Linear Instability Analysis Research Articles

EFFECT OF VISCOUS DISSIPATION AND INTERNAL HEAT SOURCE ON MONO-DIFFUSIVE THERMOCONVECTIVE STABILITY IN A HORIZONTAL POROUS MEDIUM LAYER

A mathematical model is developed for studying the onset of mono-diffusive convective fluid flow in a horizontal porous layer with temperature gradient, internal heat generation, and viscous dissipation effects. Darcy's model is used for the porous medium, which is considered to be isotropic and homogenous. A linear instability analysis is conducted, and transverse or longitudinal roll disturbances are examined. The dimensionless emerging eigenvalue problem is solved numerically with the Runge-Kutta and shooting methods for both cases of disturbances, i.e,. longitudinal and transverse rolls. Critical wave number and critical vertical thermal Rayleigh number <i>R<sub>z</sub></i> are identified. For higher values of Gebhart number, Ge, a significant destabilizing effect of Hadley-Prats flow is computed. Internal heat generation also strongly modifies the critical vertical Rayleigh number. Extensive interpretation of the solutions related to the onset of convection is provided. The study is relevant to geophysical flows and materials processing systems.

Linear instability analysis of a shear thinning gelled jet with yield stress in coaxial atomization.

This study aims to experimentally and theoretically investigate the roles of a threaded nozzle and yield stress in the coaxial air blast atomization of a gelled jet by means of rheological tests and high-speed flow visualization techniques. Similar to the breakup types of Newtonian fluids, a gelled jet will gradually exhibit an oscillation mode, membrane-type breakup, fiber-type breakup and a superpulsating submode as the airflow velocity increases. The transition conditions for each type are determined and plotted in the dimensionless map of We-Oh. For the jet spray angle, the maximum occurs when We = 600 and is clearly enlarged by the gas-side thread in the nozzle. The numerical results of the linear stability analysis indicate that a smaller dominant unstable growth rate is acquired by swelling the yield stress, while the higher dominant unstable growth rate and truncated wavenumber appear as the rotational airflow is acquired through the gas-side threaded nozzle.

A weakly nonlinear analysis of the effect of vertical throughflow on Darcy–Bénard convection

The aim of this paper is to analyze the effect of a downward vertical net mass flow on the type of instability that occurs in a horizontal fluid-saturated porous layer that is heated from below. The strength of the downflow is modelled by Pe, the Péclet number. First, we prove the validity of the principle of exchange of stabilities; consequently, we perform a linear instability analysis of the basic steady flow to determine the critical Darcy–Rayleigh number for the onset of steady convective instability as a function of Pe. Then, a weakly nonlinear stability analysis is performed to determine the smallest value of the Péclet number for which the onset of convection corresponds to a subcritical instability.

The combined effects of rotation and anisotropy on double diffusive bi-disperse convection

In the present paper, double-diffusive convection, taking into account Coriolis effects, in a horizontal layer of Brinkman-anisotropic bi-disperse porous medium is analysed. Via linear instability analysis, we find that convection can set in through stationary or oscillatory motions and the critical Rayleigh numbers for the onset of stationary secondary flow (steady convection) and overstability (oscillatory convection) are determined.

