Stability Analysis Of Convection Research Articles

Effect of noninertial acceleration on heat transfer by Rayleigh–Bénard magnetoconvection: Exploring nonlinear dynamics

AbstractBuoyancy‐driven Rayleigh–Bénard convection in the presence of a magnetic field, rotation, and temperature‐dependent viscosity has applications in the field of crystal growth, space laboratory experiments, and atmospheric convection. It also has potential applications in heat transfer and magnetohydrodynamics, which can occur in planetary and stellar interiors. The present work aims to study the combined effect of magnetic fields and rotation on the onset of Rayleigh–Bénard convection in temperature‐dependent viscosity Newtonian liquids with internal heat sources/sinks. Both linear and weak nonlinear stability analyses of convection are performed in the problem. The minimal representation of the Fourier series for the stream function and the magnetic potential allows us to derive the analytical expression for the thermal Rayleigh number () and the generalized Lorenz model. In linear theory, the critical Rayleigh number and the wave number are tabulated for different values of the Chandrasekhar number (), the Taylor number (), and the thermorheological parameter (), and the onset of stability is analyzed. It is found that the instability manifests at when for stress‐free and isothermal boundaries. In nonlinear theory, the generalized Lorenz model obtained is not amenable to analytical treatment; therefore, the classical fourth‐order Runge–Kutta method is applied to solve the Lorenz model. It is found that the internal Rayleigh number, the thermorheological parameter, and the Taylor number influence the onset of convection and heat transfer coefficient. It is also demonstrated that the joint increase in rotational force and magnetic field strength stabilizes the system strongly, thereby reducing heat transfer. However, increasing the heat source and the variable viscosity parameter have antagonistic influences.

Unstable Convection in a Vertical Double–Layer Porous Slab

A convective stability analysis of the flow in a vertical fluid-saturated porous slab made of two layers with different thermophysical properties is presented. The external boundaries are isothermal with one of them impermeable while the other is open to an external fluid reservoir. This study is a development of previous investigations on the onset of thermal instability in a vertical heterogeneous porous slab where the heterogeneity may be either continuous or piecewise as determined by a multilayer structure. The aim of this paper is investigating whether a two-layer structure of the porous slab may lead to the onset of cellular convection patterns. The linear stability analysis is carried out under the assumption that one porous layer has a thermal conductivity much higher than the other layer. This assumption may be justified for the model of a heat transfer enhancement system involving a saturated metal foam. A flow model for the natural convection based on Darcy’s momentum transfer in a porous medium is adopted. The buoyancy-induced basic flow state is evaluated analytically. Small-amplitude two-dimensional perturbations of the basic state are introduced, thus leading to a linear set of governing equations for the disturbances. A normal mode analysis allows one to formulate the stability eigenvalue problem. The numerical solution of the stability eigenvalue problem provides the onset conditions for the thermal instability. Moreover, the results evidence that the permeability ratio of the two layers is a key parameter for the critical conditions of the instability.

Linear stability analysis of convection in a solid partitioned inhomogeneous multilayered porous structure

We investigate the effect of varying permeability on the onset of convection in a system of multi-layered porous medium when heated from below. Here, the interface between any two porous sublayers is a thick solid conducting plate. The porous sublayers are inhomogeneous with respect to their permeability. Results for the homogeneous case are validated against those from the literature. The solvability condition is produced for a general N– layered inhomogeneous system to study the neutral curves. Detailed results are presented for a two layered and a three layered configuration. The effect of the involved parameters such as permeability ratio, thermal conductivity ratio, and thickness ratio on the onset of convection is analyzed using linear stability analysis. The critical values of Rayleigh number and the corresponding wave number increased with the increase in the thermal conductivity of the solid partitions, irrespective of the change in the permeability of porous sublayers. Multicellular sublayer flow circulation strength is amplified by the conductivity of the solid interfaces. Higher modes are also examined for varying permeability ratio and thermal conductivity ratio. A different flow pattern is observed for mode 3 when the permeability ratio is either very much less than one or much greater than one. Onset of the convection is sensitive to the increasing number of inhomogeneous porous sublayers of the layered porous system.

Investigation of the Polymer Coextrusion Process: A Review.

