Linear Perturbation Analysis Research Articles

Stability of a plane Couette flow over inhomogeneously stressed solids

In this paper, we investigate the stability of a plane Couette flow over non-linear solids undergoing homogeneous or inhomogeneous initial stresses. Almost all biological and engineering structures are subjected to complex initial residual stresses. We probe here the stability characteristics of a plane Couette flow over these stressed solid surfaces. Since initial stress is commonly inhomogeneous in Nature, our main focus is to analyze the stability of flow over inhomogeneously stressed solids at all Reynolds number regimes, which was not investigated earlier. We apply an initially stressed neo-Hookean solid model which precludes the necessity of using the stress-free reference. A linear perturbation analysis based on an Eulerian–Lagrangian formulation determines the stability associated with short and finite wave modes for zero, low, and high Reynolds number flows. In presence of a uniform initial stress field, the short wave modes are stabilized by tensile stresses and destabilized by compressive stresses in the limit of creeping flow. A similar influence of pre-stress is observed in buckling of columns. On the other hand, the case of spatially varying initial stress is far more intricate due to the inhomogeneous shearing of the solid in the unperturbed state. A linear perturbation introduced to this complicated unperturbed state presents several interesting observations. We find multiple unstable modes for low Reynolds number (Re≪1) creeping flows which were not reported earlier to the best of our knowledge. While literature demonstrates only downstream unstable modes for a plane Couette flow, both upstream and downstream unstable modes are observed for inhomogeneously stressed solids in various Reynolds number regimes. In the creeping flow limit (Re=0), the stability depends mainly on the tensile or compressive initial stress at the solid–fluid interface. When this initial stress is compressive at the interface, only the finite wave modes determine the stability for all thicknesses instead of the short-wave modes. At higher Reynolds number, we report many more interesting results for spatially varying stress-fields which includes an unusual scaling of wall-mode validated through eigenmode analysis.

Inverting ice surface elevation and velocity for bed topography and slipperiness beneath Thwaites Glacier

Abstract. There is significant uncertainty over how ice sheets and glaciers will respond to rising global temperatures. Limited knowledge of the topography and rheology of the ice–bed interface is a key cause of this uncertainty as models show that small changes in the bed can have a large influence on predicted rates of ice loss. Most of our detailed knowledge of bed topography comes from airborne and ground-penetrating radar observations. However, these direct observations are not spaced closely enough to meet the requirements of ice-sheet models, so interpolation and inversion methods are used to fill in the gaps. Here we present the results of a new inversion of surface elevation and velocity data over Thwaites Glacier, West Antarctica, for bed topography and slipperiness (i.e. the degree of basal slip for a given level of drag). The inversion is based on a steady-state linear perturbation analysis of the shallow-ice-stream equations. The method works by identifying disturbances to surface flow which are caused by obstacles or sticky patches in the bed and can therefore be applied wherever the shallow-ice-stream equations hold and where surface data are available, even where the ice thickness is not well known. We assess the performance of the inversion for topography with the available radar data. Although the topographic output from the inversion is less successful where the bed slopes steeply, it compares well with radar data from the central trunk of the glacier for medium-wavelength features (5–50 km). This method could therefore be useful as an independent test of other interpolation methods such as mass conservation and kriging. We do not have data to allow us to assess the success of the slipperiness results from our inversions, but we provide maps that may guide future seismic data collection across Thwaites Glacier. The methods presented here show significant promise for using high-resolution satellite datasets, calibrated by sparser field datasets, to generate high-resolution bed topography products across the ice sheets and therefore contribute to reduced uncertainty in predictions of future sea-level rise.

Electrophoresis of Dielectric and Hydrophobic Spherical Fluid Droplets Possessing Uniform Surface Charge Density.

The present article deals with the theoretical study on electrophoresis of hydrophobic and dielectric spherical fluid droplets possessing uniform surface charge density. Unlike the ideally polarizable liquid droplet bearing constant surface ζ-potential, the tangential component of the Maxwell stress is nonzero for dielectric fluid droplets with uniform surface charge density. We consider the continuity of the tangential component of total stress (sum of the hydrodynamic and Maxwell stresses) and jump in dielectric displacement along the droplet-to-electrolyte interface. The typical situation is considered here for which the interfacial tension of the fluid droplet is sufficiently high so that the droplet retains its spherical shape during its motion. The present theory can be applied to nanoemulsions, hydrophobic oil droplets, gas bubbles, droplets of immiscible liquid suspended in aqueous medium, etc. Based on weak field and low charge assumptions and neglecting the Marangoni effect, the resultant electrokinetic equations are solved using linear perturbation analysis to derive the closed form expression for electrophoretic mobility applicable for the entire range of Debye-Hückel parameter. We further deduced an alternate approximate expression for electrophoretic mobility without involving exponential integrals. Besides, we have derived analytical results for mobility pertaining to various limiting cases. The results are further illustrated to show the impact of pertinent parameters on the overall electrophoretic mobility.