Sensitivity-aided active control of flow past twin cylinders

The flow-induced vibration (FIV) of twin structures always experiences significantly larger vibration amplitude than that of a single one, which causes severe structural safety problems. However, FIV control methods specifically for twin structures are still lacking. This study investigates the FIV and fluid forces of twin square cylinders under a jet flow control for a laminar flow. The control effects are studied for stationary and free-oscillating cylinders with a gap distance of L = 6.0D in co-shedding regimes (D is the side length of the cylinder). Global linear instability, adjoint method, and sensitive analysis are conducted to find a position where the jet flow should be placed. It was found that the most sensitive region is located in the near-wake region, and we thus set at jet flow at the rear of the cylinders. The control effects show that for stationary cylinders, UC achieves its maximum mean drag (CD) reduction of 11% when Uj > 0.3, meanwhile the CD of the downstream cylinder (DC) and fluctuating lift (CL') ≈ 0. UC plays a primary role in determining the control effects on fluid forces compared to DC, which coincides with the sensitivity analysis. For the free-oscillating case, much larger control energy is required to completely suppress vortex-induced vibration (VIV) with Uj > 2.4. In addition to the splitting effects of the jet flow, the neutralization effect of jet-induced vorticity was found plays a significant role in suppressing VIV, which can neutralize the vorticity generated from the side edge of the cylinders. High-order dynamic mode decomposition (HODMD) is also applied to uncover the underlying control mechanism. It demonstrates that the global energy of the dynamic mode energy is significantly decreased, meanwhile, the primary dynamic mode (i.e., the mode shape introducing VIV) is significantly weakened by the jet flow due to splitting and neutralizing effects.

Spreading or Contraction of an Axisymmetric Viscous Drop between Two Plates in a Rapidly Rotating Frame

We analyse the motions of a axisymmetric drop expanding between two rotating discs. We restrict to the case of a highly viscous fluid and a rapid rate of rotation. Therefore, we make modelling assumptions following from both a low Reynolds number and a low Rossby number. We investigate both the squeezing problem, where the top disc is pushed down on the drop; and the contraction problem, where the top plate is pulled away from the drop. Both problems have similar solutions to the non-rotating case but we find that the rotation term in the contraction problem allows a critical rotation rate that prevents the plates from moving apart. This exists because pressure in the fluid layer is lowered by the rotation and thus there is a suction effect between the two plates which promotes adhesion. We also complete the linear instability analysis of the squeezing problem and determine the critical values where the system shifts from symmetrical to asymmetrical.

Global linear stability analysis of a flame anchored to a cylinder

This study investigates the linear stability of a laminar premixed flame, anchored on a square cylinder and confined inside a channel. Many modern linear analysis concepts have been developed and validated around non-reacting bluff-body wake flows, and the objective of this paper is to explore whether those tools can be applied with the same success to the study of reacting flows in similar configurations. It is found that linear instability analysis of steady reacting flow states accurately predicts critical flow parameters for the onset of limit-cycle oscillations, when compared to direct numerical simulation performed with a simple one-step reaction scheme in the low Mach number limit. Furthermore, the linear analysis predicts a strong stabilising effect of flame ignition, consistent with documented experiments and numerical simulations. Instability in ignited wake flows is, however, found to set in at sufficiently high Reynolds number, and a linear wavemaker analysis characterises this instability as being driven by hydrodynamic mechanisms of a similar nature as in non-reacting wake flows. The frequency of nonlinear limit-cycle flame oscillations in this unstable regime is retrieved accurately by linear eigenmode analysis performed on the time-averaged mean flow, under the condition that the full set of the reacting flow equations is linearised. If, on the contrary, unsteadiness in the density and in the reaction rate are excluded from the linear model, then the congruence between linear and nonlinear dynamics is lost.

Electro-thermo-convection in a high Prandtl number fluid: Flow transition and heat transfer

This paper aims to study the electro-thermo-convection of a high Prandtl number fluid (silicone oil) in a rectangular cavity. A high accuracy numerical solver is undertaken to solve the problem. These numerical simulations are performed for a range of Rayleigh and electric Rayleigh parameters, Ra=0,104∼106 and T=0∼1000, which correspond to regimes from laminar to chaotic flow for different cases. Under the condition of weightlessness Ra=0, a strong mixing between heat and cold liquid caused by the electric field is observed, and there is a flow motion transition from large vortices to plume oscillation with the increased value of T. The large vortices motion has a better effencify on heat transfer enhancement. A dual solution at low and moderate Ra numbers is observed. The beingness of these two solutions is validated by a global linear instability analysis based on the linearized lattice Boltzmann method. With the help of some nonlinear analysis techniques, we find that the presence of the electric field may stabilize the thermal convection system. Furthermore, a full map of the heat transfer rate with respect to electric and thermal driving parameters is provided as a reference for applications. Heat transfer enhancement appears only when the electric field force is dominant. A neutral force curve about Ra−T is presented that can characterize the threshold of electric field force to achieve heat transfer enhancement for thermal convection at different Ra numbers.

