Oscillatory Mode Of Convection Research Articles

Marginal stability analyses for thermochemical convection and its implications for the dynamics of continental lithosphere and core–mantle boundary regions

SUMMARY Significant compositional differences may exist in the lithospheric mantle and above the core–mantle boundary (CMB) relative to the ambient mantle. The intrinsic density differences may affect the development of thermal boundary layer (TBL) instabilities associated with lithospheric delamination and formation of thermochemical plumes. In this study, we explored the instability of two-layer thermochemical fluid using two different techniques: marginal stability analysis with a propagator-matrix method and finite element modelling. We investigated both the instabilities in lithospheric mantle (i.e. lithospheric instability) and the mantle above the CMB (i.e. plume-forming instability) using a background temperature Tbg(z) with the TBL. For lithospheric instability, we found that two-layer fluid with free-slip boundary conditions mainly undergoes the same three different convective modes (i.e. two oscillatory convection modes and one layered convection regime) as that with no-slip boundary conditions reported in Jaupart et al. However, with free-slip boundary conditions, the transitions between these convection modes occur at larger values of buoyancy number B. Free-slip boundary conditions lead to smaller critical Rayleigh number Rac, but larger convective wavelength and oscillation frequency ωc, compared with those with no-slip boundary conditions. Our numerical modelling results demonstrate that Rac and ωc predicted from the classical marginal stability analyses using Tbg(z) with TBL temperature may have significant errors when the oscillatory period is comparable with or larger than the timescale of lithospheric thermal diffusion that causes Tbg(z) to vary with time significantly. In this case, using a more gently sloped background temperature profile ignoring the TBL temperature, the stability analysis predicts more accurate stability conditions, thus presenting an effective remedy to the stability analysis. For plume-forming instability, because of the reduced viscosity in the hot and compositionally dense bottom layer, the transition to the layered convection occurs at significantly smaller B values, and in the oscillatory convection regime, Rac is larger but ωc is smaller, compared with those for lithospheric instability. Finally, our study provides a successful benchmark of numerical models of thermochemical convection by comparing Rac and ωc from numerical models with those from the marginal stability analysis.

Non-radial oscillations mimicking a brown dwarf orbiting the cluster giant NGC 4349 No. 127

Context. Several evolved stars have been found to exhibit long-period radial velocity variations that cannot be explained by planetary or brown dwarf companions. Non-radial oscillations caused by oscillatory convective modes have been put forth as an alternative explanation, but no modeling attempt has yet been undertaken. Aims. We provide a model of a non-radial oscillation, aiming to explain the observed variations of the cluster giant NGC 4349 No. 127. The star was previously reported to host a brown dwarf companion, but whose existence was later refuted in the literature. Methods. We reanalyzed 58 archival HARPS spectra of the intermediate-mass giant NGC 4349 No. 127. We reduced the spectra using the SERVAL and RACCOON pipelines, acquiring additional activity indicators. We searched for periodicity in the indicators and correlations between the indicators and radial velocities. We further present a simulation code able to produce synthetic HARPS spectra, incorporating the effect of non-radial oscillations, and compare the simulated results to the observed variations. We discuss the possibility that non-radial oscillations cause the observed variations. Results. We find a positive correlation between chromatic index and radial velocity, along with closed-loop Lissajous-like correlations between radial velocity and each of the spectral line shape indicators (full width at half maximum, and contrast of the cross-correlation function and differential line width). Simulations of a low-amplitude, retrograde, dipole (l = 1, m = 1), non-radial oscillation can reproduce the observed behavior and explain the observables. Photometric variations below the detection threshold of the available ASAS-3 photometry are predicted. The oscillation and stellar parameters are largely in agreement with the prediction of oscillatory convective modes. Conclusions. The periodic variations of the radial velocities and activity indicators, along with the respective phase shifts, measured for the intermediate-mass cluster giant NGC 4349 No. 127, can be explained by a non-radial oscillation.