Initial stage of nanometer metal films electrical explosion: Current action integral and electrothermal instability

An electrical explosion of nanometer (∼20 nm) metal films onto polymer substrate at the current densities of ∼1012A/m2 is studied theoretically. The initial stage of explosion is considered with respect to current action integral and development of electrothermal instability. It is revealed that the effective value of specific action integral is larger than its value for the vacuum conditions due to the heat outflow into the substrate. The paper considers a linear perturbation analysis of thermal state in order to find the marginal condition for electrothermal instability. This instability leads to the formation of strata structures arranged normally to the direction of current flow before the melting stage.

Rossby wave instabilities of protoplanetary discs with cooling

ABSTRACT Rossby wave instabilities (RWIs) usually lead to non-axisymmetric vortices in protoplanetary discs and some observed substructures of these discs can be explained well by RWIs. We explore how the cooling influences the growth rate of unstable RWI modes in terms of the linear perturbation analysis. The cooling associated with the energy equation is treated in two different ways. The first approach that we adopt is a simple cooling law. The perturbed thermal state relaxes to the initial thermal state on a prescribed cooling time-scale. In the second approach, we treat the cooling as a thermal diffusion process. The difference in the growth rate between the adiabatic and isothermal modes becomes more pronounced for discs with smaller sound speed. For the simple cooling law, the growth rates of unstable modes monotonically decrease with the shorter cooling time-scale in barotropic discs. However, the dependence of the growth rate with the cooling time-scale becomes non-monotonic in non-baratopic discs. The RWIs might even be enhanced in non-barotropic discs during the transition from the adiabatic state to the isothermal state. When the cooling is treated as thermal diffusion, even in barotropic discs, the variation of the growth rate with thermal diffusivity becomes non-monotonic. Furthermore, a maximum growth rate may appear with an appropriate value of thermal diffusivity. The angular momentum flux is investigated to understand the angular momentum transport by RWIs with cooling.

Variability in evapotranspiration shifts from meteorological to biological control under wet versus drought conditions in an alpine meadow

Abstract The Tibetan Plateau is generally referred to as the Chinese water tower, and evapotranspiration (ET) affects the water budget and stability of alpine meadows on the Tibetan Plateau. However, its variability and controlling mechanisms have not been well documented under the drier conditions induced by global warming. Therefore, this study aimed to clarify whether meteorological or biological factors primarily affected the variability in ET under contrasting water conditions in the alpine meadow ecosystem on the Tibetan Plateau. Based on 6-year (2013–2018) eddy covariance observations and the corresponding meteorological and biological data, linear perturbation analyses were employed to isolate the contributions of meteorological and biological factors to the variability in evapotranspiration (δET). The results showed that δET was mainly driven by meteorological factors in wet peak seasons (July and August), and was dominated by net radiation (Rn) and air temperature (Ta), indicating that the inadequate available energy is the factor limiting ET. However, the dominant factors affecting δET shifted from meteorological to biological in dry peak seasons when the canopy stomatal conductance (gs) and leaf area index were dominant. At this point, the ecosystem was limited by the water conditions. These results provide empirical insights into how meteorological and biological factors regulate variability in ET under contrasting water conditions. These findings can further improve our understanding of water cycle processes and can help effectively manage water resources in alpine meadow ecosystems under future climate change conditions.

Stabilization of ultrathin nanowires by self-assembly into bundles

The relative tendency of freely dispersed and bundled gold nanowires to break up along their length by the Rayleigh–Plateau instability is investigated both experimentally and theoretically. Small angle X-ray scattering, in combination with transmission electron microscopy, reveal that the bundling of nanowires can enhance their stability. The experimental observation is rationalized by a linear perturbation analysis of a representative unit cell of bundled wires. A stability map is constructed for a bundle of nanowires to display the sensitivity of the Rayleigh–Plateau instability to the number and size of contacts with nearest neighbors per nanowire, and to the ratio of interfacial energy to surface energy. Stabilisation is enhanced by allowing the bundle of wires to sinter freely: a criterion for this kinetically-based stabilisation is given in terms of the ratio of pinch-off time for the instability to the sintering time to form the necks between nanowires.

