5 × 10 15−5 × 10 16 m 2 s −1 for viscosity in the thermal boundary layer below the plate.

Nonlinear development of convective instability within slender flux tubes. II. The effect of radiative heat transport

Inclusion of radiative heat transport in the energy equation for a slender flux tube leads to oscillations of the tube. The amplitude of the oscillations depends on the radius of the tube when lateral heat exchange alone is considered. Longitudinal heat transport has a greater influence on the evolution of the instability than lateral heat exchange for the particular value of tube radius considered in the calculation. Heat transport is seen to reduce the efficiency of concentration of magnetic fields by convective collapse in the case of polytropic tubes.

Nonlinear development of convective instability within slender flux tubes. I. Adiabatic flow

The nonlinear development of convective instability within slender flux tubes is studied using the method of characteristics. It is seen that the initial magnetic field influences the development of the instability. The asymptotic state of the unstable tube depends on the boundary conditions. Flux tubes subjected to ’open’ boundary conditions show a better evidence for field amplification than those subjected to ’closed’ boundary conditions. In either case, convective instability results in the generation of significant gas flow within slender flux tubes.

Effect of Joule dissipation on the convective instability of a conducting liquid with flow in a magnetic field

The results of [1] are extended to the case when the Joule dissipation leads to a nonlinear profile of the unperturbed temperature of the liquid. Convective instability of a conducting liquid, with flow in a magnetic field directed perpendicular to the flow, with a temperature-dependent distribution of the conductivity which is nonhomogeneous in the direction of action of the electromagnetic force, was discussed in [1], neglecting Joule dissipation. This type of approach permitted investigating an energy equation without electromagnetic terms, which to a certain degree facilitated the solution of the problem. In many cases, however, the Joule dissipation is considerable and may exert a considerable effect on the development of convective instability. Thus, without taking account of Joule evolution of heat, instability can arise only with positive values of the Rayleigh number, exceeding some critical value, while, at the same time, Joule dissipation may lead to a situation in which instability will develop also with negative values of the Rayleigh number, i.e., under conditions when the state without the evolution of Joule heat is absolutely stable.