Linear Stability Research Articles

Stability, Bifurcation and Dynamics in a Network with Delays

In real-world networks, due to complex topological structures and uncertainties such as time delays, uncontrolled systems may generate instability and complexity, thereby degrading network performance. This paper provides a detailed analysis of the stability, Hopf bifurcation, and complex dynamics of a networked system under delayed feedback control. Based on the linear stability method and Hopf bifurcation theorem, the stability of the equilibrium of the error system and the existence of Hopf bifurcation are studied. The stability of periodic solutions bifurcating from the trivial equilibrium is analyzed using normal form theory and central manifold theorem. Special focus is on the effects of the network topology and time delays on the stability and Hopf bifurcation. The theoretical results are also extended to the complex networks with asymmetric adjacent matrices. In addition, the controlled model exhibits complicated dynamical behavior via three types of codimension two bifurcations and period-doubling bifurcations that eventually lead to chaos. Numerical experiments have validated the theoretical results and indicated that delayed feedback control can effectively generate or annihilate the complicated behavior of complex networks.

Growth and morphogenesis of an everted tubular biological tissue

Eversion, a ubiquitous phenomenon in living organisms, significantly contributes to the growth and morphogenesis of soft biological tissues and organs. In this study, we examine the role of elasticity and geometry in the morphogenesis of everted tubular biological tissues, both theoretically and experimentally. Our findings demonstrate that the morphogenesis of an everted tubular tissue is primarily influenced by its elasticity and geometry. Through linear stability analysis, we show that an everted bilayer tubular tissue, with isotropic differential growth, can generate a circumferential or two-dimensional pattern, but fails to produce an axial pattern, which distinguishes it from a bilayer tubular tissue without eversion. Furthermore, as the modulus ratio and layer thicknesses increase, the surface instability pattern transitions from a two-dimensional pattern to a circumferential pattern. Notably, when the modulus ratio or layer thickness reaches sufficiently high values, spontaneous instabilities emerge on the inner surface of the everted bilayer tubular tissues pre-growth. To investigate the morphological instability of everted bilayer tubes, we perform a series of quantificational finite-element simulations and swelling experiments. The results of both numerical simulations and experimental observations align well with our theoretical analysis. This study not only enhances our comprehension of everted tubular tissue morphogenesis but also offers valuable insights for biomedical engineering and growth self-assembly applications.

A Theory for Self-Sustained Multicentennial Oscillation of the Atlantic Meridional Overturning Circulation. Part II: Role of Temperature

Abstract In the first part of our research on self-sustained multicentennial oscillation of the Atlantic meridional overturning circulation (AMOC), we utilized a hemispheric box model considering only the salinity equations. In this study, we consider both thermal and saline processes in the box model to investigate the AMOC multicentennial oscillation and the role of temperature. The thermal processes have mainly two effects, shortening the oscillation period and stabilizing the system, which are caused by the fast surface temperature restoration and negative feedback between temperature advection and AMOC, respectively. Introducing nonlinearity into the system can lead to self-sustained AMOC oscillation that is controlled by ocean internal dynamics, whose mechanism is generalized as a growing oscillation restrained by nonlinearity. The nonlinearity can arise from subpolar vertical mixing, or a nonlinear relation between the AMOC anomaly and the meridional difference of density anomaly. Linear stability analyses reveal that the eigenmode of the system is sensitive to model parameters, including model geometry, mean strength of the AMOC, and the AMOC’s sensitivity to density perturbation, surface virtual salt flux, and meridional temperature gradient. A larger surface virtual salt flux enhances the positive salinity advection feedback, and a smaller meridional temperature gradient weakens the negative temperature advection feedback. Both processes destabilize the AMOC multicentennial oscillation. Such situations may be expected in the future due to more intense warming and freshwater hosing at the high latitudes of the Northern Hemisphere.

Linear stability analysis of nanofluid flow over static or moving wedge using the collocation spectral method

The major goal of the present contribution is to fully examine the linear stability analysis of two-dimensional nanofluid flow over a moving as well as a static wedge under the impact of an adverse or favorable pressure gradient. Using similarity variables, the velocity profiles for the mean flow are obtained by converting the mean flow equations into a nonlinear ODE that is numerically solved using the fourth-order Runge-Kutta method. The stability equations for the flow of nanofluids are solved using the spectral Chebyshev collocation technique. Numerous numerical simulations are performed to understand the impact and influence of various parameters such as the concentration of the selected nanoparticle, wedge velocity, pressure gradient, and types of nanofluid. The obtained outcomes indicate that the critical dimensionless value of Reynolds number of instability decreases significantly with increasing wedge velocity in the same direction of fluid motion, with increasing concentration of nanoparticles, and when the pressure is adverse, which makes the flow more stable. On the contrary, the flow becomes unstable when the wedge moves against the fluid flow's direction in the instance of a favorable pressure.