On the geometrical perturbation of a three‐shell spherical model in electroencephalography

The forward electroencephalography (EEG) problem is studied in the framework of a multilayered structure, which models the scalp, skull, cerebrospinal fluid, and brain. Both the exterior and all inner boundaries are perturbed spheres so that special localized defects in head‐brain imaging are considered in analytic fashion. Linear perturbation analysis is implemented, providing exact expression for the first significant term of the forward EEG solution of the perturbed problem. Comparison with the solution of the corresponding unperturbed spherical problem is included, together with numerical demonstration of the produced errors in special deformation cases. The results suggest that significant errors are caused when large inaccuracies in the head‐brain structure in the vicinity of EEG source are not taken into account. Moreover, the suggested procedure provides a mathematical tool for evaluating quantitatively the impact of special deformations in the head representation on the EEG forward analytic solution.

Biotic effects dominate the inter-annual variability in ecosystem carbon exchange in a Tibetan alpine meadow

AbstractThe alpine meadow ecosystem in Tibet is fragile and sensitive, and its carbon sink function with respect to climate change has become a matter of widespread concern. Therefore, this study aims to clarify the inter-annual variations (IAVs) in the carbon fluxes in an alpine meadow and to further quantify the contributions of the driving factors to the IAVs. Based on 7 years of flux data (2012–2018) and the corresponding climatic and biotic data, a set of look-up tables was used to separate and quantify the IAV sources. Furthermore, linear perturbation analyses were employed to quantify the contributions of each key factor. During 2012–2018, the net ecosystem productivity (NEP), gross primary productivity (GPP) and ecosystem respiration (Re) of this alpine meadow were 3.31 ± 26.90, 210.18 ± 48.35 and 206.88 ± 28.45 g C m−2 y−1, respectively, which indicated relatively large IAVs. When the contributions of climatic and biotic effects were distinguished and quantified, the dominant effects of biotic factors emerged. Additionally, negative interactions between climatic and biotic effects were detected. Among the climatic factors, only soil water content contributed relatively more to the IAVs and played a role in regulating the interactions between climatic and biotic effects. These results suggest that biotic effects must be carefully considered to reduce the uncertainties associated with future carbon flux estimates.

Thermal instability in the presence of the cosmic ray and ambipolar diffusion

ABSTRACT The aim of this study is to investigate the impact of the cosmic ray (CR) and ambipolar diffusion (AD) on the thermal instability in a weakly ionized gas. The CR propagates along the magnetic-field line, where the AD is present. The weakly ionized gas and the CR are considered as two different interacting fluids. Using the linear perturbation analysis, we consider the occurrence of the thermal instability to obtain a dispersion relation in the presence of both phenomena. This equation describes the stable and unstable modes in the terms of some parameters, which depend on the CR and AD. The results show that the angle between the direction of the perturbation propagation and the magnetic-field lines plays an important role on the domains of stability and instability. Furthermore, we found that there is a characteristic wavelength related to the ambipolar diffusivity that explains whether the AD increases the magnetic support of the cloud against the collapse, or decreases it against the collapse. Finally, the results address some new points in the study of the structure formation within the clumpy molecular clouds, as well as the formation of the low-mass stars in the interstellar medium.

Fragmentation of ring galaxies and transformation to clumpy galaxies

ABSTRACT We study the fragmentation of collisional ring galaxies (CRGs) using a linear perturbation analysis that computes the physical conditions of gravitational instability, as determined by the balance of self-gravity of the ring against pressure and Coriolis forces. We adopt our formalism to simulations of CRGs and show that the analysis can accurately characterize the stability and onset of fragmentation, although the linear theory appears to underpredict the number of fragments of an unstable CRG by a factor of 2. In addition, since the orthodox ‘density-wave’ model is inapplicable to such self-gravitating rings, we devise a simple approach that describes the rings propagating as material waves. We find that the toy model can predict whether the simulated CRGs fragment or not using information from their pre-collision states. We also apply our instability analysis to a CRG discovered at a high redshift, z = 2.19. We find that a quite high-velocity dispersion is required for the stability of the ring, and therefore the CRG should be unstable to ring fragmentation. CRGs are rarely observed at high redshifts, and this may be because CRGs are usually too faint. Since the fragmentation can induce active star formation and make the ring bright enough to observe, the instability could explain this rarity. An unstable CRG fragments into massive clumps retaining the initial disc rotation, and thus it would evolve into a clumpy galaxy with a low surface density in an interclump region.