A review of the different coextrusion processes and the related processing problems is presented. A one-dimensional bilayer coextrusion Poiseuille flow model is first developed with Newtonian and shear-thinning rheological behaviors. A transitory computation at the convergence between the two independent polymer layers shows that stationary interface position and velocity profile are established after a short distance of the order of the die gap which justifies the validity of the 1D stationary model. This model is then applied to multilayer temperature dependent coextrusion flows which correspond to realistic industrial coextrusion conditions. Marked interface instabilities may be observed depending on the rheology of the coextruded polymers and of their flow rate ratios. Experiments point clearly out that these instabilities may be amplified along the die land. Convective stability analysis as well as direct numerical computation discriminate flow situations which amplify or damp down instabilities. These 1D models are unable to account for the complex feedblock coat-hanger die geometries. A thin layer coextrusion model is then developed, based on the Hele-Shaw lubrication approximations already used for single layer extrusion problems. It allows to predict the location of the interfaces between the different layers in the whole die, and especially at die exit. This represents a major issue in feedblock die coextrusion. These thin layer approaches are unable to address the encapsulation of one polymer by the other in these complex die geometries with important gap thicknesses. Experiments conducted in dies of square section allow identifying the dynamics of encapsulation. 3D models are required to account for this phenomenon but the management of the sticking contact at the die wall poses difficult numerical problems.

Triggering Vortex Shedding for the Freestream Flow of Nanofluids Around Bluff Objects

Abstract The freestream flow around a bluff object shows steady symmetric nature in the low Reynolds number laminar regime. However, when the Reynolds number increases to a critical value, the flow shows unsteadiness with alternate shedding of vortices. We show here numerically that the vortex shedding could be initiated for flow of a nanofluid over a bluff object even when the Reynolds number is lying in the steady regime (10≤Re≤30). Cu–H2O and Ag–H2O nanofluids are used and the volume fractions of Cu and Ag nanoparticles are gradually increased. At some critical values of the volume fractions, the flow shows unsteadiness with vortex shedding. The critical solid volume fraction is estimated from the convective stability analysis following the extended Landau model. The shedding phenomenon is established through contour plots, phase diagrams, and analysis of the time signals of lift coefficient. The critical volume fractions for the two different nanofluids for transition of steady to unsteady flow over circular and square-shaped bluff objects are observed to decrease with increasing Reynolds number.

Two-phase magnetohydrodynamics: Theory and applications to planetesimal cores

Core freezing and resultant compositional convection are likely important drivers for dynamo activity in large terrestrial bodies like the Earth. The solidification of compositional mixtures, such as iron and sulfur, generates mush zones of partial melt at the freezing front, which can eject chemically buoyant or heavy liquid that then drives convection. For smaller bodies such as planetesimals in the asteroid belt, conditions for generating a dynamo are harder to achieve. Nevertheless, evidence for magnetization of achondrite meteorites is abundant, suggesting that many planetesimal cores were somehow magnetized. As such small bodies cool rapidly under low gravity they likely spend much of their evolution with a large poorly compacted partial melt mushy zone. The magneto-hydrodynamic behavior of a deformable partial melt zone can induce magnetism via separation of solid and liquid phases, and conversely magnetism can impose extra forces on the phase separation in the mush zone. To this end, we have developed a new two-phase magneto-hydrodynamic theory for deformable mushes and slurries. The model includes the standard effects of Lorentz forces, and the competition between magnetic field stretching and diffusion. There are additional effects at the liquid pore or solid grain scale, which involve the interaction between phases, akin to Darcy or Stokes drag; these include Lorentz drag, as well as pore/grain-scale diffusive exchange of magnetism between solid and fluid phases, and field stretching due to relative motion between phases. Magnetic induction by gravitational phase separation is most significant after extensive mixing of liquid and solid phases, such as induced by vigorous mechanical stirring due to, for example, tidal and elliptic instabilities, or impacts with other bodies, all of which are conceivably common in the asteroid belt, especially in the early solar system. Gravitational phase separation following such events can induce significant magnetism in the liquid and solid phases, and much more rapidly than can magnetization by large-scale circulation. Magnetic field variances can be at first orders of magnitude larger than an imposed background field, during initial gravitational phase separation of the well-mixed slurry. As the phases separate toward the top or bottom, the solid phase compacts, the separation velocity decreases, and the magnetic field variance likewise diminishes. However, solitary waves in the compacting region can cause an additional large magnetic induction in the liquid, taking the form of strongly magnetized wave packets that can be trapped in the solid. Thus, phase separation, segregation and compaction potentially induce large magnetic field anomalies. A linear stability analysis for convection in a porous medium (with a rigid matrix) is also explored. Pore and grain scale effects, such as field stretching due to phase separation, are found to enhance the influence of the magnetic field on convective instability.