Stability analysis of Poiseuille flow in an annulus partially filled with porous medium

The linear stability analysis of fluid flow, driven by an axial pressure gradient, inside the annular region partially filled with porous medium is investigated. The porous layer is attached to the inner cylinder. The flow is governed by the unsteady Darcy model in the porous region and the Navier–Stokes equation in the viscous region. The effect of the curvature parameter C (ratio of the inner cylinder radius to the gap between cylinders), the ratio of the fluid to the porous layer thickness (t̂), and the Darcy number (Da) on the stability characteristics are explored. In addition, the help of the radial velocity contours and the kinetic energy balance is taken to get an insight into the mode and the cause of instability, respectively. The results show that depending upon the value of t̂, a decrease in the value of C causes a shift in the neutral stability curve from bimodal to trimodal. For low values of t̂, when the onset of instability is dominated by a porous mode, C destabilizes the flow, whereas it has a stabilizing impact on the flow stability for the odd-fluid mode and the even-fluid mode. At high values of t̂, C has again destabilizing characteristics and instability is dominated by even-fluid mode. When axisymmetric disturbances are dominant, it is observed that the value of t̂ for which similar instability characteristics are found varies directly as the square root of Da. However, the correlation between t̂ and Da does not hold when the non-axisymmetric disturbances are least stable. Contrary to the unconditional stability of the annular Poiseuille flow under non-axisymmetric disturbances for C < 0.1325, the present system is unstable even for C = 0.005 and t̂≤1. This shows the significant impact of the curved fluid–porous interface on the stability characteristics.

Coupled instability modes at a solvent/non-solvent interface to decorate cellulose acetate flowers

Dispensing a water drop on the thin film of a solution composed of cellulose acetate (CA) in dimethyl formamide (DMF) forms a thin and porous CA layer at the water–DMF interface. While a denser water drop on a rarer CA–DMF film manifests a Rayleigh–Taylor instability—RTI, the dynamically forming porous layer at the water–DMF interface triggers a Saffman–Taylor instability—STI. The combined effects of RTI and STI enable the formation, growth, coalescence, and branching of an array of periodic finger patterns to finally develop into a flower-like morphology. A general linear stability analysis (GLSA) of a thin bilayer composed of a Newtonian and incompressible water layer resting on a Darcy–Brinkman porous medium could predict the length and the time scales of such a finger formation phenomenon. The GLSA uncovers the crucial roles of pressure gradients originating from the gravitational effects, osmotic forces, the Marangoni effect, and capillary forces on the dynamics of the finger formation. While the density difference between water and CA–DMF layer plays a crucial role in deciding the initial finger spacing, the osmotic pressure dictates the formation, growth, branching, and coalescence of fingers. The length-FL and number-Navg of fingers are found to scale as FL∼We0.33Re−0.25 and Navg∼We0.33Re0.25. Further, an inverse relationship of the concentration of CA (C) with ∼We−0.3 and ∼Re−0.7 highlights its role in the formation and growth of fingers. The loading of CA in DMF, the viscosity and density of the CA–DMF film, and the curvature of the fingers are found to be other parameters that decide morphologies.

Stability of buoyancy–thermocapillary convection in molten silicon liquid bridge between two disks with different radii under gravity

By employing a linear stability analysis based on the spectral element method, we investigated the impact of radius ratio (Γr) on the stability of buoyancy–thermocapillary convection in a molten silicon liquid bridge (Pr = 0.011). This liquid bridge was located between two coaxial disks with different radii under the influence of gravity. The aspect ratio of the liquid bridge was maintained at Γ = 1, with a volume ratio Γv = 1 and a fixed height. To explore the physical mechanisms behind convection instability, a perturbation energy analysis was adopted. The free surface shape was determined using the Young–Laplace equation, and two distinct heating strategies were employed. In a top-heated liquid bridge, the convection stability under gravity is always stronger than under zero-gravity. However, in a bottom-heated liquid bridge, the convection stability under gravity is not consistently stronger than under zero-gravity; specifically, when 0.522 < Γr < 0.673, the convection stability under gravity is weaker than under zero-gravity. Despite the small height of the liquid bridge (approximately 2 mm), the maximum relative difference of the critical Marangoni number (Mac) between gravity and zero-gravity conditions reaches as high as 227.8%. In a bottom-heated liquid bridge, oscillatory instability occurs at larger radius ratios (Γr = 0.8) compared to the zero-gravity condition. Furthermore, all instabilities for various radius ratios and heating strategies were found to be of hydrodynamic in nature.

