Oscillatory Modes Of Instability Research Articles

Soret-induced convection of ternary fluid in a horizontal porous layer heated from below

The present paper deals with the investigation of the Soret-induced convection of a three-component mixture of hydrocarbons in a horizontal porous layer. This problem is important for geological applications. The first part of the paper is devoted to the linear stability of the conductive state. The longwave instability is studied analytically by the expansion into the power series with respect to the wave number. A new long-wave oscillatory instability mode existing at negative separation ratios is found out. It is shown that this mode is more dangerous than the long-wave monotonic instability mode in the entire range of its existence. The instability to the perturbations with nonzero wave numbers is studied numerically by the shooting method. Stability map is obtained. The results confirm the predictions of the longwave analysis. In the second part of the paper, the nonlinear convection regimes are studied by the finite difference method. The calculations give the results consistent with the linear stability analysis. It is found that the primary bifurcation as a result of which the conductive state losses its stability is supercritical. The transformations of the convective flow structure with the change in the Rayleigh–Darcy number are accompanied by the hysteresis phenomena.

Oscillatory Porous Medium Ferroconvection in a Viscoelastic Magnetic Fluid with Non-Classical Heat Conduction

The classical stability analysis is used to examine the combined effect of viscoelasticity and the second sound on the onset of porous medium ferroconvection. The fluid and solid matrix are assumed to be in local thermal equilibrium. Considering the boundary conditions appropriate for an analytical approach, the critical values pertaining to both stationary and oscillatory instabilities are obtained by means of the normal mode analysis. It is observed that the oscillatory mode of instability is preferred to the stationary mode of instability. It is shown that the oscillatory porous medium ferroconvection is advanced through the magnetic forces, nonlinearity in magnetization, stress relaxation due to viscoelasticity, and the second sound. On the other hand, it is observed that the presence of strain retardation and porous medium delays the onset of oscillatory porous medium ferroconvection. The dual nature of the Prandtl number on the Rayleigh number with respect to the Cattaneo number is also delineated. The effect of various parameters on the size of the convection cell and the frequency of oscillations is also discussed. This problem may have possible implications for technological applications wherein viscoelastic magnetic fluids are involved.

Predicting the effect of inertia, rotation, and magnetic field on the onset of convection in a bidispersive porous medium using machine learning techniques

Effects of the magnetic field and inertia on the onset of thermal convection in a horizontal bidispersive porous layer, rotating about a vertical axis, are analyzed. The Darcy equation with same temperature in the micro- and macrophases is used to characterize the fluid motion. The Vadasz number is taken into account in a generalized Darcy equation for the macrophase. The eigenvalue problem obtained from the linear stability analysis is solved analytically for free–free boundaries. Moving one step further from the traditional linear stability analysis, machine learning tools are introduced in this paper to include the effect of multiple parameters on the marginal state of the system. Machine learning techniques have been implemented to identify the mode of instability with respect to different parameters. In particular, classification algorithms, namely, Artificial Neural Networks (ANN) and Support vector machine, are used to examine the onset of oscillatory convection and stationary convection. The required data for training of the algorithms are generated from the results of linear stability analysis. It is found that ANN with the sufficient number of hidden layers along with good choice of training dataset can predict the mode of instability even on the small variation in a given parameter. The combined effect of rotation, magnetic field, and inertia is to reduce the oscillatory mode of instability; hence, the system exhibits the steady mode of instability for a significant region in the three dimensional space comprising the Taylor number, the Hartman number, and the Vadasz number.

Linear stability analysis of horizontal convection under a Gay-Lussac type approximation

A linear stability analysis is conducted for horizontal natural convection under a Gay-Lussac (GL) type approximation in a relatively shallow enclosure cavity. The GL type approximation is developed based on extending density variations to the advection term as well as gravity term through the momentum equation. Such a treatment invokes the GL parameter (Ga=βΔθ) as the non-Boussinesq parameter with a physical value ranging 0≤Ga≤2, characterising deviation from the classic Boussinesq approximation. Results are compared against the Boussinesq approximation in terms of the Nusselt number and skin friction. Extreme values of Ga are found to produce a counter-rotating convection cell at the hot end of the enclosure at higher Rayleigh numbers - a feature absent from Boussinesq natural convection modeling. For stability analysis purposes, linearized perturbation equations under the GL type approximation are derived and solved to characterise the breakdown of the steady two-dimensional solution to infinitesimal three-dimensional disturbances. Stability results predict that the flow remains stable up to Racr1=6.46×108 for the Boussinesq case (Ga=0), and then with increasing Ga the flow briefly stabilises to Ga≅0.2, then becomes progressively more unstable with futher increases in Ga, yielding a critical Rayleigh number Racr2=4.23×108 at Gamax=2. Three-dimensional transition is predicted to be via an oscillatory instability mode of the steady base flow having a spanwise wavelength that increases as Rayleigh number increases. 3D-DNS simulations verify the linear stability analysis predictions in terms of growth rate, and elucidate the mode shapes achieved at saturation.

