Predictions Of Linear Analysis Research Articles

Dynamical analysis of arbitrary dark energy and coincidence problem

This paper reviews a systematic dynamical analysis on a general form of scalar field as Dark Energy (DE) with dark matter (DM) to sort out the “cosmic coincidence” problem. Here the autonomous system of differential equations is two-dimensional (2D) as well as nonlinear. So we have utilized nonlinear dynamical theory to explain various cosmological implications of this model. Nowadays, we have noted that some works are undertaking this nonlinear systems theory. Although we have seen that most of the works are simplifying the underlying nonlinear dynamical systems similar to a linear one, that can lead to flawed conclusions about the evolution of the universe. Since an important theorem, Poincare–Bendixson theorem asserts linearization of the nonlinear system and does not give “global” stability, unlike the linear one if the dimension is more than two. Anyway, our work is different from others in this regard. Here the dimension of the system is two, and we have obtained some interesting stuffs also. We have applied the above theorem of nonlinear dynamical systems and others to find the “global” stability. This theorem offers completely different stable solutions, contrary to the prediction of linear analysis. As a result, we have obtained two fixed points; one of them is a stable “attractor” (it is attracting “node” actually), and thereafter, we have analyzed the stability. To investigate the dynamical system behavior, we have drawn different figures. These figures include vector field and a new plotting strategy (explained later). These investigations suggest a way out of the coincidence problem (or, precisely speaking, what should be the mathematical form of the term “[Formula: see text]”, which indicates interaction between DE and DM to reduce coincidence). In this scenario, if the equation of state (EoS) of DE and DM obeys [Formula: see text], then coincidence problem may be avoided.

Numerical Simulations of Instability in the Shell of a Supernova Remnant Expanding in a Weakly in Inhomogeneous Interstellar Medium

Astronomical observations show that the supernova remnants, even with a close to spherical shape, usually have multiscale ripple-like distortions. For example 15 bends on the shock front are clearly visible in the remnant 0509-67.5. The global instability of the flow is considered as one of the possible mechanisms for generating such structures. In the frame of linear analysis [26] was shown that this instability has a resonance character. It means that the perturbations with a certain wavelength number should grow faster, therefore ripples in the remnant shell will manifest itself predominantly in a certain range of scales. In this paper we present the results of numerical simulations of the nonlinear stage of this instability, caused by small perturbations in the external environment, depending on their scale and intensity. The unpertubed gas is supposed to has a power-law spartial dependence ρ0(r) ~ r-ω, where ω is a constant. The blast wave generated by a supernova expolosion is descibed by a Sedov type similarity solution. We have developed two-dimensional numerical model of adiabatic flow with a blast wave in a comoving frame of reference based on parallel code AstroChemHydro [1]. It was shown that, according to the predictions of linear analysis, perturbations in the external flow amplify behind the front of the shock wave, which leads to the development of convective instability and the development of turbulence. The results of numerical simulations demonstrated that in shell-type flows (for omega = 2,7 and gamma = 4/3) external disturbances along with the characteristic rearrangement of the shock front and turbulization of the flow behind it, cause the formation of radially elongated filaments with a vortex structure behind the shock, the number of which is determined by the harmonic number of the perturbation l.

Stability of the anabatic Prandtl slope flow in a stably stratified medium

In the Prandtl model for anabatic slope flows, a uniform positive buoyancy flux at the surface drives an upslope flow against a stable background stratification. In the present study, we conduct linear stability analysis of the anabatic slope flow under this model and contrast it against the katabatic case as presented in Xiao & Senocak (J. Fluid Mech., vol. 865, 2019, R2). We show that the buoyancy component normal to the sloped surface is responsible for the emergence of stationary longitudinal rolls, whereas a generalised Kelvin–Helmholtz (KH) type of mechanism consisting of shear instability modulated by buoyancy results in a streamwise-travelling mode. In the anabatic case, for slope angles larger than to the horizontal, the travelling KH mode is dominant whereas, at lower inclination angles, the formation of the stationary vortex instability is favoured. The same dynamics holds qualitatively for the katabatic case, but the mode transition appears at slope angles of approximately . For a fixed slope angle and Prandtl number, we demonstrate through asymptotic analysis of linear growth rates that it is possible to devise a classification scheme that demarcates the stability of Prandtl slope flows into distinct regimes based on the dimensionless stratification perturbation number. We verify the existence of the instability modes with the help of direct numerical simulations, and observe close agreements between simulation results and predictions of linear analysis. For slope angle values in the vicinity of the junction point in the instability map, both longitudinal rolls and travelling waves coexist simultaneously and form complex flow structures.

