Linear Instability Analysis Research Articles

Proton Cyclotron Waves and Pickup Ion Ring Distribution Instabilities Upstream of Mars

AbstractProton cyclotron waves (PCWs) upstream of Mars are thought to be associated with the instabilities of pickup ions. The instabilities of pickup ions of ring distributions have larger maximum growth rates compared with beam distributions. However, observations revealed a notably‐reduced occurrence rate of PCWs for pickup ions of ring distributions. Linear instability analysis and a corresponding two‐dimensional particle‐in‐cell (PIC) simulation are performed to investigate the instabilities of pickup ion ring distributions upstream of Mars. Linear instability analysis indicates that a pickup ion ring distribution is unstable to the ion cyclotron, ion Bernstein, and mirror instabilities. The corresponding PIC simulation confirms the linear analysis results and further demonstrates that the pickup ions are scattered toward an isotropic shell distribution by the waves excited. Interestingly, the saturation energy of the waves is much lower than that driven by a corresponding pickup ion beam distribution, and mirror waves eventually dominate the system.

Simultaneous Observations of Mirror Mode Structures and Electromagnetic Ion Cyclotron Waves in the Earth's Outer Magnetosphere

AbstractElectromagnetic ion cyclotron (EMIC) waves and mirror mode structures respectively produced by the cyclotron and mirror instabilities are both generated by the temperature anisotropy of ions. However, whether these two instabilities can grow simultaneously in the magnetosphere is still unclear, as well as how they distribute if they can. In this paper, we first report an event of simultaneous observation of mirror mode structures and EMIC waves and then perform a statistical survey in the magnetosphere based on the measurements from Magnetospheric Multiscale (MMS) satellites. The event is observed in the duskside (MLT ∼ 18 hr) of Earth's outer magnetosphere (L ∼ 12) near the magnetic equator (MLAT ∼ 12°). The further linear instability analyses demonstrate that mirror mode structures and EMIC waves can be simultaneously and locally generated in the magnetosphere. The statistical results show that simultaneous mirror mode structures and EMIC waves are mainly distributed in the dawnside and duskside of the outer magnetosphere with slightly larger numbers in the dawnside but no significant differences in the proton number density, and plasma beta, proton temperature anisotropy, power‐weighted wave frequency, and wave normal angle.

Drawing of fibres composed of shear-thinning or shear-thickening fluid with internal holes

We explore the drawing of a shear-thinning or shear-thickening thread with an axisymmetric hole that evolves due to axial drawing, inertia and surface tension effects. The stress is assumed to be proportional to the shear rate raised to the $n$ th power. The presence of non-Newtonian rheology and surface tension forces acting on the hole introduces radial pressure gradients that make the derivation of long-wavelength equations significantly more challenging than either a Newtonian thread with a hole or shear-thinning and shear-thickening threads without a hole. In the case of weak surface tension, we determine the steady-state profiles. Our results show that for negligible inertia the hole size at the exit becomes smaller as $n$ is decreased (i.e. strong shear-thinning effects) above a critical draw ratio, but surprisingly the opposite is true below this critical draw ratio. We determine an accurate estimate of the critical draw ratio and also discuss how inertia affects this process. We further show that the dynamics of hole closure is dominated by a different limit, and we determine the asymptotic forms of the hole closure process for shear-thinning and shear-thickening fluids with inertia. A linear instability analysis is conducted to predict the onset of draw resonance. We show that increased shear thinning, surface tension and inlet hole size all act to destabilise the flow. We also show that increasing shear-thinning effects reduce the critical Reynolds number required for unconditional stability. Our study provides valuable insights into the drawing process and its dependence on the physical effects.