Oscillatory porous medium ferroconvection with Maxwell-Cattaneo law of heat conduction

The scheme of small perturbation is used to address the problem of buoyancy driven convection of Darcy-Brinkman type in a ferromagnetic fluid invoking the Maxwell-Cattaneo law. An analytical solution of the eigenvalue problem involved, encompassing stationary and overstable modes, is obtained by adopting simplified boundary conditions. The mathematical application package MATHEMATICA is adopted to determine the eigenvalue expressions and the critical numbers. It is established that the threshold of Darcy-Brinkman ferroconvection is amplified through the stresses of magnetic and second sound mechanisms and the opposite influence is found to be true due to the presence of porous medium. It is delineated that both the critical frequency of oscillations and aspect ratio of cells of convective heat transfer are susceptible to the different parameters of the study. It is also shown that, as long as the Cattaneo and Prandtl numbers are pretty high, the oscillatory mode of instability is preferred to the stationary mode of ferroconvection.

Stability of slowly evaporating thin liquid films of binary mixtures

We consider the evaporation of a thin liquid layer which consists of a binary mixture of volatile liquids. The mixture is on top of a heated substrate and in contact with the gas phase that consists of the same vapour as the binary mixture. The effect of thermocapillarity, solutocapillarity and the van der Waals interactions are considered. We derive the long-wave evolution equations for the free interface and the volume fraction that govern the two-dimensional stability of the layer subject to the above coupled mechanisms and perform a linear stability analysis. Our results demonstrate two modes of instabilities, a monotonic instability mode and an oscillatory instability mode. We supplement our results from stability analysis with transient simulations to examine the dynamics in the nonlinear regime and analyse how these instabilities evolve with time. More precisely we discuss how the effect of relative volatility along with the competition between thermal and solutal Marangoni effect defines the mode of instability that develops during the evaporation of the liquid layer due to preferential evaporation of one of the components.

Effect of rotating magnetic field on instabilities of thermocapillary flow in a Czochralski silicon melt pool

The effect of the rotating magnetic field on the thermocapillary flow instabilities in a Czochralski silicon melt pool was numerically investigated by means of linear stability analysis. We considered three cases, namely, the non-rotating crystal, the crystal rotating in the same direction as the magnetic field (co-rotating crystal), and the crystal rotating in the opposite direction of the magnetic field (counter-rotating crystal). The linear stability analysis results show that the basic flow becomes unstable against two oscillatory instability modes. With the increase of the magnetic Taylor number (Ta), the overall trend of the critical Marangoni number decreases in the cases of a non-rotating crystal and counter-rotating crystal. However, in the case of a co-rotating crystal, the rotating magnetic field significantly improves the flow stability, and an inversion of the propagation direction of the oscillatory wave is observed with the increase of Ta. Energy analyses reveal that the instabilities are hydrodynamic in nature, which are caused by the combination of the strain-induced instability mechanism and the centrifugal instability mechanism, and the rotating magnetic field only provides little energy for the flow instabilities even at large Ta.

Onset of Thermal Buoyancy Convection in a Two-Layer System with Deformable Interface and Fixed Heat Flux at the Boundaries under Terrestrial and Microgravity Conditions

The stability of a conductive state of a system of two horizontal layers of immiscible fluids of close densities with a fixed heat flux at external boundaries is studied. The problem is solved in the framework of the generalized Boussinesq approximation, which correctly takes into account the deformations of the interface. The onset of convection at heating from above or below is investigated. Main attention is paid to the effect of layer thickness ratio and gravity level on instability. Two long-wave instability modes related the presence of a deformable interface and to a fixed heat flux at the external boundaries are found out. They are analyzed analytically. The behavior of the system with respect to monotonic and oscillatory perturbations of finite wavelength is studied numerically. The complete stability map is obtained. It is found that for very different layer thicknesses, as well as for single layers with fixed heat flux at the boundaries, a long-wavelength instability is observed. The presence of a long-wave instability domain at close thicknesses of layers is also found, which is associated with a strong stabilization of finite-wavelength instability in this parameter range. In the case of Earth gravity, the monotonic perturbations are most dangerous at any ratios of layer thicknesses, the oscillatory mode of instability is responsible for the onset of convection only in microgravity conditions.