Конвективные режимы псевдопластической жидкости в квадратной полости при воздействии высокочастотных вибраций в условиях пониженной гравитации

In this paper, we investigate convective modes of pseudoplastic fluid in a square cavity with solid perfectly heat-conducting boundaries under reduced gravity. The cavity performs vertical linearly polarized high-frequency vibrations. Between the vertical walls of the cavity a temperature drop is prescribed in the direction perpendicular to vibrations and the field of gravity. The fluid rheology is described using the Williamson model. The problem is solved based on the averaged thermovibration convection equations for nonlinear viscous fluids. The vibration intensity is determined by the vibration parameter V, which is proportional to the ratio of the amplitude of the vibration acceleration to the free-fall acceleration and does not depend on the temperature difference. The problem is found to have two types of solutions, which we call Newtonian and non-Newtonian modes. These solutions correspond to different convective structures, for which the dependences of the maximum of the stream function and the Nusselt number on the Grashof number are derived. We use these dependences for determining at different values of rheological parameters the threshold values of the Grasgof numbers, corresponding to the change in the modes of stationary averaged convection and the critical Grasgof numbers, corresponding to the loss of stability of stationary averaged flow and the occurrence of oscillatory modes of averaged convection. The structures of different modes of averaged stationary and oscillatory convection are studied. The results obtained for the Newtonian mode have shown that at small values of the Grasgoff number, a slow one-vortex stationary convective flow is realized in the cavity. With increasing the Grasgoff number it transforms into a three-vortex flow. A further increase in the Grasgoff number results in the appearance of a four-vortex convective flow, which eventually transforms again into a three-vortex motion. At even greater Gr, the stationary averaged motion becomes unstable and the oscillatory modes appear in the cavity. For the non-Newtonian mode, a stationary convective flow is the basic flow pattern detected in the cavity; the oscillatory modes are not observed in the whole investigated range of the Grasgoff numbers.

Throughflow effect on bi-disperse convection in Rivlin-Ericksen fluid

In this investigation, we delve into the influence of throughflow on the phenomenon of bi-disperse convection within Rivlin-Ericksen fluid. In the context of examining bi-disperse convection within this specific type of fluid, the throughflow effect is considered to have a uniform vertical distribution. The primary focus of this study centers on evaluating the system’s linear stability. To achieve this, we employ normal mode analysis to compute the Darcy-Rayleigh number at the onset of convection. This Darcy-Rayleigh number is computed for both stationary and oscillatory convection modes. Furthermore, we conduct a comprehensive analysis and present the results in graphical form to illustrate the impact of various parameters, including Peclet number and kinematic viscoelastic parameter, on both stationary and oscillatory convection. Our research findings demonstrate that when Peclet number Pr1 < 0, it leads to destabilising effect on both stationary and oscillatory convections. Conversely, when Peclet number Pr1 > 0, it induces stabilising effect on both stationary as well as oscillatory convections.

Возбуждение релаксационных колебаний на искривленной межфазной границе в условиях внутренней задачи

The oscillatory mode of solutal Marangoni convection during the absorption of a surfactant from a homogeneous external solution into a water droplet is studied numerically. This is caused by the effect of gravity, which promotes the sedimentation of surfactant molecules in an aqueous medium. This version of oscillatory convection arising under the conditions of an internal problem was recently discovered experimentally. In the present paper, we consider the case of a chemically inert system, in which there are no reactions. The effects of interfacial deformation are assumed to be insignificant and thus they are neglected. The mathematical model includes the Navier—Stokes equations written in the Hele-Shaw and Boussinesq approximations, and the equations of surfactant transport in the system. We assume that the characteristic time of surfactant adsorption is shorter than the time of its diffusion in both solutions, which makes it possible to ignore the formation of a surface phase. The boundary value problem includes the equilibrium condition of the system, which takes into account different values of the chemical potential in the phases. It is shown that a water droplet is a surfactant accumulator that diffuses from the organic phase. The problem is solved in dimensional form using the COMSOL Multiphysics package and based on a set of physical constants for acetic acid which, like many other members of the carboxylic acid family, has the properties of surfactant in water. It was found that direct numerical simulation of the system is able to reproduce the relaxation oscillations observed in the experiment only under the additional phenomenological assumption of non-Newtonian rheology of the interface, which was previously proposed for the external problem. The physical mechanism which may be responsible for the delayed onset of Marangoni instability is discussed. We demonstrate that periodic oscillations are generated inside the drop due to the competition between the Marangoni effect and the gravity-dependent convective instability of the solution. Using direct numerical simulation, we identified the structures of convective motion at the interface and in its neighborhood, determined the flow intensity as a function of time, and obtained the range of change in the oscillation period.