Perspectives on local thermal non-equilibrium (LTNE) Darcy–Bénard convection: Variable permeability and viscosity effects

The interplay between variations in permeability and viscosity on the onset of local thermal non-equilibrium in Darcy–Bénard convection has been investigated. Specifically, permeability is modeled as decreasing linearly with depth, while viscosity decreases exponentially. The validity of the principle of exchange of stabilities is confirmed. A linear instability analysis of the quiescent state is conducted through normal mode decomposition of disturbances, with threshold values for instability onset computed numerically using the Galerkin method. The individual and combined effects of increasing the variable permeability and viscosity parameters on the instability characteristics of the system are examined in detail, highlighting both commonalities and distinctions. It is observed that increasing each parameter individually hastens the onset of convection. However, their combined influence produces both stabilizing and destabilizing effects under certain parametric conditions. In all scenarios, an increase in the scaled interphase heat transfer coefficient consistently delays the onset of convection, whereas a higher ratio of porosity-modified conductivities has the opposite effect. Furthermore, the size of the convection cells remains unchanged at the extreme values of the scaled interphase heat transfer coefficient.

Fluctuation-driven dynamics of liquid nano-threads with external hydrodynamic perturbations

Instability and rupture dynamics of a liquid nano-thread, subjected to external hydrodynamic perturbations, are captured by a stochastic lubrication equation (SLE) incorporating thermal fluctuations via Gaussian white noise. Linear instability analysis of the SLE is conducted to derive the spectra and distribution functions of thermal capillary waves influenced by external perturbations and thermal fluctuations. The SLE is also solved numerically using a second-order finite difference method with a correlated noise model. Both theoretical and numerical solutions, validated through molecular dynamics, indicate that surface tension forces due to specific external perturbations overcome the random effects of thermal fluctuations, determining both the thermal capillary waves and the evolution of perturbation growth. The results also show two distinct regimes: (i) the hydrodynamic regime, where external perturbations dominate, leading to uniform ruptures, and (ii) the thermal-fluctuation regime, where external perturbations are surpassed by thermal fluctuations, resulting in non-uniform ruptures. The transition between these regimes, modelled by a criterion developed from linear instability theory, exhibits a strong dependence on the amplitudes and wavenumbers of the external perturbations.

An Improved Comprehensive Atomization Model for Pressure Swirl Atomizers

This study presents an improved comprehensive atomization model for a pressure swirl atomizer. The model integrates internal flow predictions, linear instability analysis of a swirling annular liquid sheet, primary atomization sub-model, and droplet velocity sub-model. Measurement data combined with the inviscid theory model predict the internal flow, providing liquid sheet velocity and thickness at the atomizer outlet. The dispersion relation of surface disturbances is obtained through linear instability analysis. A primary breakup predictive model for particle size distribution is constructed based on the wavelength and growth rate within the full unstable wavenumber range of the dispersion relation. Assuming uniform circumferential distribution and a normal distribution of spray angles, the droplet velocity is assigned according to the liquid sheet velocity. The model is implemented into Eulerian–Lagrangian simulations as initial conditions for discrete phase droplets to simulate the spray field. Results show the model can accurately predict the Sauter mean diameter with an error of less than 6% and effectively predicts the spray structure and spray cone angle. The dependency of the model on its parameters is also studied, determining that the values of the ligament constant and dispersion angle have an obvious impact on the prediction of Sauter mean diameter and spray structure.

Coaxial air blast atomization of a particulate gel suspension jet

The objective of this study is to experimentally and theoretically explore the impact of particle volume fractions and nozzle thread structure on the coaxial air blast atomization of particulate gel suspension jets through the high-speed flow visualization technology. The four breakup modes of particulate gel suspension jets are confirmed, namely oscillation mode, membrane-type breakup, fiber-type breakup, and superpulsating submode. However, the fiber-type breakup of jets disappears as the particle concentration reaches 40%, primarily attributed to the numerous particles sharply promoting the instability on the free jet surface. The experimental statistical results demonstrate that an increase in particle concentration and the adoption of the threaded nozzle both result in an expansion of the jet spray angle. The breakup length exhibits a reduction trend with the increase of particle concentration. However, it remains nearly the same value as that in the case of volume fraction 20% when the particle concentration continues to increase, contributed by the surge of viscous dissipation counteracting the enhanced instability on the jet interface caused by a large number of particles. The breakup length of a pure gel jet is predicted through the linear instability analysis, which is further modified by the viscosity model of particle suspensions to derive the breakup lengths of particulate gel suspension jets with different concentrations. The theoretical predictions align well with the statistical results in the experiment. The research is poised to have potential implications for numerous industrial processes and engineering applications including gel propellants.