Oblique Electromagnetic Instabilities Driven by Pickup Ion Ring-beam Distributions in the Outer Heliosheath. I. Linear Instability Analysis

The energetic neutral atom ribbon observed by the Interstellar Boundary Explorer spacecraft is believed to originate from the pickup ions in the outer heliosheath. The outer heliosheath pickup ions generally have a ring-beam velocity distribution at a certain pickup angle, α, the angle at which these ions are picked up by the interstellar magnetic field. The pickup ion ring-beam distributions can drive unstable waves of different propagation angles with respect to the background interstellar magnetic field, θ. Previous studies of the outer heliosheath pickup ion dynamics were mainly focused on ring-like pickup ion distributions with α ≈ 90° and/or the parallel- and antiparallel-propagating unstable waves (θ = 0° and 180°). The present study carries out linear kinetic instability analysis to investigate both the parallel and oblique unstable modes (0° ≤ θ ≤ 180°) driven by ring-beam pickup ion distributions of different pickup angles between 0° and 90°. Our linear instability analysis reveals that ring-beam pickup ion distributions can excite oblique mirror waves as well as parallel/quasi-parallel and oblique right- and left-helicity waves. The maximum growth rate among all the instabilities belongs to the parallel-propagating left-helicity waves at most pickup angles. Furthermore, the evolution of the unstable mirror waves by varying pickup angle indicates that as the pickup angle increases, the maximum growth rate of the mirror modes increases, while its propagation angle decreases.

Oblique Electromagnetic Instabilities Driven by Pickup Ion Ring-beam Distributions in the Outer Heliosheath. II. Hybrid Simulations

Hybrid simulations are carried out to study the instabilities driven by the outer heliosheath pickup ions of ring-beam velocity distributions. Following the linear instability analysis in the companion paper, the electromagnetic instabilities at various propagation directions (0° ≤ θ ≤ 180°) are studied for pickup ion ring-beam distributions of different pickup angles (0° ≤ α ≤ 90°). Previous studies of the ring-beam pickup ion distributions have been limited to the parallel- and antiparallel-propagating modes (θ = 0° and 180°). In this paper, several one-dimensional hybrid simulations are performed to investigate the most unstable modes of different propagation angles predicted by the linear instability analysis. Moreover, two-dimensional hybrid simulations are conducted to examine the simultaneous excitation of the parallel, antiparallel, and oblique unstable modes and their role in the scattering of the outer heliosheath pickup ions. Our recent one-dimensional hybrid and particle-in-cell simulations with simulation domains along the background magnetic field demonstrated that at small pickup angles, the outer heliosheath pickup ions experience an incomplete scattering limited to the hemisphere of positive parallel velocities with respect to the background magnetic field. In contrast, the present two-dimensional hybrid simulation results show that when both parallel/antiparallel and oblique unstable modes are acting together, the pickup ions can be scattered into the hemisphere of negative parallel velocities and approach a near-isotropic velocity distribution. This is in conflict with the spatial retention scenario of the energetic neutral atom ribbon observed by the Interstellar Boundary Explorer, which requires pickup ions at small pickup angles to stream away from the ribbon directions.