Effects of Pr and pool curvature on thermocapillary flow instabilities in annular pool

We linearly analyzed the effect of the Prandtl number (Pr) on the stability of thermocapillary flow in shallow annular pools with an aspect ratio Γ = (Ro - Ri)/d = 20 and two radius ratios ΓR = Ri/Ro= 0.50 and 0.98039, where d, Ro, and Ri are the liquid depth and the radii of the heated outer wall and the cooled inner wall, respectively. The results for Pr ∈ [0, 102] show that the steady axisymmetric thermocapillary flows in these annular pools become unstable against oscillatory instability modes, OSC1, OSC2 and hydrothermal wave (HTW). Two co-dimension-two bifurcations occur at Pr*1 and Pr*2. The critical mode for Pr ≤ Pr*1 is OSC2 with almost constant critical Reynolds number Rec, large wave number and high frequency. OSC1 with smaller wave number and lower frequency is the critical mode for Pr*1 ≤ Pr ≤ Pr*2 and Rec slightly decreases with increasing Pr. HTW is the critical mode for Pr ≥ Pr*2 and Rec decreases with increasing Pr. The values of (Pr*1, Pr*2) are (0.00953, 0.03054) for ΓR = 0.50, and (0.01919, 0.01946) for ΓR = 0.98039. Energy-budget analyses reveal that the instabilities in low-Pr range are caused by instabilities in the steady toroidal vortex (or vortices) near the cold wall.

Strong shear-flow modulation of instabilities in rapid directional solidification

We examine the effect of a strong shear flow on morphological instabilities that occur in the directional solidification of a dilute, binary alloy when the interface departs from local thermal equilibrium in a frozen-temperature, one-sided model. In particular, the flow velocity U∞ is much larger than the rate of solidification V and the Schmidt number is arbitrary. In contrast to solidification processes under small or no flow, for which both a cellular and an oscillatory mode of instability appear, the liquid-solid interface under flows of large magnitude is susceptible to a single mode of instability. All experiments on banding occur in the overlap region between these modes under no flow and since these two primary modes coalesce into one time-dependent, spatially-dependent mode, there is no mechanism for the production of bands anymore in the large-flow regime. No experiments have been conducted to date on this rapid solidification under flow and this suggests that such experiments would show no bands. The flow significantly stabilizes and selects higher wavenumbers for the preferred form of instability in comparison to systems involving small or no flow. The introduction of a strong flow provides an effective mechanism to eliminate instabilities at high solidification rates. However, stability thresholds at low solidification rates remain mainly indifferent to the presence of flow.

Strong Shear-Flow Modulation of Instabilities in Rapid Directional Solidification

We examine the effect of a strong shear flow on morphological instabilities that occur in the directional solidification of a dilute, binary alloy when the interface departs from local thermal equilibrium in a frozen-temperature, one-sided model. In particular, the flow velocity U∞ is much larger than the rate of solidification V and the Schmidt number is arbitrary. In contrast to solidification processes under small or no flow, for which both a cellular and an oscillatory mode of instability appear, the liquid-solid interface under flows of large magnitude is susceptible to a single mode of instability. All experiments on banding occur in the overlap region between these modes under no flow and since these two primary modes coalesce into one time-dependent, spatially-dependent mode, there is no mechanism for the production of bands anymore in the large-flow regime. No experiments have been conducted to date on this rapid solidification under flow and this suggests that such experiments would show no bands. The flow significantly stabilizes and selects higher wavenumbers for the preferred form of instability in comparison to systems involving small or no flow. The introduction of a strong flow provides an effective mechanism to eliminate instabilities at high solidification rates. However, stability thresholds at low solidification rates remain mainly indifferent to the presence of flow.