Instability analysis for spiral arms of local galaxies: M51, NGC 3627, and NGC 628

ABSTRACT We investigate dynamical states of grand-design spiral arms in three local galaxies: M51, NGC 3627, and NGC 628. Based on linear perturbation analysis considering multiple components in the galaxies, we compute instability parameters of the spiral arms using their observational data and argue whether the arms will fragment by their self-gravity. Our analysis utilizes observations of carbon monoxide (CO), 21-cm line emission, and multiband photometric images for molecular gas, atomic gas, and stellar components in the arms, respectively. We find that the grand-design arms of these galaxies indicate marginally stable states, and hence they are not on the way to fragment. We consider this to be consistent with the commonness of spiral galaxies and the relative rarity of fragmented discs at low redshifts. In the analysis, molecular gas is the dominant component to determine the (in)stability of the arms, whereas atomic gas and stars are far less important. Therefore, the results of our analysis are sensitive to an assumed CO-to-H2 conversion factor. If we assume a typical scatter of the measurements and admit nearly twice as large a conversion factor as our fiducial value, our analysis results in predicting the instability for the spiral arms. More sophisticated determination of the conversion factor is required for more accurate analysis for the (in)stability of spiral arms.

Simple Algebraic Expressions for the Prediction and Control of High-Temperature Annealed Structures by Linear Perturbation Analysis

The prediction and control of the transformation of void structures with high-temperature processing is a critical area in many engineering applications. In this work, focused on the void shape evolution of silicon, a novel algebraic model for the calculation of final equilibrium structures from initial void cylindrical trenches, driven by surface diffusion, is introduced. This algebraic model provides a simple and fast way to calculate expressions to predict the final geometrical characteristics, based on linear perturbation analysis. The obtained results are similar to most compared literature data, especially, to those in which a final transformation is reached. Additionally, the model can be applied in any materials affected by the surface diffusion. With such a model, the calculation of void structure design points is greatly simplified not only in the semiconductors field but in other engineering fields where surface diffusion phenomenon is studied.

Two-dimensional hybrid model of gradient drift instability and enhanced electron transport in a Hall thruster

An axial–azimuthal two-dimensional Hall thruster discharge model was developed for analyzing gradient drift instability (GDI) and cross field electron transport enhancement induced solely by the GDI. A hybrid particle-fluid model was used for the partially ionized plasma, where the inertialess electron fluid in the quasineutral plasma was assumed. A nonoscillatory numerical method was proposed for the potential solver in the electron fluid model to avoid numerical instability and analyze the physics of GDI accurately. A simulation is performed for a 1 kW-class anode-layer-type Hall thruster, and the flow field with plasma instability is presented. Plasma instability with vortex-like structures is observed in the acceleration and plume regions. The generated plasma instability enhances the cross field electron transport in the axial direction around the channel exit and in the plume region. Grid convergence is confirmed regarding the effect of electron transport enhancement, which indicates that cross field electron transport enhancement is based on the plasma instability. Furthermore, the comparison between the simulation results and linear perturbation analyses demonstrates that the simulated plasma instability reflects the theory of GDI. Thus, it is concluded that the hybrid model is useful for the analyses of GDI, and the GDI can enhance the cross field electron transport in Hall thrusters.

Magneto-acoustic waves in magnetic twisted flux tubes

At present, many works about MHD wave diagnostics in magnetic flux tubes are based on some pioneering works not considering the contributions of magnetic twist. Other works considered the effect on MHD waves, but the dispersion relationship they presented only gave the wave modes of m = 0,1,2… The kink mode of m = –1 was absent. Therefore, in this work we present a complete dispersion relationship that includes both magnetic twist and the wave mode of m = –1. Analogous to the m = +1 wave mode, the mode of m = –1 also exhibits the mode change at finite kr 0, from body to surface mode. The phase speeds of this mode are usually less than those of m = +1 mode. The harmonic curves of m = ± 1 modes in dispersion relationship diagrams are approximately symmetric in respect to a characteristic velocity, e.g. the tube velocity in flux tubes. Based on the present dispersion relationship, we revisit the issue of spiral wave patterns in sunspots and find that the magnetic twist has no great influence on their morphology in the frame of linear perturbation analysis.

