Stability Of Steady Solutions Research Articles

Existence and uniqueness of solutions to the Peierls–Nabarro model in anisotropic media

We study the existence and uniqueness of solutions to the vector field Peierls–Nabarro (PN) model for curved dislocations in a transversely isotropic medium. Under suitable assumptions for the misfit potential on the slip plane, we reduce the 3D PN model to a nonlocal scalar Ginzburg–Landau equation. For a particular range of elastic coefficients, the nonlocal scalar equation with explicit nonlocal positive kernel is derived. We prove that any stable steady solution has a one-dimensional profile. As a result, we obtain that solutions to the scalar equation, as well as the original 3D system, are characterized as a one-parameter family of straight dislocations. This paper generalizes results found previously for the full isotropic case to an anisotropic setting.

On the existence and monotonicity of pulsating traveling waves for reaction–diffusion–advection systems in high dimensional and periodic media

This paper is concerned with reaction–diffusion–advection systems posed in high dimensional and periodic media. Under certain conditions, we first prove that such systems admit pulsating traveling waves if and only if c≥c+∗(e), where c+∗(e) is the minimal wave speed. We also give a lower bound of the minimal wave speed. By establishing a comparison principle for pulsating traveling waves near the stable periodic steady solution, we further prove the monotonicity of pulsating traveling waves in the co-moving frame coordinate. These results generalize the known results for scalar equations and in the case when the system is invariant by translation along the direction of propagation. The obtained results are then applied to two species competition systems.

Stratospheric vacillations in a minimal contour dynamics model

AbstractA low-dimensional dynamical system that describes dynamical variability of the stratospheric polar vortex is presented. The derivation is based on a linearized, contour-dynamics representation of quasigeostrophic shallow water flow on a polar f-plane. The model consists of a single linear wave mode propagating on a near-circular patch of constant potential vorticity (PV). The PV jump at the vortex edge serves as an additional degree of freedom. The wave is forced by surface topography, and interacts with the vortex through a simplified parameterization of diabatic wave/mean flow interaction. The approach can be generalized to other geometries.The resulting three-component system depends on four non-dimensional parameters, and the structure of the steady state solutions can be determined analytically in some detail. Despite its extreme simplification, the model exhibits variability that is closely analogous to the Holton-Mass model, a well-known and more complex dynamical model of stratospheric variability. The present model exhibits two stable steady solutions, one consisting of a strong vortex with a small amplitude wave and the second consisting of a weak vortex with a large amplitude wave. Periodic and aperiodic limit cycles are also identified, analogous to similar solutions in the Holton-Mass model. Model trajectories also exhibit a number of behaviors that have been identified in observations. A key insight is that the time-mean state of the vortex is predominantly controlled by the properties of the linear mode, while the strength of the topographic forcing plays a far weaker role away from bifurcations.

A center of pressure progression model for walking with non heeled and heeled footwear

Background:Heeled footwear benefits people with movement disorder in the form of shoe lifts, wedges and inserts while its prolonged use causes foot injury in healthy people. There lies a need to detect parameters that affect COP progression of the foot and gait stability due to footwear. Research question:Do we have bipedal models that can estimate gait parameters corresponding to different center of pressure (COP) trajectories? Method:In this study, we propose a COP translation model that can account for non heeled to heeled footwear. We describe the COP progression as a function of the center of mass (COM) state. This model is used to generate stable steady state walking solutions for different COP profiles. We compare these model solutions with experimental data on non-heeled and heeled-gait. Results:The bipedal model shows stability across different COP profiles. The model estimates GRF profile (R2=0.83 for 1.3 m/s ) for non heeled normal walking qualitatively and on the temporal scale. It estimates GRF due to heeled gait (R2=0.83 for 1.08 m/s) but is limited in estimation of heeled gait parameters. Significance:A bipedal model that can generate stable steady state walking solutions for different forward progressing COP profiles can help in design of foot orthotics for patients with gait disorder and understand injuries occurring due to prolonged wear of rigid heeled footwear.

A Theory of Growth and Threshold Inflation with Estimates

For an inflation targeting Central Bank, a precise estimate of the threshold inflation in the economy is important. Existing studies provide estimates without any coherent theory of growth and threshold inflation and hence suffer from several limitations about concept and measurement. The present paper attempts to develop such a theory to establish a stable steady state growth solution. It also operationalizes the theory through a model with support from the Indian data for specific components of the model to derive the required functional form. Final estimates in India with annual data from 1995–96 to 2017–18 show that the threshold inflation and associated optimal growth vary considerably as rates of fiscal deficit and current account deficit on the balance of payments vary. The current combinations of the long term four policy targets of 4% inflation; 8% growth; 6% fiscal deficit (to GDP); and 2% current account deficit (to GDP) are internally inconsistent and hence not achievable. Now that there is an opportunity to revise the inflation target for the period after March 2021, the present paper argues for choosing from the menu of internally consistent options for all these four policy targets to avoid unnecessary costs.