How memory of direct animal interactions can lead to territorial pattern formation

Mechanistic home range analysis (MHRA) is a highly effective tool for understanding spacing patterns of animal populations. It has hitherto focused on populations where animals defend their territories by communicating indirectly, e.g. via scent marks. However, many animal populations defend their territories using direct interactions, such as ritualized aggression. To enable application of MHRA to such populations, we construct a model of direct territorial interactions, using linear stability analysis and energy methods to understand when territorial patterns may form. We show that spatial memory of past interactions is vital for pattern formation, as is memory of 'safe' places, where the animal has visited but not suffered recent territorial encounters. Additionally, the spatial range over which animals make decisions to move is key to understanding the size and shape of their resulting territories. Analysis using energy methods, on a simplified version of our system, shows that stability in the nonlinear system corresponds well to predictions of linear analysis. We also uncover a hysteresis in the process of territory formation, so that formation may depend crucially on initial space-use. Our analysis, in one dimension and two dimensions, provides mathematical groundwork required for extending MHRA to situations where territories are defended by direct encounters.

A weighted residual method for two-layer non-Newtonian channel flows: steady-state results and their stability

AbstractWe study buoyant displacement flows in a plane channel with two fluids in the long-wavelength limit in a stratified configuration. Weak inertial effects are accounted for by developing a weighted residual method. This gives a first-order approximation to the interface height and flux functions in each layer. As the fluids are shear-thinning and have a yield stress, to retain a formulation that can be resolved analytically requires the development of a system of special functions for the weight functions and various integrals related to the base flow. For displacement flows, the addition of inertia can either slightly increase or decrease the speed of the leading displacement front, which governs the displacement efficiency. A more subtle effect is that a wider range of interface heights are stretched between advancing fronts than without inertia. We study stability of these systems via both a linear temporal analysis and a numerical spatiotemporal method. To start with, the Orr–Sommerfeld equations are first derived for two generalized non-Newtonian fluids satisfying the Herschel–Bulkley model, and analytical expressions for growth rate and wave speed are obtained for the long-wavelength limit. The predictions of linear analysis based on the weighted residual method shows excellent agreement with the Orr–Sommerfeld approach. For displacement flows in unstable parameter ranges we do observe growth of interfacial waves that saturate nonlinearly and disperse. The observed waves have similar characteristics to those observed experimentally in pipe flow displacements. Although the focus in this study is on displacement flows, the formulation laid out can be easily used for similar two-layer flows, e.g. co-extrusion flows.

Three-dimensional instabilities in compressible flow over open cavities

Direct numerical simulations are performed to investigate the three-dimensional stability of compressible flow over open cavities. A linear stability analysis is conducted to search for three-dimensional global instabilities of the two-dimensional mean flow for cavities that are homogeneous in the spanwise direction. The presence of such instabilities is reported for a range of flow conditions and cavity aspect ratios. For cavities of aspect ratio (length to depth) of 2 and 4, the three-dimensional mode has a spanwise wavelength of approximately one cavity depth and oscillates with a frequency about one order of magnitude lower than two-dimensional Rossiter (flow/acoustics) instabilities. A steady mode of smaller spanwise wavelength is also identified for square cavities. The linear results indicate that the instability is hydrodynamic (rather than acoustic) in nature and arises from a generic centrifugal instability mechanism associated with the mean recirculating vortical flow in the downstream part of the cavity. These three-dimensional instabilities are related to centrifugal instabilities previously reported in flows over backward-facing steps, lid-driven cavity flows and Couette flows. Results from three-dimensional simulations of the nonlinear compressible Navier–Stokes equations are also reported. The formation of oscillating (and, in some cases, steady) spanwise structures is observed inside the cavity. The spanwise wavelength and oscillation frequency of these structures agree with the linear analysis predictions. When present, the shear-layer (Rossiter) oscillations experience a low-frequency modulation that arises from nonlinear interactions with the three-dimensional mode. The results are consistent with observations of low-frequency modulations and spanwise structures in previous experimental and numerical studies on open cavity flows.