Water vapor emissions from urban landscapes in Sakai, Japan

Water vapor emissions in urban areas are important processes in urban hydrology and urban climate. We evaluated the spatial and temporal characteristics of water vapor fluxes, based on measurements at an urban center for six years, suburban landscape for five years, and urban park for one year in Sakai, Osaka, Japan. The magnitudes of the water vapor emissions were approximately half for the urban center and one-fourth for the suburb compared with evapotranspiration in a planted forest near the city. Anthropogenic water vapor emissions were estimated to be high in the urban center, resulting in higher water vapor emissions in the urban center than in the suburb. The water vapor emissions at the three landscapes were controlled by net radiation and surface wetness, where surface conductance decreased with days after a rain event. Evapotranspiration of the urban park was comparable to that of the planted forest due to the dry urban atmosphere, known as the oasis effect. The green fraction explained the spatial variation in the annual water vapor emissions. Based on the linear perturbation analysis, the changes in water vapor emissions by urbanization could be determined by a balance between increased anthropogenic water vapor emissions and decreased surface conductance.

Numerical analysis of steel columns subject to eccentric loadings

Buckling of framed and plated structures has been a great concern that researchers try to handle over the past decades. In most developing nations such as ours, fewer or no experimental trials are available to obtain requisite information for the proper understanding of this phenomenon. It is on this premise that an attempt is made to conduct a preliminary study to numerically evaluate the buckling of steel columns under eccentric loadings. To achieve this, a static, linear perturbation analysis was initially performed on a pin-ended steel column using the subspace Eigen solver for the different buckled mode shapes to illustrate the likely behaviour of the column when subjected to compressive actions. Then, the static, general analysis was conducted with the column subjected to varying magnitudes of eccentric loadings. It was required to determine the load level at which the column would fail when subjected to these eccentric loadings. Consequently, a base load value equivalent to 10 % of Euler's critical buckling load was used. This load value was thereafter increased by 20 % in sequence. It was discovered that 10 % of the Euler's critical buckling load can alter the stiffness of the column when loaded eccentrically. It was further observed that the steel column finally failed at a load greater than 20 % of the Euler's critical buckling load and 40.1% of Rankine's critical buckling load. This is because the permissible deflection for unbraced columns may be taken as the quotient of effective length of column to 250, which translates to 13.8 mm. Therefore, the maximum deflection of 14.72 mm reached by applying an eccentric load of 514 kN exceeds the allowable limit of 13.8 mm.

Computing Covariant Lyapunov Vectors in Hilbert spaces

<p style='text-indent:20px;'>Covariant Lyapunov Vectors (CLVs) are intrinsic modes that describe long-term linear perturbations of solutions of dynamical systems. With recent advances in the context of semi-invertible multiplicative ergodic theorems, existence of CLVs has been proved for various infinite-dimensional scenarios. Possible applications include the derivation of coherent structures via transfer operators or the stability analysis of linear perturbations in models of increasingly higher resolutions.</p><p style='text-indent:20px;'>We generalize the concept of Ginelli's algorithm to compute CLVs in Hilbert spaces. Our main result is a convergence theorem in the setting of [<xref ref-type="bibr" rid="b19">19</xref>]. The theorem relates the speed of convergence to the spectral gap between Lyapunov exponents. While the theorem is restricted to the above setting, our proof requires only basic properties that are given in many other versions of the multiplicative ergodic theorem.</p>

Model design and finite element analysis of a traction link of a railcar

The traction link is a key component of the railcar suspension system which is used to transmit force and provide acoustic and vibration isolation. To ensure satisfactory performance of the railcar suspension system, the traction link should be properly designed. In this study, the traction link is considered as a component for local manufacturing. The aim of this study is to provide a guided approach on the design and simulation of the part to ensure that it meets the service and functional requirements. To clearly understand the effect of vibration and variable alternating loading on the mechanical integrity of the railcar suspension system, the designed traction link was analysed using finite element analysis. Dynamic analysis on the traction link was done using a linear perturbation analysis tool in the commercial FEM software ABAQUS©. The results show that the component is designed to a safe limit for the required service condition and that the material chosen is unlikely to deform under extreme conditions of the system. It is envisaged that the findings of this study will assist manufacturers in the development of railcar suspensions system that will meet the required service conditions.

Landau-like theory for buckling phenomena and its application to the elastica hypoarealis

Bifurcation phenomena are ubiquitous in elasticity, but their study is often limited to linear perturbation or numerical analysis since second or higher variations are often beyond an analytic treatment. Here, we review two key mathematical ideas, namely, the splitting lemma and the determinacy of a function, and show how they can be fruitfully used to derive a reduced function, named Landau expansion in the paper, that allows us to give a simple but rigorous description of the bifurcation scenario, including the stability of the equilibrium solutions. We apply these ideas to a paradigmatic example with potential applications to various softly constrained physical systems and biological tissues: a stretchable elastic ring under pressure. We prove the existence of a tricritical point and find bistability effects and hysteresis when the stretching modulus is sufficiently small. These results seem to be in qualitative agreement with some recent experiments on heart cells.