The influence of thermocapillary effect on the onset of convection in a two-layer system with deformable interface and perfectly conductive boundaries

The work is devoted to studying the excitation of the Rayleigh-Benard-Marangoni convection in a two-layer system of immiscible fluids with a deformable interface and rigid perfectly heat-conductive external boundaries. The influence of thermocapillary effect on the linear stability of the conductive state of the system under vertical temperature gradient is analyzed. The problem is solved in the framework of the generalized Boussinesq approximation, which allows us to correctly take into account the deformations of the interface. It has been found that, depending on the properties of fluids, a thermocapillary effect can either stabilize or destabilize the longwave mode of instability associated with the deformations of the interface. In the case of a two-layer system of formic acid and transformer oil, strong destabilization takes place. Numerical calculations performed for this pair of fluids at arbitrary values of wave number have shown that the influence of thermocapillary effect on finite-wavelength monotonic and oscillatory instability modes is also destabilizing. However, the lowering of instability threshold to oscillatory finite-wavelength perturbations due to the thermocapillary effect is much weaker than that for both longwave and finite-wavelength monotonic instability modes. As the result, at sufficiently large values of the Marangoni number the conductive state becomes unstable only with respect to monotonic perturbations. Further enhancement of the thermocapillary effect gives rise to a thermocapillary finite-wavelength oscillatory instability mode, which occurs at Galileo numbers large in absolute value (Ga<-80). Good agreement between the results of calculations and available experimental data has been obtained.

Oscillatory instability in a weakly non-Boussinesq fluid layer with a deformable surface

We investigate the oscillatory convective instability in a liquid layer with a deformable free surface. We provide linear stability analysis within a non-Boussinesq approach as our aim is to examine the influence of deformability of the free surface on the Rayleigh–Bénard–Marangoni instability. Within this approach, fluid is assumed to be isothermally incompressible; the density variations are accounted for in the continuity equation and in the buoyancy and inertial terms of the momentum equations. The numerical results show significant differences in stability behavior as compared to results obtained using the Boussinesq approach. Moreover, a novel oscillatory mode of instability is revealed for the zero Marangoni number and zero gravity. This result could not be obtained within the framework of the conventional Boussinesq approximation. Thorough investigation of the novel oscillatory mode let us propose the mechanism of this mode. It is connected with the capillary wave that enforced with thermal expansion of fluid. Weakly nonlinear analysis shows that supercritical branching of standing rolls is possible.

Marangoni instability in a thin film heated from below: Effect of nonmonotonic dependence of surface tension on temperature.

We investigate Marangoni instability in a thin liquid film resting on a substrate of low thermal conductivity and separated from the surrounding gas phase by a deformable free surface. Considering a nonmonotonic variation of surface tension with temperature, here we analytically derive the neutral stability curve for the monotonic and oscillatory modes of instability (for both the long-wave and short-wave perturbations) under the framework of linear stability analysis. For the long-wave instability, we derive a set of amplitude equations using the scaling k∼(Bi)^{1/2}, where k is the wave number and Bi is the Biot number. Through this investigation, we demonstrate that for such a fluid layer upon heating from below, both monotonic and oscillatory instability can appear for a certain range of the dimensionless parameters, viz., Biot number (Bi), Galileo number (Ga), and inverse capillary number (Σ). Moreover, we unveil, through this study, the influential role of the above-mentioned parameters on the stability of the system and identify the critical values of these parameters above which instability initiates in the liquid layer.

Stability of a stationary plane-parallel flow of a ternary fluid between two vertical plates maintained at constant different temperatures.

The linear stability of a steady convective flow of a ternary mixture placed between differently heated vertical rigid plates is studied. The applied temperature gradient induces concentration gradients due to the Soret effect. The analysis is done for the case when separation ratios of ternary mixture, i.e., Soret coefficients, have different signs but the net separation ratio is negative. The stability maps in terms of the Grashof number and net separation ratio are obtained and discussed for monotonic and oscillatory modes of instability. The previous results for long-wave instability of a binary mixture were recovered in the limit when one of the Soret coefficients tends to zero. For finite-wavelength perturbations the previous results were extended by discovering the oscillatory instability.

Parametric Excitation of Marangoni Instability in a Heated Thin Layer Covered by Insoluble Surfactant

The paper presents the analysis of the impact of vertical periodic vibrations on the long-wavelength Marangoni instability in a liquid layer with poorly conducting boundaries in the presence of insoluble surfactant on the deformable gas-liquid interface. The layer is subject to a uniform transverse temperature gradient. Linear stability analysis is performed in order to find critical values of Marangoni numbers for both monotonic and oscillatory instability modes. Longwave asymptotic expansions are used. At the leading order, the critical values are independent on vibration parameters; at the next order of approximation we obtained the rise of stability thresholds due to vibration.