Aerodynamic breakup of gel suspension droplets loaded with aluminum particles

The persistent and frequent space explorations are affected by the performance of propellants, among which the closely related gel propellants have been widely used due to their unique characteristics. Currently, research efforts primarily concentrate on the metallization and atomized combustion of gel propellants, emphasizing the combustion performance of metallized gel propellants. However, little attention has been given to the secondary breakup process that occurs just before combustion. By utilizing high-speed flow visualization technology, this study examines the key aerodynamic breakup process of gel suspension droplets loaded with micro aluminum particles, experimentally and theoretically investigating the droplet dynamics induced by particle volume fraction. The experimental confirmation of the three breakup behaviors exhibited by gel suspension droplets, namely oscillation mode, bag-stamen breakup, and hemline breakup, is presented along with the construction of a phase diagram. The impact of heterogeneous effects resulting from an increase in particle volume fraction on the critical breakup Weber number is revealed, with a concentration of 9 % as the turning point. Within the range of moderate Weber numbers, the tail flick deformation of suspension droplets in airflow is observed and reported for the first time. At different particle concentrations, there keeps a linear correlation between the dimensionless tail length and Weber number. Spherical suspension droplets will deform into thin disks in the airflow, with a nearly constant deformation rate regardless of changes in particle concentration. Additionally, the relationships between characteristic times and droplet parameters are discussed through statistical analysis of the results. As Ohnesorge number and particle volume fraction increase, the characteristic times both show a monotonically increasing trend. Moreover, a model of unstable growth rate associated with the particle volume fraction is established through linear instability analysis, accurately predicting the variation in critical Weber number for gel suspension droplet breakup. The congruence between our theoretical framework and experimental findings not only enhances comprehension of the impact of micro aluminum particles on the secondary breakup of gel suspension droplets but also provides valuable guidance for the aerospace applications of metallized gel propellants.

A zero-net-mass-flux wake stabilization method for blunt bodies via global linear instability

A rectangular cylinder, with an aspect ratio of 5, is a widely used bluff body in engineering practice. It undergoes intricate dynamical behavior in response to minute alterations in the flow angle of attack (α). These modifications invariably precipitate the failure of wake control for classical flow control methods with various α values. In this study, global linear instability, adjoint method, and sensitivity analysis are employed to identify the optimal position for flow control. It is found that the sensitive region gradually transitions from the leeward side to the downwind side of the model as α and Reynolds number (Re) increase. So, we set up airflow orifices for flow control in both positions. Jet flow control on the leeward side effectively inhibits vortex shedding (α ≤ 2°). High-order dynamic mode decomposition is employed to reveal the inherent mechanism of control. Suction control on the downside effectively mitigates the shear layer separation phenomenon induced by the altered spatial structure associated with higher α. A novel zero-net-mass-flux wake control, bionics-based breathe-valve control (BVC), is proposed to optimize the control effect. BVC is applicable for various α and Re, with optimal effectiveness achievable through jet velocity adjustments. The prediction-control approach in this investigation provides a targeted method to mitigate flow-induced vibration.

Double-diffusive thermosolutal Hadley–Prats flow: stability analysis

ABSTRACT A mathematical model is formulated to study the onset of double-diffusion fluid flow through an infinite permeable channel with internal heat source and viscous dissipation effects under the influence of convection conditions. Numerically, the dimensionless emerging eigenvalue problem is tackled using the fourth-order Runge–Kutta scheme, specifically applied to longitudinal roll disturbances. Critical values of wave number and vertical thermal Rayleigh number are determined. A higher value of Gebhart number is observed to correlate significantly with the destabilizing phenomena in Hadley–Prats flow. Concentration-based internal heat generation also strongly modifies the critical thermal Rayleigh number. Also, it is found that the horizontal mass flow and viscous dissipation exert a substantial influence on the onset of instability in the flow regime. Linear instability analysis indicates that greater values of Lewis number in the porous medium stabilize the convection process for both values of mass diffusion parameter ( C z ) . Enhancing C z from negative to positive values diminishes the critical thermal Rayleigh ( R z ) value and consequently induces instability in the porous medium. Increased concentration-based internal heat generation generates the destabilization process in the flow region.