Nonlinear Stability of Natural Convection in an Inclined Fluid Layer

A nonlinear stability analysis of convection in an inclined layer of a viscous, incompressible fluid is carried out using energy method. The nonlinear stability boundary determined by the Rayleigh number $$\text {Ra}$$ for the underlying dynamical system is found to depend upon the inclination of the layer with respect to the horizontal and Prandtl number of the fluid. A comparison with the linear instability boundary for the considered hydrodynamic system indicates a parametric region where the nonlinear stability of the basic flow is uncertain.

Stability analysis of convection–diffusion equations of different finite-element spaces at discrete times

The convection-controlled diffusion problem is hyperbolic in nature and its solutions tend to have numerical shocks. To solve the problem of time dependence, it is common to use finite-element method in the spatial region, while the algorithm proposed in the past with finite differential procedure is mostly limited to the fixed finite-element grid of the spatial region. We often need to use different finite-element spaces at different times, such as the spread of flame, oil and water frontier problems; so many mathematicians and engineers have set their sights on the use of dynamic finite-element space, but also put across a lot of dynamic finite-element methods in giving the general parabola problem of the variable-mesh finite-element method. The principal purpose of this paper is to adopt different spatial grids for different time layers, and project the approximate solution of the previous time layer to the present time layer to act as the initial value of the current layer to enable us deduce the stability at discrete times.

Flow induced by a rotating cone: Base flow and convective stability analysis

The boundary layer over a cone rotating in a still fluid is investigated. A self-similar correction to the classical von K\'arm\'an solution, taking into account the effect of the outer flow, is proposed and validated. Finally, the stability properties of the corrected base flow are assessed.

Stability analysis of convection in the intracluster medium

We use the machinery usually employed for studying the onset of Rayleigh–Bénard convection in hydro- and magnetohydro-dynamic settings to address the onset of convection induced by the magnetothermal instability and the heat-flux-buoyancy-driven-instability in the weakly-collisional magnetized plasma permeating the intracluster medium. Since most of the related numerical simulations consider the plasma being bounded between two ‘plates’ on which boundary conditions are specified, our strategy provides a framework that could enable a more direct connection between analytical and numerical studies. We derive the conditions for the onset of these instabilities considering the effects of induced magnetic tension resulting from a finite plasma beta. We provide expressions for the Rayleigh number in terms of the wave vector associated with a given mode, which allow us to characterize the modes that are first to become unstable. For both the heat-flux-buoyancy-driven-instability and the magnetothermal instability, oscillatory marginal stable states are possible.

Three dimensional simulations and stability analysis for convection induced by absorption of radiation

Purpose – The purpose of this paper is to investigate a model for convection induced by the selective absorption of radiation in a fluid layer. The concentration based internal heat source is modelled quadratically. Both linear instability and global nonlinear energy stability analyses are tested using three dimensional simulations. The results show that the linear threshold accurately predicts on the onset of instability in the basic steady state. However, the required time to arrive at the steady state increases significantly as the Rayleigh number tends to the linear threshold. Design/methodology/approach – The author introduce the stability analysis of the problem of convection induced by absorption of radiation in fluid layer, then the author select a situations which have very big subcritical region. Then, the author develop a three dimensions simulation for the problem. To do this, first, the author transform the problem to velocity – vorticity formulation, then the author use a second order finite difference schemes. The author use implicit and explicit schemes to enforce the free divergence equation. The size of the Box is evaluated according to the normal modes representation. Moreover, the author adopt the periodic boundary conditions for velocity and temperature in the $x, y$ dimensions. Findings – This paper explores a model for convection induced by the selective absorption of radiation in a fluid layer. The results demonstrate that the linear instability thresholds accurately predict the onset of instability. A three-dimensional numerical approach is adopted. Originality/value – As the author believe, this paper is one of the first studies which deal with study of stability of convection using a three dimensional simulation. When the difference between the linear and nonlinear thresholds is very large, the comparison between these thresholds is very interesting and useful.