Impact of an insoluble surfactant on the thresholds of evaporative Bénard-Marangoni instability under air.

It is known that the addition of an insoluble surfactant to a Bénard-Marangoni (BM) layer heated from below or cooled from above can give rise to a supplementary, oscillatory mode of instability. Here the objective is to see how exactly this plays out in the framework of a recently studied and experimentally tested case of a non-long-wavelength BM instability driven by diffusion-limited evaporation into air in isothermal surroundings. Linear stability analysis is accomplished within a now standard reduction to a one-sided model. In the absence of surfactant, we just recover the classical Pearson problem, albeit with an evaporation-specific wavenumber-dependent Biot number potentially attaining large values for strongly volatile liquids. Adding a surfactant not only sharply stabilizes the monotonic Pearson-like mode, but also leads to a more dangerous oscillatory mode, a parametric study of which is here undertaken. Although slanted towards the evaporative case, the present study is also of interest from the general viewpoint of the Pearson problem + an insoluble surfactant, the results for which are scarce in the literature without being obscured by further effects. In particular, an asymptotic analysis based on small interfacial Lewis numbers (diffusion coefficients) is undertaken near the codimension-2 point.

Magnetohydrodynamic stability of natural convection in a vertical porous slab

The stability of the conduction regime of natural convection in an electrically conducting fluid saturated porous vertical slab is investigated in the presence of a uniform external transverse magnetic field. The flow in the porous medium is described by modified Brinkman-extended Darcy equation with fluid viscosity different from effective viscosity. The boundaries of the vertical porous slab are assumed to be rigid-isothermal and electrically non-conducting. The resulting stability equations are solved numerically using Galerkin method. The critical Grashof number Gc, the critical wave number αc and the critical wave speed cc are computed for a wide range of porous parameter σp, the ratio of effective viscosity to the fluid viscosity Λ, the Prandtl number Pr and the Hartmann numberM. Based on these parameters, the stability characteristics of the system are discussed in detail. The presence of advective inertia is to instill instability on the flow in a porous medium and found that the magnetic field, porous parameter and ratio of viscosities have a stabilizing effect on both stationary and oscillatory wave instabilities. Besides, the value of Pr at which transition occurs from stationary to oscillatory mode of instability decreases with increasing M,σpandΛ.

Stability of homogeneous seepage of a liquid mixture through a closed region of the saturated porous medium in the presence of the solute immobilization

The linear stability problem of homogeneous vertical seepage of the liquid mixture through a closed region of porous medium is investigated. The problem is solved taking into account immobilization of the solute by the porous medium, which is described within the MIM approach based on the second-order kinetics model allowing for the saturation of the solid matrix of porous medium with immobilized solute. The stability of the homogeneous vertical seepage mode to two-dimensional perturbations has been considered for the case of square porous domain. It is assumed that the lower and upper boundaries are maintained at fixed different concentrations of solute and the lateral boundaries are impermeable to the mixture. It has been found that due to consideration for the solute immobilization the perturbation spectrum of the homogeneous vertical seepage mode exhibits oscillatory perturbations which are absent in the case of using the classical diffusion model. Neutral curves for the space of the respective parameters and plots of neutral perturbation frequency as the function of governing parameters have been obtained numerically. It has been shown that in the presence of the immobilization the stability of the examined seepage mode decreases as compared to the classical diffusion model. The numerical investigation has shown that the oscillatory mode of convection exists only in a certain range of parameters characterizing the intensity of sorption. The maps of oscillatory instability mode existence in the space of governing parameters have been obtained.

The effect of thermal boundary conditions on a stability of convective flow in a horizontal layer subjected to the longitudinal temperature gradient and acoustic wave

The effect of thermal boundary conditions on linear stability of stationary convective flow in a horizontal layer subjected to the longitudinal temperature gradient and propagating acoustic wave is studied. Two-dimensional shear instability and three-dimensional monotonic and oscillatory instabilities are analyzed. The stability maps for different acoustic Reynolds number values with respect to different perturbation types are obtained. The stabilization under the influence of acoustic wave at low Prandtl numbers is demonstrated. It takes place when two-dimensional shear or three-dimensional oscillatory instability modes are most dangerous. Three-dimensional monotonic mode is destabilized by acoustic influence. Energy analyses for two-dimensional and monotonic three-dimensional modes of instability is carried out.