Inertia and slip effects on the instability of a liquid film coated on a fibre

To investigate the influence of inertia and slip on the instability of a liquid film on a fibre, a theoretical framework based on the axisymmetric Navier–Stokes equations is proposed via linear instability analysis. The model reveals that slip significantly enhances perturbation growth in viscous film flows, whereas it exerts minimal influence on flows dominated by inertia. Moreover, under no-slip boundary conditions, the dominant instability mode of thin films remains unaltered by inertia, closely aligning with predictions from a no-slip lubrication model. Conversely, when slip is introduced, the dominant wavenumber experiences a noticeable reduction as inertia decreases. This trend is captured by an introduced lubrication model with giant slip. Direct numerical simulations of the Navier–Stokes equations are then performed to further confirm the theoretical findings at the linear stage. For the nonlinear dynamics, no-slip simulations show complex vortical structures within films, driven by fluid inertia near surfaces. Additionally, in scenarios with weak inertia, a reduction in the volume of satellite droplets is observed due to slip, following a power-law relationship.

Pollution overturning instability in an incompressible fluid with a Maxwell-Cattaneo-Mariano model for the pollutant field

We develop a model for a pollutant dissolved in or dispersed in an incompressible Navier–Stokes fluid when the diffusion theory for the pollutant obeys a second order in time system of equations rather than the first order in time system obtained from Fourier’s law. A detailed analysis is performed for a layer of fluid where a pollutant is such that the top of the layer will be heavier in concentration. A detailed expression for the critical pollutant Rayleigh number is found indicating precise conditions under which a convective overturning motion will arise. The investigation is performed by a linear instability analysis, but additionally we provide a completely nonlinear energy stability analysis. Diffusion of flux is also added to a Cattaneo-like equation and this leads to surprising results.

Stability and instability of thermosolutal convection in a Brinkman–Darcy–Kelvin–Voigt fluid with couple stress effect

In this article, the phenomenon of thermosolutal convection within a fluid characterized by the Brinkman–Darcy–Kelvin–Voigt (BDKV) model is delved into, while the impact of couple stresses on this process is considered. Both linear instability and nonlinear stability analyses are encompassed in our investigation. Several noteworthy observations have been made. When the fluid layer is heated from below and salt is introduced from above, it is found that the points at which stability and instability thresholds are reached coincide. This alignment is supported by the validity of the linear theory in predicting the initiation of convection under these conditions. However, the scenario changes when the layer is salted from the bottom while being heated. In this case, the stability thresholds remain constant, regardless of variations in the salt Rayleigh number. This discrepancy between the thresholds of linear instability and nonlinear stability is deemed significant. To gain a deeper understanding, numerical computations were conducted to identify and thoroughly discuss the thresholds of linear instability. These findings offer valuable insights into the behavior of the system under study. It is indicated by our results that parameters such as Brinkman, couple stresses, and Kelvin–Voigt contribute to stabilizing the system. Additionally, it was noted that the salt Rayleigh number has a stabilizing effect when the layer is salted from below, whereas it has a destabilizing effect when salt is introduced from above.