STUDY OF GLOBAL STABILITY OF ROTATING PARTIALLY IONIZED PLASMA SATURATING A POROUS MEDIUM

The importance of thermal convection in rotating partially ionized plasma has been observed in various laboratory and astrophysical plasmas. The focus of this work is on the investigation of the effect of rotation on the thermal convection of partially ionized plasma within a porous medium by using nonlinear and linear analyzes. For porous medium, the Darcy-Brinkman model has been used. The eigenvalue problems for linear and nonlinear analyzes have been developed using the normal mode method and energy method, respectively. For numerical analysis, the Galerkin-weighted residual method has been employed to determine the Rayleigh-Darcy number. The effects of rotation, medium permeability, compressibility, and collisional frequency have been observed on the stability of the system. It has been found that the subcritical region does not exist, and hence global stability prevails. The rotation is found to induce oscillatory modes of convection. Rotation, medium permeability, and compressibility are found to delay the onset of convection. The collisional frequency doesn't influence the stability of the system for stationary convection; however, it does influence energy decay and oscillatory convection. All the findings of our study have been discussed and presented graphically.

Internal heat source effects on thermosolutal convection of Casson nanofluids embedded by Darcy-Brinkman model

The current study investigates Darcy-Brinkman convective instability for a non-Newtonian binary nanofluid layer in the presence of internal heat source. Non-Newtonian fluid behavior is characterized by Casson model. Normal mode technique is employed to convert the relevant PDEs into ODEs and Boussinesq approximation is used. The phenomenon is explored for both stationary as well as oscillatory modes of convection. The novelty and significance of the work lie in the fact that oscillatory motions have come into existence due to the presence of internal heat source term in the thermal energy equation which were non-existent otherwise for top-heavy configuration of nanoparticles. The impact of various nanofluid and solute parameters except solute Rayleigh number is found to destabilize the nanofluid layer while solute Rayleigh number, Darcy number and porosity parameters are found to postpone the onset of convection. Non-Newtonian Casson parameter and internal heat source strength are found to hasten the convection for the present arrangement of nanoparticles.

Onset of triply diffusive thermo-bio-convection in the presence of gyrotactic microorganisms and internal heating into an anisotropic porous medium: Oscillatory convection

This paper accentuates different aspects of an important and novel triply-diffusive thermo-bio-convection instability problem in the presence of inconsistent internal heating and the gyrotactic microorganism species into an anisotropic porous fluid layer. This instability study may be important from the application point of view in microfluidic devices, enhanced oil recovery, antibiotics, and many other areas. The normal mode analysis and linear stability analysis along with the weighted Galerkin method are used to establish the instability criteria in terms of the thermal Rayleigh number for both the oscillatory and non-oscillatory modes of convection as a secular equation among prominent parameters used in this study. It is perceived that the presence of internal heating and gyrotactic microorganism species destabilize the suspension and fast forward the triply-diffusive thermo-bio-convection. It is also established that the increment of mechanical anisotropy hastens the convection and advances the heat transfer in the porous medium while mounting thermal anisotropy and the concentration of solutes constitute a stable system that postpones the onset of triply-diffusive thermo-bio-convection.