Instability, Dynamics, and Morphology of Thin Slipping Films

Based on the linear stability and nonlinear simulations, we show that the surface instability, dynamics, and morphology of supported thin liquid films are profoundly altered by the presence of slippage on the substrate. A general dispersion equation for flow in slipping thin films is derived and simplified to identify three different regimes of slippage (weak, moderate, and strong) and obtain the length and time scales of instability in them. For illustration, the ubiquitous van der Waals interactions have been employed. Different regimes of slip-flow can be predicted based on a nondimensional parameter, xi, which is a function of slip length, film thickness, intermolecular potential, and interfacial tension. Two distinct transitions from weak to moderate slip and from moderate to strong slip occur at xiT1 approximately 0.01 and xiT2 approximately 500, respectively. More specifically, a decrease in film thickness causes transitions from weak to moderate to strong slip regime. Even a weak slippage causes faster breakup of a thin film, whereas slippage beyond a transition value (slip length, bT1) increases the length scale of instability and reduces the number density of holes compared to the nonslipping case. Strong slippage produces holes faster, and the holes are fewer in number and have less developed rims. The exponents for the length scale (lambdam infinity h0n; h0 is film thickness) and time scale of instability (tr infinity h0m) change nonmonotonically with slippage (for nonretarded van der Waals instability, n E (1.25, 2), m E (3, 6)). Retardation in van der Waals potential increases the exponents (n E (1.5, 2.5), m E (5, 8)). The initial stage of evolution of a slipping film, simulated based on nonlinear equations, follows the length scale and time scale of instability, close to the prediction of linear analysis. It is hoped that the present analysis will help in better interpretation of thin film experiments, in estimation of slippage, and in the determination of intermolecular forces from the length and time scales of the instability.

Nonlinear Analysis of Juxtacrine Patterns

We investigate a discrete mathematical model for a type of cell-cell communication in early development which has the potential to generate a wide range of spatial patterns. Our previous work on this model has highlighted surprising differences between the predictions of linear analysis and the results of numerical simulations. In particular, there is no quantitative agreement between the unstable modes derived from linear analysis and the patterns observed numerically. In this paper, we look at the nonlinear model on a domain of two cells with the aim of gaining an insight into behavior in larger systems. We study the existence and stability of spatially heterogeneous steady-state solutions, which correspond to patterns of alternating cell fate on larger domains, as we vary two key parameters. These parameters are measures of the strength of positive feedback in the biological system. By reducing the problem to two coupled nonlinear algebraic equations, we show that a patterned solution exists and is stable on a 2-cell domain for a significant part of parameter space. We compare these results to those obtained from linear analysis and conclude that the behavior of the nonlinear 2-cell system gives a better insight into the results of numerical simulations on large arrays of cells. Furthermore, we conduct a bifurcation analysis of the model on domains of various sizes: we demonstrate that as the domain size increases, the 2-cell pattern becomes unstable for certain parameters, and overall the number of stable patterns increases. This leads us to speculate that on large domains there are many stable patterned solutions to the model of approximately the same periodicity, which is typical of the fine-grained patterns that one sees during early development. Our work predicts that this is a feature of the patterning dynamics rather than a consequence of environmental heterogeneity.

Direct numerical simulation of isotropic turbulence interacting with a weak shock wave

Interaction of isotropic quasi-incompressible turbulence with a weak shock wave was studied by direct numerical simulations. The effects of the fluctuation Mach number Mt of the upstream turbulence and the shock strength M21 — 1 on the turbulence statistics were investigated. The ranges investigated were 0.0567 ≤ Mt ≤ 0.110 and 1.05 ≤ M1 ≤ 1.20. A linear analysis of the interaction of isotropic turbulence with a normal shock wave was adopted for comparisons with the simulations.Both numerical simulations and the linear analysis of the interaction show that turbulence is enhanced during the interaction with a shock wave. Turbulent kinetic energy and transverse vorticity components are amplified, and turbulent lengthscales are decreased. The predictions of the linear analysis compare favourably with simulation results for flows with M2t < a(M21 — 1) with a ≈ 0.1, which suggests that the amplification mechanism is primarily linear. Simulations also showed a rapid evolution of turbulent kinetic energy just downstream of the shock, a behaviour not reproduced by the linear analysis. Investigation of the budget of the turbulent kinetic energy transport equation shows that this behaviour can be attributed to the pressure transport term.Shock waves were found to be distorted by the upstream turbulence, but still had a well-defined shock front for M2t < a(M21— 1) with a ≈ 0.1). In this regime, the statistics of shock front distortions compare favourably with the linear analysis predictions. For flows with M2t > a(M21— 1 with a ≈ 0.1, shock waves no longer had well-defined fronts: shock wave thickness and strength varied widely along the transverse directions. Multiple compression peaks were found along the mean streamlines at locations where the local shock thickness had increased significantly.