Penetrative magneto-convection of a rotating Boussinesq flow in f-planes

In this study, we conducted linear instability analysis of penetrative magneto-convection in rapidly rotating Boussinesq flows within tilted f-planes, under the influence of a uniform background magnetic field. We integrated wave theory and convection theory to elucidate the penetration dynamics in rotating magneto-convection. Our findings suggest that efficient penetration in rapidly rotating flows with weakly stratified stable layers at low latitudes can be attributed to the resonance of wave transmission near the interface between unstable and stable layers. In the context of strongly stratified flows, we derived the scaling relationships of penetrative distances Δ with the stability parameter δ. Our calculation shows that, for both rotation-dominated and magnetism-dominated flows, Δ obeys a scaling of Δ∼O(δ−1/2). In rotation-dominated flows, we noted a general decrease in penetrative distance with an increased rotational effect, and a minor decrease in a penetrative distance with an increased latitude. When a background magnetic field is introduced, we observed a significant shift in the penetrative distance as the Elsasser number Λ approaches one. The penetrative distance tends to decrease when Λ≪1 and increase when Λ≫1 with the rotational effect, indicating a transition from rotation-dominated to magnetism-dominated flow. We have further investigated the impact of the background magnetic field when it is not aligned with the rotational axis. This presents a notable contrast to the case where the magnetic field is parallel to the rotational axis.

Global nonlinear stability of bidispersive porous convection with throughflow and depth-dependent viscosity

The linear instability and the nonlinear stability analyses have been performed to examine the combined impact of a uniform vertical throughflow and a depth-dependent viscosity on bidispersive porous convection using the Darcy theory with a single temperature field. The validity of the principle of exchange of stability is proved. The eigenvalue problems resulting from both linear instability and nonlinear stability analyses with variable coefficients are numerically solved using the Chebyshev pseudo-spectral method. The equivalence of linear instability and nonlinear stability boundaries is established in the absence of throughflow, while in its presence, the subcritical instability is shown to be evident. The stability of the system is independent of the direction of throughflow in the case of constant viscosity, whereas upflow is found to be more stabilizing than downflow when the viscosity is varying with depth. While the viscosity parameter offers a destabilizing influence on the onset of convection in the absence of throughflow, it imparts both stabilizing and destabilizing effects on the same in its presence. The influence of the ratio of permeabilities and the interphase momentum transfer parameter is to make the system more stable. The findings of a mono-disperse porous medium are presented as a specific case within the broader context of this investigation.

Modulation of coaxial cone-jet instability in active co-flow focusing

The breakup of coaxial cone-jet interfaces to compound droplets in axisymmetric co-flow focusing (CFF) upon actuation is studied through numerical simulations. Due to the coupling effect of double interfaces, the response behaviours of coaxial cone-jet flow to actuation are more complex than those of a single-layered interface structure. Particularly, the coaxial jet presents totally different response modes between weak and strong interface coupling situations. In this work, the phase diagrams of response modes for coaxial jet breakup are depicted, considering the effect of perturbation frequency, amplitude and liquid flow rates. In particular, the breakup of a coaxial jet can be synchronized with actuation within a frequency range containing the natural breakup frequency, resulting in uniform compound droplets with a single core inside the shell, and the size of droplets can be adjusted by frequency. As the perturbation frequency exceeds the upper critical value, the external perturbation is unable to dominate the jet breakup, while below the lower critical frequency, the jet breaks up with multiple droplets generated in one period. The perturbation amplitude mainly affects the jet breakup length and also leads to the transition between different response modes. The coaxial cone upstream of the orifice can act as a buffer layer, regulating the perturbation amplitude of the coaxial jet downstream. The degree of buffering effect is affected by the perturbation frequency and amplitude. As the perturbation amplitude approaches unity, the decrease of perturbation frequency leads to the intermittent jet behaviour from the cone tip with a vibrating manner of the coaxial cone. Based on the linear instability analysis on the simplified single jet models for weak-coupled and strong-coupled jets, scaling analyses are carried out, which predict the jet breakup length and the natural frequency and critical frequency for the synchronized breakup. Finally, a strong pulse is added on the perturbation to produce compound droplets with a controllable number of cores. The present work provides valuable guidance for the practical application of on-demand compound droplet generation through active CFF.