Thermoconvective instability and local thermal non-equilibrium in a porous layer with isoflux-isothermal boundary conditions

The effects of lack of local thermal equilibrium between the solid phase and the fluid phase are taken into account for the convective stability analysis of a horizontal porous layer. The layer is bounded by a pair of plane parallel walls which are impermeable and such that the lower wall is subject to a uniform flux heating, while the upper wall is isothermal. The local thermal non-equilibrium is modelled through a two-temperature formulation of the energy exchange between the phases, resulting in a pair of local energy balance equations: one for each phase. Small-amplitude disturbances of the basic rest state are envisaged to test the stability. Then, the standard normal mode procedure is adopted to detect the onset conditions of convective rolls. Beyond the Darcy-Rayleigh number, playing the role of order parameter for the transition to instability, the relevant dimensionless parameters are the inter-phase heat transfer parameter and the thermal conductivity ratio. The disturbance governing equations, formulated as an eigenvalue problem, are solved numerically by a shooting method. Results are reported for the neutral stability curves and for the critical values for the onset of instability.

Magnetoconvection in a plane layer rotating about the horizontal axis: The effect of anisotropic diffusivities

Abstract A linear stability analysis of convection arising in a horizontal plane layer rotating about the horizontal axis and permeated by a homogeneous horizontal magnetic field perpendicular to the rotation axis is performed. Resulting horizontal convective rolls are inclined to the magnetic field at an angle dependent on the dimensionless numbers − the Elsasser, Ekman and Roberts numbers, and moreover on the anisotropy parameter, the ratio of horizontal and vertical diffusion coefficients (which are the viscosity and thermal diffusivity; magnetic diffusivity is considered isotropic). Two types of anisotropies, SA andBM, are considered and compared with the isotropic case of diffusion coefficients. In the stratification anisotropy, SA, of the Sa and So types, diffusivities in the horizontal directions are, respectively, smaller and greater than the vertical ones. In the BM anisotropy (Braginsky and Meytlis, 1990), the diffusivities in the directions of rotation axis and magnetic field - in the horizontal directions are greater than in vertical direction, thus identically as in So type anisotropy. Results of this H case, the model with the horizontal rotation axis, are compared with the V case of a similar model with the vertical rotation axis. The modes of instabilities are much more sensitive to viscosity and various anisotropies in the H case than in the V case. Results indicate that the effects of anisotropic diffusivities on the Earth’s core magnetoconvection and geodynamo processes should be studied more thoroughly in simpler models than is usually done.

A Nonlinear Stability Analysis of Convection in a Porous Vertical Channel Including Local Thermal Nonequilibrium

The problem is considered of thermal convection in a saturated porous medium contained in an infinite vertical channel with differentially heated sidewalls. The theory employed allows for different solid and fluid temperatures in the matrix. Nonlinear energy stability theory is used to derive a Rayleigh number threshold below which convection will not occur no matter how large the initial data. A generalized nonlinear analysis is also given which shows convection cannot occur for any Rayleigh number provided the initial data is suitably restricted.

Convective stability analysis of a micropolar fluid layer by variational method

This paper studies Rayleigh-Bénard convection of micropolar fluid layer heated from below with realistic boundary conditions. A specific approach for stability analysis of a convective problem based on variational principle is applied to characterize the Rayleigh number for quite general nature of bounding surfaces. The analysis consists of replacing the set of field equations by a variational principle and the expressions for Rayleigh number are then obtained by using trial function satisfying the essential boundary conditions. Further, the values of the Rayleigh number for particular cases of large and small values of the microrotation coefficient have been obtained. The effects of wave number and micropolar parameter on the Rayleigh numbers for onset of stationary instability for each possible combination of the bounding surfaces are discussed and illustrated graphically. The present analysis establishes that the nature of bounding surfaces combination and microrotation have significant effect on the onset of convection.