Linear stability of viscoplastic flows down an incline

The stability analysis of viscoplastic flows down an inclined plane is done by comparing results obtained through theoretical and numerical studies of “regularized” models. The theoretical analysis is performed for Regularized Bingham and Casson-like fluids via the long-wave approximation method. In particular, the Bingham and the Casson flow have different stability characteristics, for Bingham-type materials an increase in yield stress leads to flow destabilization, while Casson-type materials show the opposite behaviour. The numerical study is performed by using the Papanastasiou and the “exact” Bingham model via a spectral method. The comparison between theoretical and numerical results shows excellent agreement. Our findings highlight that “regularized” and “exact” flow and can have stability characteristics, although they are “practically indistinguishable”. We validate our approach with the Regularized Bingham-like model, which is in rather satisfactory agreement with the experimental data.

Separation-induced transition on a T106A blade under low and elevated free stream turbulence

The separation-induced transition on the suction surface of a T106A low pressure turbine blade is a complex phenomenon with implications for aerodynamic performance. In this numerical investigation, we explore an adverse pressure gradient-dominated flow subjected to varying levels of free stream excitation, as the underlying separation-induced transition is a critical factor in assessing blade profile loss. By comprehensively analyzing the effects of free stream turbulence (FST) on the transition process, we delve into the various mechanisms which govern the instabilities underlying bypass transition by studying the instantaneous enstrophy field. This involves solving the two-dimensional (2D) compressible Navier–Stokes equation through a series of numerical simulations, comparing a baseline flow to cases where FST with varying turbulent intensity (Tu=4% and 7%) is imposed at the inflow. Consistent with previous studies, the introduction of FST is observed to delay flow separation and trigger early transition. We explore the different stages of bypass transition, from the initial growth of disturbances (described by linear stability theory) to the emergence of unsteady separation bubbles that merge into turbulent spots (due to nonlinear interactions), by examining the vorticity dynamics. Utilizing the compressible enstrophy transport equation for the flow in a T106A blade passage, we highlight the various routes of bypass transition resulting from different levels of FST, emphasizing the relative contributions from baroclinicity, compressibility, and viscous terms.

Nonlinear Analysis of an Extended Heterogeneous Lattice Hydrodynamic Model Considering on/off-Ramps

To study the impact of on/off-ramps on traffic stability in heterogeneous traffic flow, a novel lattice hydrodynamic model was presented. The new model’s stability condition was determined using the linear stability analysis method. The theoretical results reveal that traffic flow stability is influenced by the proportion of vehicles with different maximum speeds and safe headway, as well as the presence of on-ramps and off-ramps to a certain degree. Through the approximate perturbation method, the mKdV equation and the kink–antikink solution of the traffic density at the jam area are obtained. In order to verify the effectiveness and feasibility of the new model, numerical simulations were conducted to demonstrate on/off ramp effect and different proportions of vehicles which possess bigger maximum velocity or safe space headway affect the traffic stability. The numerical results indicate that in heterogeneous traffic flow scenarios, increasing the ratio of vehicles which possess bigger maximum velocity or bigger safe space headway can lead to a deterioration of traffic stability. The effect of on-ramp could cause traffic instability, while the effect of off-ramp is beneficial for easing traffic congestion.

Predicting the upper bound of two-dimensional flow regimes of symmetric objects through two-dimensional computations

The onset of secondary wake instability is generally predicted via experiments, linear stability analysis, and three-dimensional direct numerical simulations. The current work stems from an open question that is very intriguing and fundamental: Can the upper bound of a two-dimensional flow be predicted purely on the basis of two-dimensional computational results? It is found that spatial distribution of a field variable, i.e., streamwise velocity in the vortex formation region, aids in determining the upper limit of a two-dimensional flow regime of a symmetric object. The vortex formation length attains its least value at the second critical Reynolds number. In addition, streamwise extents of mean wake and vortex formation region along wake axis become the same. Under this circumstance, the streamwise velocity at the terminal point of vortex formation region is such that its mean value vanishes while intensity of fluctuations or corresponding Reynolds stress becomes the maximum. The predicted values of critical Reynolds numbers for circular, square, and diamond cross sections exhibit excellent agreement with the results available in the literature.