Weak Nonlinear Oscillatory Double Diffusive Convection in a Viscoelastic Fluid-Saturated Porous Layer Under Gravity Modulation

A thermal instability caused by buoyancy force is investigated in an initially quiescent indefinitely extended horizontal porous layer saturated with non - Newtonian fluid. The characteristics of fluid motion are explained by using Modified Darcy’s law. Here, we considered the time periodic gravity field, and its effect on the system has been investigated. For the oscillatory mode of convection, a weakly non-linear stability analysis has been performed to evaluate Heat and Mass transfers in terms of the Nusselt number and Sherwood Number. To compute the results, the complex non-autonomous Ginzburg – Landau equation is used. It has been studied how viscoelastic fluid relaxation and retardation times impacts on heat and mass transmission. Further, the research confirms that heat and mass transfer can be successfully regulates by a technique that is external to the system. Finally, it has been discovered that excessive stability delays the onset of convection, hence increasing in the heat and mass transmission.

Nanofluid gravity-driven oscillatory mode of convection in a porous medium

Nanofluid gravity-driven oscillatory mode of convection in a porous medium

Nonlinear stability analysis for the rheology of Oldroyd-B nanofluids embedded by Darcy–Brinkman porous media using a two-phase model

An alternative to the Darcy model called the Darcy–Brinkman is employed to investigate a linear and nonlinear analysis of thermal convection in porous media saturated by Oldroyd-B nanofluids. The two-phase model is utilized for a nanofluid which is based on the theory of nanoparticles–fluid relative velocity. The analytical expressions for oscillatory and stationary thermal Rayleigh numbers have been derived using the Galerkin scheme and normal mode technique to study the impacts of viscoelastic and other non-dimensional parameters governing the stability of the system. Nonlinear analysis is made using minimal double Fourier series with a weekly period. Both time-dependent and independent variations of the Nusselt number and Sherwood number for emerging parameters are represented to study heat and mass transport. The novelty of the paper lies in the fact that due to the viscoelasticity of the fluid oscillatory mode of convection came into existence for top-heavy nanoparticle arrangement which was otherwise not possible for nanofluids with Newtonian base fluids. The results of the mathematical model revealed that heat and mass transfer is enhanced with the higher values of viscoelastic parameters, namely stress relaxation and strain retardation numbers. The increasing Darcy number is observed to have substantial control over the stability as it inhibits heat and mass transfer rate. Graphical interpretations reveal that viscoelastic non-Newtonian nanofluids have higher heat and mass transfer rates than Newtonian nanofluids for the system under consideration.

Modified stability analysis of double‐diffusive convection in viscoelastic fluid layer saturating porous media

AbstractThe present analysis mathematically investigates the thermohaline convection problem in viscoelastic fluid layer saturating porous media by utilizing the modified Boussinesq approximation. By performing linear stability analysis, the Darcy–Rayleigh numbers for stationary and oscillatory modes of convection are derived. The effects of different parameters describing the problem are studied numerically. In nonlinear stability analysis, the heat and mass transfer rates in the form of Nusselt and Sherwood numbers, respectively, are obtained for oscillatory convection using the derived Ginzburg–Landau equation. From the results, it is observed that overstability is the preferred mode of instability in linear stability. It is found that in linear double‐diffusive convection problems, the stress relaxation imparts a destabilizing effect whereas the strain retardation time, the coefficient of specific heat variation due to temperature, and the concentration gradient have a stabilizing effect on the system's stability. The numerical values of heat and mass transfer rates varied with the coefficient of specific heat showing that the heat transport decreases while the mass transport increases. Also, the stress relaxation time, the concentration gradient, and the gravity modulation's amplitude increase while the strain retardation time decreases the heat and mass transfer rates. The wavelength of oscillations remains unaltered with the variation of specific heat variation due to temperature. The modulation frequency does not affect the heat/mass transfer rate; though, the wavelength of oscillations decreases with increasing frequency.