Global stability of flow past a cylinder with centreline symmetry

Global absolute and convective stability analysis of flow past a circular cylinder with symmetry conditions imposed along the centreline of the flow field is carried out. A stabilized finite element formulation is used to solve the eigenvalue problem resulting from the linearized perturbation equation. All the computations carried out are in two dimensions. It is found that, compared to the unrestricted flow, the symmetry conditions lead to a significant delay in the onset of absolute as well as convective instability. In addition, the onset of absolute instability is greatly affected by the location of the lateral boundaries and shows a non-monotonic variation. Unlike the unrestricted flow, which is associated with von Kármán vortex shedding, the flow with centreline symmetry becomes unstable via modes that are associated with low-frequency large-scale structures. These lead to expansion and contraction of the wake bubble and are similar in characteristics to the low-frequency oscillations reported earlier in the literature. A global linear convective stability analysis is utilized to find the most unstable modes for different speeds of the disturbance. Three kinds of convectively unstable modes are identified. The ones travelling at very low streamwise speed are associated with large-scale structures and relatively low frequency. Shear layer instability, with relatively smaller scale flow structures and higher frequency, is encountered for disturbances travelling at relatively larger speed. For low blockage a new type of instability is found. It travels at relatively high speed and resembles a swirling flow structure. As opposed to the absolute instability, the convective instability appears at much lowerReand its onset is affected very little by the location of the lateral boundaries. Analysis is also carried out for determining the convective stability of disturbances that travel in directions other than along the free stream. It is found that the most unstable disturbances are not necessarily the purely streamwise travelling ones. Disturbances that move purely in the cross-stream direction can also be convectively unstable. The results from the linear stability analysis are confirmed by carrying out direct time integration of the linearized disturbance equations. The disturbance field shows transient growth by several orders of magnitude confirming that such flows act as amplifiers. Direct time integration of the Navier–Stokes equation is carried out to track the time evolution of both the large-scale low-frequency oscillations and small-scale shear layer instabilities. The criticalRefor the onset of convective instability is compared with earlier results from local analysis. Good agreement is found.

Elastic effects on Rayleigh-Bénard convection in liquids with temperature-dependent viscosity

A linear stability analysis of convection in viscoelastic liquids with temperature-dependent viscosity is studied using normal modes and Galerkin method. Stationary convection is shown to be the preferred mode of instability when the ratio of strain retardation parameter to stress relaxation parameter is greater than unity. When the ratio is less than unity then the possibility of oscillatory convection is shown to arise. Oscillatory convection is studied numerically for Rivlin-Ericksen, Maxwell and Jeffreys liquids by considering free-free, rigid-rigid and rigid-free isothermal/adiabatic boundaries. The effect of variable viscosity parameter is shown to destabilize the system. The problem reveals the stabilizing nature of strain retardation parameter and destabilizing nature of stress relaxation parameter, on the onset of convection. The Maxwell liquids are found to be more unstable than the one subscribing to Jeffreys description whereas the Rivlin-Ericksen liquid is comparatively more stable. Free-free adiabatic boundary combination is found to give rise to a most unstable system, whereas the rigid isothermal rigid adiabatic combination gives rise to a most stable system. The problem has applications in non-isothermal systems having viscoelastic liquids as working media.

Stability Analysis of Convection & Diffusion equation

The Stability Analysis of Convection& Diffusion equation by using Fourier mode Stability analysis in two cases has been considered , the first one when the amplitude is constant and the second one when the amplitude is variable . In the first case, the solution is always stable and in the second case the solution is conditionally stable .

Convective stability of multicomponent fluids in the thermogravitational column

A comprehensive linear stability analysis of convection in the thermogravitational column is first performed for multicomponent fluids. Two types of perturbations are investigated: Longitudinal waves propagating in vertical direction of the column and transversal waves propagating perpendicular to the vertical axis and temperature gradient. The stability problems are reduced to those without cross-diffusion effect by a special transformation. The calculations are performed for binary and ternary mixtures by the Galerkin method. It is found that in binary fluids, the onset of longitudinal instability can be monotonic or oscillatory depending on the separation ratio, which characterizes the Soret effect. The difference between stability characteristics of binary and ternary fluids is associated with different diffusion times of components in a ternary system. It is shown that the mechanism of transversal instability is related to the unstable density stratification in the column (in total or due to individual components). The unstable stratification can only be realized in fluids with negative Soret effect. The analogue of exchange of stabilities principle for a plane column with a multicomponent fluid is proved. The obtained results indicate that the thermogravitational column can be used for measuring diffusion and thermal diffusion coefficients in ternary and higher mixtures with one or several components having negative Soret effect.

Linear stability analysis of convection in two-layer system with an evaporating vapor-liquid interface

Classical theories have successfully provided an explanation for convection in a liquid layer heated from below without evaporation. However, these theories are inadequate to account for the convective instabilities in an evaporating liquid layer, especially in the case when it is cooled from below. In the present paper, we study the onset of Marangoni convection in a liquid layer being overlain by a vapor layer. A new two-sided model is put forward instead of the one-sided model in previous studies. Marangoni-Bénard instabilities in evaporating liquid thin layers are investigated with a linear instability analysis. We define a new evaporation Biot number, which is different from that in previous studies and discuss the influences of reference evaporating velocity and evaporation Biot number on the vapor-liquid system. At the end, we explain why the instability occurs even when an evaporating liquid layer is cooled from below.