Linear stability analysis of a droplet under an axisymmetric thermal gradient

We study the linear stability of a droplet placed at the center of a horizontal disk under the effect of surface stress promoted by an axisymmetric thermal gradient. Since the fluid volume is constant, we solve the non-steady base flow and the perturbation simultaneously as they evolve over time. The numerical results show that the base state migrates from a droplet to a ring shape, with the front position and maximum thickness following power laws with time. The perturbations travel with the same velocity as the advancing front and develop their maxima close to the contact line. All of them initially decrease their amplitudes, later showing an increment with the growth rates depending on the wavenumber and time. The dominant wavenumber increases with time, in agreement with recent experimental work.

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.

An alternative numerical solution for the Orr–Sommerfeld problem

A simple tutorial for the numerical solution of the Orr–Sommerfeld problem is presented. The aim is the determination of the neutral stability threshold to the linear instability for the Poiseuille flow within a plane parallel channel. Such an instability is hydrodynamic in nature as it is driven by the advection term in the local momentum balance equation. The numerical solution is achieved by a spectral method. The basis functions are chosen as Chebyshev polynomials multiplied by a suitable overall factor meant to allow the fulfilment of the boundary conditions. The details of the numerical code implemented via the Mathematica software environment are also provided. A comparison with other numerical solutions of the same problem implemented via the tau-method is presented.

Linear instability and weakly nonlinear effects in eastward dipoles

The linear instability and weakly nonlinear dynamics of eastward-propagating, steady-state Larichev–Reznik vortex dipoles are explored in terms of two-dimensional normal-mode analysis. To extract the fastest growing normal modes, we apply both breeding methodology based on solving the initial-value problem, as well as a direct-solution approach through the full-spectrum eigenproblem involving large matrices. We find that the amplification rate of dipole instability decreases with respect to increase in dipole intensity. In our study, both approaches yield consistent results and are systematically compared to provide guidance for further studies of vortex structures.We consider nonlinear self-interaction of the fastest growing mode, along with the induced eddy fluxes, their divergence and mechanical energy balance. Through this analysis, we find that the unstable mode leads to weakening of the dipole by extracting its energy and exchanging potential vorticity content in the down-gradient sense, thus, providing a nonlinear physical mechanism for the dipole destruction. In particular, we highlight the fundamental importance of the west-east asymmetry of the normal mode for destruction to be realized. In summary, we consider this work to be a foundational demonstration of useful methodology for future studies of dynamics and stability of isolated vortices without simplifying spatial symmetries, such as ubiquitous vortices in geophysical fluids.

Prediction of Onset Temperature in Standing Wave Thermoacoustic Engine with Mesh Screen Stack

A thermoacoustic engine is a engine that converts thermal energy into acoustic energy, which can be used to generate electricity or cooling. This engine is attractive because it consists only of a stack, heat exchangers, and a resonator. The stack serves as the primary component for the energy conversion process and consists of porous materials like an array of stainless steel mesh screens. To generate the acoustic energy, a minimum temperature difference is necessary between the two sides of the stack, called the onset temperature difference. However, the calculation for prediction of onset temperature on the stack made of mesh screen has not been addressed. Therefore, the objective of this paper is to propose a method that can be used to estimate the onset temperature difference in standing wave thermoacoustic engine with stacks made of mesh screen arrays. The onset temperature difference is predicted numerically using linear stability theory and matrix transfer methods. Experimental verification is carried out by using standing wave thermoacoustic engine from pervious study. The results showed that the lowest onset temperature difference (TH - TC = 140ºC) is obtained when rh = 0.497 mm. Furthermore, the numerical and experimental onset temperature difference comparisons show a qualitative agreement, allowing the onset temperature prediction method to be used in designing standing wave thermoacoustic engines with stacks made of mesh screens.

Saturation of spiral instabilities in disc galaxies

ABSTRACT Spiral density waves can arise in galactic discs as linear instabilities of the underlying stellar distribution function. Such an instability grows exponentially in amplitude at some fixed growth rate β before saturating non-linearly. However, the mechanisms behind saturation, and the resulting saturated spiral amplitude, have received little attention. Here, we argue that one important saturation mechanism is the non-linear trapping of stars near the spiral’s corotation resonance. Under this mechanism, we show analytically that an m-armed spiral instability will saturate when the libration frequency of resonantly trapped orbits reaches $\omega _\mathrm{lib} \sim \mathrm{a\, \, few}\times m^{1/2} \beta$. For a galaxy with a flat rotation curve, this implies a maximum relative spiral surface density $\vert \delta \Sigma /\Sigma _0\vert \sim \mathrm{a\, \, few} \times (\beta /\Omega _\mathrm{p})^2 \cot \alpha$, where Ωp is the spiral pattern speed and α is its pitch angle. This result is in reasonable agreement with recent N-body simulations, and suggests that spirals driven by internally-generated instabilities reach relative amplitudes of at most a few tens of per cent; higher amplitude spirals, like in M51 and NGC 1300, are likely caused by very strong bars and/or tidal perturbations.