First Observation of Electron Cyclotron Harmonic Waves Inside Mirror Mode Structures in the Earth's Outer Magnetosphere

AbstractElectron cyclotron harmonic (ECH) waves are one of the most common plasma waves in the Earth's magnetosphere, and they are considered to be excited by the electron loss‐cone distributions near the magnetic equator. Using the Magnetospheric Multiscale data, we report an unusual event of ECH waves at L = ∼12.4 and MLT = ∼16.0 hr, where the ECH waves are locally excited inside non‐propagating magnetic oscillations, that is, mirror mode structures. The observation reveals that the mirror mode structures are generated by anisotropic ions ( ∼ 1.3 – 1.7), and electron mirror loss‐cone distributions are formed due to the magnetic configuration of the mirror mode structures. Combing with the linear instability analysis, we confirm that the ECH waves are locally excited by the electron mirror loss‐cone distributions inside the mirror mode structures. Therefore, we propose that the mirror mode structures provide another potential source region for ECH waves in the Earth's magnetosphere.

Natural convection in a fluid saturating an anisotropic porous medium in LTNE: effect of depth-dependent viscosity

Thermal convection in a fluid saturating an anisotropic porous medium in local thermal nonequilibrium (LTNE) is investigated, with specific attention to the effect of variable viscosity on the onset of convection. Many fluids show a remarkable dependence of viscosity on temperature that cannot be neglected. For this reason, we take into account a fluid whose viscosity decreases exponentially with depth, according to Straughan (Acta Mech. 61:59–72, 1986), Torrance and Turcotte (J. Fluid Mech. 47(1):113–125, 1971). The novelty of this paper is to highlight how variable viscosity coupled with the LTNE assumption affects the onset of convection. A numerical procedure shows the destabilising effect of depth-dependent viscosity. Moreover, it comes out that the LTNE hypothesis makes the influence of viscosity more intense. Linear instability analysis of the conduction solution is carried out by means of the Chebyshev-tau method coupled to the QZ algorithm, which provides the critical Rayleigh number for the onset of convection in a straightforward way. The energy method is employed in order to study the nonlinear stability. The optimal result of coincidence between the linear instability threshold and the global nonlinear stability threshold is obtained. The influence of anisotropic permeability and conductivity, weighted conductivity ratio, and interaction coefficient on the onset of convection is highlighted.

Whistler Waves Associated With Electron Beams in Magnetopause Reconnection Diffusion Regions

AbstractWhistler waves are often observed in magnetopause reconnection associated with electron beams. We analyze seven MMS crossings surrounding the electron diffusion region (EDR) to study the role of electron beams in whistler excitation. Waves have two major types: (a) Narrow‐band waves with high ellipticities and (b) broad‐band waves that are more electrostatic with significant variations in ellipticities and wave normal angles. While both types of waves are associated with electron beams, the key difference is the anisotropy of the background population, with perpendicular and parallel anisotropies, respectively. The linear instability analysis suggests that the first type of wave is mainly due to the background anisotropy, with the beam contributing additional cyclotron resonance to enhance the wave growth. The second type of broadband waves are excited via Landau resonance, and as seen in one event, the beam anisotropy induces an additional cyclotron mode. The results are supported by particle‐in‐cell simulations. We infer that the first type occurs downstream of the central EDR, where background electrons experience Betatron acceleration to form the perpendicular anisotropy; the second type occurs in the central EDR of guide field reconnection. A parametric study is conducted with linear instability analysis. A beam anisotropy alone of above ∼3 likely excites the cyclotron mode waves. Large beam drifts cause Doppler shifts and may lead to left‐hand polarizations in the ion frame. Future studies are needed to determine whether the observation covers a broader parameter regime and to understand the competition between whistler and other instabilities.

Density inversion phenomenon in porous penetrative convection

Penetrative convection occurs in many natural phenomena where an unstable stratified fluid moves into a stable one. This topic is of interest in many research fields like, for example, in geophysics and astrophysics. In the present paper, on taking into account for quadratic density law, the onset of penetrative convection in a horizontal porous layer is investigated. For the problem at stake, since the principle of exchange of stabilities has been proved, the convection can occur only through a secondary stationary motion. The critical Rayleigh numbers for the onset of stationary convection have been found via the linear instability analysis of the conduction solution. Moreover, the nonlinear stability of the motionless state has been investigated via the energy method. Numerical simulations have been performed through Chebyshev-τ spectral method in order to analyse the behaviour of stability and instability thresholds with respect to the upper plane temperature.