Swirling instability of coaxial liquid jet in gas surroundings

Linear instability analysis of an inviscid coaxial swirling jet is carried out by deriving an analytical dispersion relation of perturbation growth. The azimuthal Rankine vortex and the axial discontinuous velocity distribution are utilized as the jet basic flow. Due to the existence of double interfaces, the instability mechanisms of the coaxial swirling jet are much more complex than those of the single-layered swirling jet. The effects of control parameters (including the swirling ratio, the Weber number, the jet radius ratio, the velocity ratios between different fluids, and the azimuthal velocity jump at the inner interface) on the temporal instability of coaxial swirling jet with different azimuthal modes are studied. By comparing the growth rate of different azimuthal modes, the predominant mode that determines the jet breakup is identified. The results indicate that an increase in the swirling ratio, the Weber number, and the radius ratio can lead to predominant mode transition to larger azimuthal wavenumbers. The velocity ratio between the inner jet and the annular jet and that between the surrounding fluid and the annular jet mainly affect the axial Kelvin–Helmholtz (KH) instability. An enhancement of the KH instability leads to the jet breakup with smaller azimuthal wavenumbers. The azimuthal velocity jump affects the azimuthal KH instability, the centrifugal instability, and the Coriolis instability simultaneously, thus leading to a multiple influence on modes transition. The phase-diagram of the predominant modes is further given, showing that the relative importance between the centrifugal force and the interfacial tension plays a significant role on the transition of predominant modes.

The Impact of Pickup Ion Thermal Spread on Pickup Ion Ring-beam-driven Instabilities and Scattering in the Outer Heliosheath

The present study investigates the unstable waves driven by the pickup ions of ring-beam distributions with various pickup angles and pickup ion temperatures in the outer heliosheath, using both linear instability analysis and hybrid simulations. While previous papers have generally assumed specific pickup ion temperatures, this study takes a more comprehensive approach by examining a wide range of pickup ion temperatures that may occur in the outer heliosheath. Our one-dimensional simulations demonstrate that regardless of the initial pickup ion temperature, the pitch-angle scattering of the pickup ions at small pickup angles saturates before the pickup ions can reach the hemisphere of negative parallel velocities with respect to the background magnetic field in velocity space, while at a 90° pickup angle, the pickup ions of ring distributions attain a significant level of isotropy. In contrast, the two-dimensional simulation results show that at all pickup angles, increasing the pickup ion temperature only slightly reduces the pickup ion scattering rate, but does not prevent the pickup ions from reaching the hemisphere of negative parallel velocities. Overall, the results do not align with the requirements of the earlier proposed scenarios for the secondary energetic neutral atom (ENA) mechanism of the Interstellar Boundary Explorer ENA ribbon, which demand either weak pickup ion scattering in the outer heliosheath or at least incomplete pickup ion scattering in the off-ribbon directions.

Variable gravity effects on penetrative porous convection

The effect of variable gravity on the onset of penetrative convection in a horizontal fluid-saturated porous layer is studied, performing the linear instability analysis and the nonlinear stability analysis (via the weighted energy method) of the thermal conduction solution. The linear and nonlinear critical Rayleigh numbers for the onset of penetrative porous convection are determined and their behaviour with respect to the fundamental physical parameters of the model is studied. In particular, we found that there exists a region of subcritical instabilities and that the increase of the upper bounding plane temperature and the decrease of gravity have a stabilizing effect on the onset of convection.

Falling liquid film down a non-uniformly heated slippery inclined plane with odd viscosity effects

The study examines the impact of the odd component of the Cauchy stress tensor, linked to the breakdown of time-reversal symmetry, on the linear instability analysis of a viscous fluid layer flowing down a non-uniformly heated and slippery inclined plane. Using a long-wave series expansion within an Orr-Sommerfeld-type boundary value problem, the critical Reynolds number is derived. The results of the linear long wavelength instability in the case of infinitesimal wavenumbers reveal that odd viscosity stabilizes the flow, while wall slip destabilizes it. The noteworthy finding is that thermocapillarity significantly enhances stability, provided gas temperature matches wall temperature.