Gravitational Modulation Effect on Double-Diffusive Oscillatory Convection in a Viscoelastic Fluid Layer

The complex Ginzburg-Landau model has been employed to derive the temporal amplitude of convection in a viscoelastic fluid layer under gravity modulation. The finite amplitude is based on a weakly nonlinear thermal convection. The perturbation technique is applied to simplify the nonlinear system of partial differential equations into a non-autonomous amplitude equation. Heat and mass transfer obtained in terms of the Nusselt and Sherwood numbers, and the corresponding results are presented for the small amplitude of modulation. The moderate values of amplitude and frequencies of gravity modulation are considered, their effect is to enhance the heat and mass transfer in the layer. Thus, the modulation plays dual role in heat and mass transfer. The variations of Nusselt and Sherwood numbers with slow time becomes rapid on either increasing the parameters Rs, Pr, λ, δ or decreasing the parameters Γ, ε, Ω. Each parameter effect has been presented graphically with their effect on Nu and Sh. Further, it is found that an oscillatory mode of convection enhances the heat and mass transfer than a stationary mode.

Electroconvection of a weakly conducting liquid subjected to unipolar injection and heated from above

The problem of a weakly conducting viscous incompressible liquid placed in an infinite horizontal capacitor is solved numerically by the two-field method using finite difference schemes. It is assumed that a free charge arises in the liquid (heated from above) due to a homogeneous unipolar autonomous injection from the cathode. In mechanical equilibrium, the Coulomb force acting on the charge injected from the cathode and the buoyancy force are oppositely directed, and the oscillatory instability and wave supercritical modes of convection occur. The problem is investigated in full statement, i.e., the electric field redistribution created by the charge redistribution is taken into account. Periodic boundary conditions are used on the lateral boundaries of the studied area, which makes it possible to detect and analyze not only the mode of stationary electroconvection, but also the solutions in the form of traveling waves. A mixed convection regime that arises as a result of a direct Hopf bifurcation from the state of mechanical equilibrium was found and investigated. In this regime, the phases of standing and traveling waves happen one after another. Traveling waves, modulated traveling waves and stationary convection successively replace each other with an increase in the control parameter (the electric Rayleigh number) proportional to the voltage on the capacitor plates. A bifurcation diagram that characterizes the intensity and phase velocity of supercritical fluid flow modes is plotted. The intensity of the stationary electroconvective flow is an order of magnitude higher than the intensity of the flow in the traveling wave mode. The influence of the grid scale on the observed regimes is studied.

The Horton–Rogers–Lapwood problem in a Jeffrey fluid influenced by a vertical magnetic field

In this work, the impact of a magnetic field on the onset of the Jeffrey fluid convection through a porous medium is investigated theoretically. The layer of Jeffrey fluid is heated from below and is operated by a consistent upright magnetic field. Using the normal mode procedure, a dispersion equation is obtained analytically and this dispersion relation is utilized to derive the critical conditions for the onset of stationary and oscillatory patterns of convection. The results reveal that the stability of the system diminished with the augmentation of the Jeffrey parameter, while an opposite result is obtained with magnetic field parameters (magnetic Chandrasekhar–Darcy number and magnetic Prandtl number). The size of convective cells decreases with Jeffrey and magnetic field parameters. It is also found that the existence of a magnetic field indicates the possibility of the survival of the oscillatory mode of convection.

Triple diffusive convection in a viscoelastic Oldroyd-B fluid layer

The stability of a triply diffusive viscoelastic fluid layer in which the fluid density depends on three stratifying agencies possessing different diffusivities is investigated. The viscoelastic fluid is modeled by means of the Oldroyd-B constitutive equation. Analytical expressions are obtained for steady and oscillatory onset by carrying out the linear instability analysis and the crossover boundary between them is demarcated by identifying a codimension-two point in the viscoelastic parameters plane. The occurrence of disconnected closed oscillatory neutral curve lying well below the stationary neutral curve is established for some values of governing parameters indicating the requirement of three critical values of thermal Rayleigh number to specify the linear instability criteria. However, the possibility of quasiperiodic bifurcation from the motionless basic state is not perceived and this is in contradiction to the case of inelastic couple stress and Newtonian fluids. The corresponding weakly nonlinear stability of stationary and oscillatory modes has been carried out using a perturbation method. The cubic Landau equations are derived and the stability of bifurcating solution is discussed. The viscoelastic parameters influence the stability of stationary bifurcation despite their effect is not felt on the stationary onset. The stationary and oscillatory finite amplitude solution is found to bifurcate either subcritical or supercritical depending on the choice of governing parameters. The effect of Prandtl number and viscoelastic parameters on stationary and oscillatory convection modes of heat and mass transfer is analyzed.