Mixing and Flow Transition in an Optimized Electrokinetic Turbulent Micromixer.

Micromixer is a key element in a lab on a chip for broad applications in the analysis and measurement of chemistry and engineering. Previous investigations reported that electrokinetic (EK) turbulence could be realized in a "Y" type micromixer with a cross-sectional dimension of 100 μm order. Although the ultrafast turbulent mixing can be generated at a bulk flow Reynolds number on the order of unity, the micromixer has not been optimized. In this investigation, we systematically investigated the influence of electric field intensity, AC frequency, electric conductivity ratio, and channel width at the entrance on the mixing effect and transition electric Rayleigh number in the "Y" type electrokinetic turbulent micromixer. It is found that the optimal mixing is realized in a 350 μm wide micromixer, under 100 kHz and 1.14 × 105 V/m AC electric field, with an electric conductivity ratio of 1:3000. Under these conditions, a degree of mixedness of 0.93 can be achieved at 84 μm from the entrance and 100 ms. A further investigation of the critical electric field and the critical electric Rayleigh number indicates that the most unstable condition of EK flow instability is inconsistent with that of the optimal mixing in EK turbulence. To predict the evolution of EK flow under high Raσ and guide the design of EK turbulent micromixers, it is necessary to apply a computational turbulence model instead of linear instability analysis.

Nonlinear magneto convection in an inclined porous layer with artificial neural network prediction

The onset of magnetoconvection in an inclined porous layer is investigated. The effects of physical parameters, such as the Rayleigh number, the inclination angle, and the Hartmann number, are examined. The system becomes more stable by increasing the inclination angle and Hartmann number. It is noted that the transverse rolls are more stable than the longitudinal rolls. A comparison between the linear and nonlinear instability analysis is discussed in more detail. The threshold values for the longitudinal and transverse rolls provide the subcritical instability region. It is noted that the subcritical instability region increases as the inclination angle also increases, whereas it remains unchanged as the Hartmann number increases. Besides, an artificial neural network (ANN) model using the Levenberg‐Marquardt backpropagation algorithm is employed to predict the distribution of critical Rayleigh numbers for both linear and nonlinear stability analyses. The optimal number of neurons in the hidden layer is selected on the basis of coefficient of determination, root mean square error, and root mean relative error. The simulated critical Rayleigh numbers obtained by the numerical study and the predicted critical Rayleigh numbers by the ANN coincides.

Fixed and Distributed Gene Expression Time Delays in Reaction–Diffusion Systems

Time delays, modelling the process of intracellular gene expression, have been shown to have important impacts on the dynamics of pattern formation in reaction–diffusion systems. In particular, past work has shown that such time delays can shrink the Turing space, thereby inhibiting patterns from forming across large ranges of parameters. Such delays can also increase the time taken for pattern formation even when Turing instabilities occur. Here, we consider reaction–diffusion models incorporating fixed or distributed time delays, modelling the underlying stochastic nature of gene expression dynamics, and analyse these through a systematic linear instability analysis and numerical simulations for several sets of different reaction kinetics. We find that even complicated distribution kernels (skewed Gaussian probability density functions) have little impact on the reaction–diffusion dynamics compared to fixed delays with the same mean delay. We show that the location of the delay terms in the model can lead to changes in the size of the Turing space (increasing or decreasing) as the mean time delay, tau , is increased. We show that the time to pattern formation from a perturbation of the homogeneous steady state scales linearly with tau , and conjecture that this is a general impact of time delay on reaction–diffusion dynamics, independent of the form of the kinetics or location of the delayed terms. Finally, we show that while initial and boundary conditions can influence these dynamics, particularly the time-to-pattern, the effects of delay appear robust under variations of initial and boundary data. Overall, our results help clarify the role of gene expression time delays in reaction–diffusion patterning, and suggest clear directions for further work in studying more realistic models of pattern formation.