On the multiple solutions of coating and rimming flows on rotating cylinders

We consider steady solutions of the Stokes equations for the flow of a film of fluid on the outer or inner surface of a cylinder that rotates with its axis perpendicular to the direction of gravity. We find that previously unobserved stable and unstable steady solutions coexist over an intermediate range of rotation rates for sufficiently high values of the Bond number (ratio of gravitational forces relative to surface tension). Furthermore, we compare the results of the Stokes calculations to the classic lubrication models of Pukhnachev (J. Appl. Mech. Tech. Phys., vol 18, 1977, pp. 344–351) and Reisfeld & Bankoff (J. Fluid Mech., vol. 236, 1992, pp. 167–196); an extended lubrication model of Benilov & O’Brien (Phys. Fluids, vol. 17, 2005, 052106) and Evans et al. (Phys. Fluids, vol. 16, 2004, pp. 2742–2756); and a new lubrication approximation formulated using gradient dynamics. We quantify the range of validity of each model and confirm that the gradient-dynamics model is most accurate over the widest range of parameters, but that the new steady solutions are not captured using any of the simplified models because they contain features that can only be described by the full Stokes equations.

On the critical free-surface flow over localised topography

Flow over bottom topography at critical Froude number is examined with a focus on steady, forced solitary wave solutions with algebraic decay in the far field, and their stability. Using the forced Korteweg–de Vries (fKdV) equation the weakly nonlinear steady solution space is examined in detail for the particular case of a Gaussian dip using a combination of asymptotic analysis and numerical computations. Non-uniqueness is established and a seemingly infinite set of steady solutions is uncovered. Non-uniqueness is also demonstrated for the fully nonlinear problem via boundary-integral calculations. It is shown analytically that critical flow solutions have algebraic decay in the far field both for the fKdV equation and for the fully nonlinear problem and, moreover, that the leading-order form of the decay is the same in both cases. The linear stability of the steady fKdV solutions is examined via eigenvalue computations and by a numerical study of the initial value fKdV problem. It is shown that there exists a linearly stable steady solution in which the deflection from the otherwise uniform surface level is everywhere negative.

Nonlinear stable steady solutions to the Ostroumov problem

We investigate the linear and weakly nonlinear solutions to a convection problem that was first studied by Ostroumov in 1947. The problem pertains to the stability of the equations governing convective motion in an infinite vertical fluid layer that is heated from below. Ostroumov’s linear stability analysis yields instability threshold conditions that are characterized by zero wavenumber for the Fourier mode in the vertical direction and by eigenfunctions that are independent of the vertical coordinate. Thus, any undertaking at determining the supercritical nonlinear solutions and their stability through a small amplitude expansion fails. This failure is attributed to the fact that the nonlinear interaction of the linear modes vanish identically. In this paper, we put forth exact and stable similarity type solutions to the Ostroumov problem. These solutions are characterized by the same linear threshold conditions as Ostroumov’s solutions. Moreover, we are able to extend the analysis to the supercritical regime through a small amplitude analysis to obtain steady two-dimensional solutions for a small range of Prandtl numbers. These solutions are found to be stable to general two-dimensional, time-dependent disturbances. Furthermore, when the analysis is extended to the case where the fluid layer thickness is also allowed to be infinite, we find that the infinite two-dimensional fluid region becomes linearly unstable through a Batchelor–Nitsche (BN) instability mechanism. Thus, the nonlinear solutions are obtained through a long wavelength expansion, and consequently our analysis provides the nonlinear development of the BN instability for the case of an unstable and uniform density profile. Finally, numerical solutions of the nonlinear problem are also presented which shed light on the flow patterns and temperature distributions for both the vertical channel and the infinitely extended stratified region problems.