Scaling and modeling of the heat transfer across the free surface of a thermocapillary liquid bridge

PurposeThis paper aims to derive a reduced-order model for the heat transfer across the interface between a millimetric thermocapillary liquid bridge from silicone oil and the surrounding ambient gas.Design/methodology/approachNumerical solutions for the two-fluid model are computed covering a wide parametric space, making a total of 2,800 numerical flow simulations. Based on the computed data, a reduced single-fluid model for the liquid phase is devised, in which the heat transfer between the liquid and the gas is modeled by Newton’s heat transfer law, albeit with a space-dependent Biot function Bi(z), instead of a constant Biot number Bi.FindingsAn explicit robust fit of Bi(z) is obtained covering the whole range of parameters considered. The single-fluid model together with the Biot function derived yields very accurate results at much lesser computational cost than the corresponding two-phase fully-coupled simulation required for the two-fluid model.Practical implicationsUsing this novel Biot function approach instead of a constant Biot number, the critical Reynolds number can be predicted much more accurately within single-phase linear stability solvers.Originality/valueThe Biot function for thermocapillary liquid bridges is derived from the full multiphase problem by a robust multi-stage fit procedure. The derived Biot function reproduces very well the theoretical boundary layer scalings.

Controlling viscous fingering instabilities of complex fluids

Despite their aesthetic elegance, wavy or fingering patterns emerge when a fluid of low viscosity pushes another immiscible fluid of high viscosity in a porous medium, producing an incomplete sweep and hampering several crucial technologies. Some examples include chromatography, printing, coating flows, oil-well cementing, as well as large-scale technologies of groundwater and enhanced oil recovery. Controlling such fingering instabilities is notoriously challenging and unresolved for complex fluids of varying viscosity because the fluids’ mobility contrast is often predetermined and yet the predominant drive in determining a stable, flat or unstable, wavy interface. Here we show, experimentally and theoretically, how to suppress or control the primary viscous fingering patterns of a common type of complex fluids (of shear-thinning with a low yield stress) using a radially tapered cell of linearly varying gap thickness, h(r). Experimentally, we displace a complex viscous (PAA) solution with gas under a constant flow rate (Q), varied between 0.02 and 2 slpm (standard liter per minute), in a radially converging cell with a constant gap-thickness gradient, α=dh/dr<0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha = dh/dr < 0$$\\end{document}. A stable, uniform interface emerges at low Q and in a steeper cell (i.e., greater |α|\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|\\alpha |$$\\end{document}) for the complex fluids, whereas unstable fingering pattern at high Q and smaller |α|\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|\\alpha |$$\\end{document}. Our theoretical predictions with a simplified linear stability analysis show an agreeable stability criterion with experimental data, quantitatively offering strategies to control complex fluid-fluid patterns and displacements in microfluidics and porous media.

Global dynamics and spatiotemporal heterogeneity of a preytaxis model with prey-induced acceleration

Abstract Conventional preytaxis systems assume that prey-tactic velocity is proportional to the prey density gradient. However, many experiments exploring the predator–prey interactions show that it is the predator’s acceleration instead of velocity that is proportional to the prey density gradient in the prey-tactic movement, which we call preytaxis with prey-induced acceleration. Mathematical models of preytaxis with prey-induced acceleration were proposed by Arditi et al. ((2001) Theor. Popul. Biol. 59(3), 207–221) and Sapoukhina et al. ((2003) Am. Nat. 162(1), 61–76) to interpret the spatial heterogeneity of predators and prey observed in experiments. This paper is devoted to exploring the qualitative behaviour of such preytaxis systems with prey-induced acceleration and establishing the global existence of classical solutions with uniform-in-time bounds in all spatial dimensions. Moreover, we prove the global stability of spatially homogeneous prey-only and coexistence steady states with decay rates under certain conditions on system parameters. For the parameters outside the stability regime, we perform linear stability analysis to find the possible patterning regimes and use numerical simulations to demonstrate that spatially inhomogeneous time-periodic patterns will typically arise from the preytaxis system with prey-induced acceleration. Noticing that conventional preytaxis systems are unable to produce spatial patterns, our results imply that the preytaxis with prey-induced acceleration is indeed more appropriate than conventional preytaxis to interpret the spatial heterogeneity resulting from predator–prey interactions.