An overstability analysis of vertically vibrated suspension of active swimmers subjected to thermal stratification

The present article focuses on the analytical approach to discuss the thermo–vibrational convection in a suspension of the active (gyrotactic) swimmers. The onset of instability criterion is investigated for the stationary and oscillatory modes of convection in a shallow fluid layer with no–slip and rigid–free walls. The eigenvalue problem is tackled by Galerkin scheme to get the desired stability diagram and the correlation between the critical Rayleigh numbers. The overstability in suspension is possible when the unstable density gradient of the gyrotactic particles is opposed by the density variation due to thermo–vibrational influence. The suspension is destabilized due to gyrotactic up–swimming while the increase in Péclet number stabilizes the system. The stabilizing influence of vertical vibration is considerably affected due to thermal gradient which destabilizes the suspension. An interesting result of this study is the influence of thermo–vibrational parameter which is associated with applied thermal and vibrational properties. We reported that the destabilizing nature of thermo–vibrational parameter becomes thermally or vibrationally governed when the suspension is heated or cooled from below. When compared to the rigid–rigid boundaries, the displayed profiles for rigid–free walls yielded less stableness in the suspension.

Magnetic convection in a nonuniformly rotating electrically conductive medium in an external spiral magnetic field

The research is devoted to the stability of convective flow in a nonuniformly rotating layer of an electrically conducting fluid in a spiral magnetic field. The stationary and oscillatory modes of magnetic convection are considered depending on the profile of the angular rotation velocity (Rossby number Ro) and on the profile of the external azimuthal magnetic field (magnetic Rossby number Rb). The non-linear dynamic system of Lorentz type equations is obtained by using the Galerkin method. Numerical analysis of these equations has shown the presence of chaotic behavior of convective flows. The criteria of the occurrence of chaotic movements are found. It depends on the parameters of convection: dimensionless numbers of Rayleigh Ra, Chandrasekhar Q, Taylor Ta, and external azimuthal magnetic field with the Rossby magnetic number Rb for Rayleigh (Ro = −1) and Kepler profiles of the angular rotation velocity of the medium.

Oscillatory thermal–inertial flows in liquid metal rotating convection

Abstract

Couple stress effects on the stability of three‐component convection‐diffusion in a porous layer

AbstractIn this study, the stability (linear and weak nonlinear) of triple‐diffusive convection in a couple stress fluid‐saturated porous layer has been studied. A normal mode analysis yields an exact dispersion equation of fourth degree, and the criterion for the onset of stationary and oscillatory convection is obtained accordingly. The numerical computations are carried out for diffusivity elements experimentally determined for an aqueousNaCl‐KCl‐Sucrosesystem. Contrary to the double‐diffusive couple stress fluid system, it is found that (i) oscillatory convection occurs even if the diffusivity ratios are greater than unity and (ii) a disconnected heart‐shaped oscillatory neutral curve exists for certain choices of physical parameters, demonstrating the requirement of three critical values of Darcy‐Rayleigh number to specify the linear instability criteria. The impact of a couple stress parameter on some of these unusual behaviors is emphasized. The cubic Landau equations are derived by performing a weak nonlinear stability analysis and the stability of stationary and oscillatory bifurcating solutions is discussed. The heat and mass transfer for stationary and oscillatory convection modes is presented, and the influence of the couple stress parameter on the same is analyzed.