Energy stability of thermally modulated inclined fluid layer

The stability of natural convection in thermally modulated inclined fluid layer is analyzed using linear instability analysis and generalized energy stability theory. A sufficient condition for the global stability of the fluid layer is obtained. The stability boundaries are found in terms of the Rayleigh number. Shooting method is used to find the stability limits numerically. Uncertain stability region is observed between the linear and the nonlinear stability boundaries. The onset of instability depends upon the frequency and the amplitude of modulation.

Wavelength Measurements of Electron Cyclotron Harmonic Waves in Earth's Magnetotail

AbstractElectron cyclotron harmonic (ECH) waves, potential drivers for diffuse aurora precipitation, have been extensively investigated for decades. The generation mechanism of ECH waves, however, remains an open question. Theoretical work in 1970s has demonstrated that ECH waves can be excited by loss cone distributions of hot plasma sheet electrons. Recent THEMIS spacecraft observations, however, indicate that the waves can also be excited by low energy electron beams. Utilizing interferometry techniques to analyze the phase difference between electric potentials measured by individual probes on Electric Field Instrument antenna pairs on THEMIS spacecraft, we compute the wavenumber of both beam‐driven ECH waves and loss‐cone‐driven ECH waves. These wavenumber measurements as well as other wave properties obtained from spacecraft measurements prove to be consistent with expectation from linear instability analysis. This provides us with independent verification of the generation mechanism and linear dispersion relation of beam‐driven and loss‐cone‐driven ECH waves. Our statistical results demonstrate that the median value of the wave vectors of beam‐driven ECH waves, characterized by wave normal angles () less than 80°, is 0.011 m−1; and that of loss‐cone‐driven ECH waves, characterized by wave normal angles larger than 85°, is 0.00765 m−1. Direct wavenumber measurements of ECH waves allow us to better understand the interaction between ECH waves and electrons in Earth's magnetosphere.

Linear Instability and Numerical Analysis of Surface Morphology Changes of Epitaxially Strained Thin Films Due to Elastic Interactions

Bu sistematik sayısal analiz çalışmasında ince film yüzeyinin kararlılığı, elastik dipol etkileri de göz önünde bulunduran matematiksel model ile farklı çeki kuvvetleri altında kararlılığı incelenmiştir. Bu modelde yüzey difüzyonuna gerilmenin etkisi birinci ve ikinci dereceden iki terim ile ifade edilmektedir. İnce film yüzeyinin sinüs dalgası şekillinde olduğu varsayılmış ve bu dalga yapısının yüzeydeki gerilme nedenli difüzyon ile sönümlenme veya büyüme dinamikleri incelenerek kararlılığı test edilmiştir. Elastik dipol etkileşimlerin basma ve çekme gerilmeleri altında farklı yüzey dinamiklerine sebep olduğu, bu farkın kritik basma gerilmesi (σ>100 MPa) üzerindeki basma gerilmesi durumunda çatlak benzeri yapıların oluşmasına neden olduğu gösterilmiştir. Çekme gerilmesine maruz kalan yüzeydeki dalgalı yapının sönümlendiği gözlemlenmiştir. Çatlak yapıların oluşumu sonrası yüzey kinetiği doğrusal kararlılık analizlerinden uzaklaşmaktadır. Uygulanan gerilmeye göre 3 farklı davranış saptanmıştır. Bunlar; sönümlenmenin gerçekleştiği (Ξ >0) çekme kuvveti uygulanan bölge, düşük basma kuvveti (-1,12>Ξ > 0) uygulanan yüzeyin karalı olduğu bölge ve çatlak benzeri oluşumların gözlemlendiği yüksek basma gerilmesinin (Ξ<-1,12) uygulandığı bölgedir. Yapılan simülasyonlarda, çatlak oluşumu sürecinde, çatlak bölgeden difüzyon ile kaçan maddenin tepelerde birikerek yeni tepeler oluşturduğu gözlemlenmiştir.