A Numerical Study on Non-Isothermal Flow Through a Rotating Curved Duct with Square Cross Section

Non-isothermal flow through a rotating curved duct with square cross section is studied numerically by using the spectral method, and covering a wide range of the Taylor number, Tr, 0 ≤ Tr ≤ 2000 and Dean number, Dn, 0 ≤ Dn ≤ 2000. A temperature difference is applied across the vertical sidewalls for Grashof number Gr=500, where the outer wall is heated and the inner one cooled. The rotation (Coriolis force) of the duct about the center of curvature is imposed. The Steady solutions are obtained by using Newton-Raphson iteration method and the Dean numbers are also discussed in detail. Then, in order to investigate the non-linear behavior of the steady and unsteady solutions, there is no stable steady solution, time evolution calculations as well as power spectrum of the periodic oscillations are obtained, and it is also found that for larger Dn, the unsteady flow undergoes in the scenario ＂steady→periodic→chaotic→steady＂, if Tr is increased. Finally spectral analysis and phase space are found to be very useful investigation of the Non-isothermal fluid flow behavior.

Rayleigh-Taylor instability of steady fronts described by the Kuramoto-Sivashinsky equation.

We study steady thin reaction fronts described by the Kuramoto-Sivashinsky equation that separates fluids of different densities. This system may lead to hydrodynamic instabilities as buoyancy forces interact with the propagating fronts in a two-dimensional slab. We use Darcy's law to describe the fluid motion in this geometry. Steady front profiles can be flat, axisymmetric, or nonaxisymmetric, depending on the slab width, the density gradient, and fluid viscosity. Unstable flat fronts can be stabilized having a density gradient with the less dense fluid on top of a denser fluid. We find the steady front solutions from the nonlinear equations executing a linear stability analysis to determine their stability. We show regions of bistability where stable nonaxisymmetric and axisymmetric fronts can coexist. We also consider the stability of steady solutions in large domains, which can be constructed by dividing the domain into smaller parts or cells.

Reaction diffusion equation with spatio-temporal delay

We investigate reaction–diffusion equation with spatio-temporal delays, the global existence, uniqueness and asymptotic behavior of solutions for which in relation to constant steady-state solution, included in the region of attraction of a stable steady solution. It is shown that if the delay reaction function satisfies some conditions and the system possesses a pair of upper and lower solutions then there exists a unique global solution. In terms of the maximal and minimal constant solutions of the corresponding steady-state problem, we get the asymptotic stability of reaction–diffusion equation with spatio-temporal delay. Applying this theory to Lotka–Volterra model with spatio-temporal delay, we get the global solution asymptotically tend to the steady-state problem’s steady-state solution.

Multiple flow solutions in buoyancy induced convection in a porous square box

The convection induced by buoyancy effects in a porous square cavity has been investigated numerically using a spectral element code with bifurcation tools. The complex bifurcation diagram initiated from the first six primary bifurcation points corresponding to the onset of convection from the trivial no‐flow solution has been calculated. Four branches of stable steady solutions have been found, corresponding to one‐roll, two‐roll, three‐roll and four‐roll flow structures. The domain of existence of these stable solutions, i.e., the Rayleigh number (Ra) range in which such solutions can potentially be observed, has been precisely determined. It is shown that there exist Ra ranges where different flow solutions can be stable together. The stable branches all terminate at Hopf bifurcation points beyond which oscillatory solutions have been computed.

Quasi 1D Bose–Einstein condensate flow past a nonlinear barrier

The problem of a quasi 1D repulsive BEC flow past wide and narrow nonlinear barriers is investigated. It is shown that in contrast to the linear barrier case, for a wide nonlinear barrier an interval of velocities 0<v<v− always exists, where the flow is superfluid regardless of the barrier potential strength. In the case of a short range barrier stable and unstable steady solutions exist below some critical velocity. An unstable solution is shown to decay into a gray soliton moving downstream and a stable solution located at the barrier position.

Efficient non‐linear proper orthogonal decomposition/Galerkin reduced order models with stable penalty enforcement of boundary conditions

SUMMARYAn efficient, stability‐preserving model reduction technique for non‐linear initial boundary value problems whose solutions exhibit inherently non‐linear dynamics such as metastability and periodic regimes (limit cycles) is developed. The approach is based on the ‘continuous’ Galerkin projection approach in which the continuous governing equations are projected onto the reduced basis modes in a continuous inner product. The reduced order model (ROM) basis is constructed via a proper orthogonal decomposition (POD). In general, POD basis modes will not satisfy the boundary conditions of the problem. A weak implementation of the boundary conditions in the ROM based on the penalty method is developed. Asymptotic stability of the ROM with penalty‐enforced boundary conditions is examined using the energy method, following linearization and localization of the governing equations in the vicinity of a stable steady solution. This analysis, enabled by the fact that a continuous representation of the reduced basis is employed, leads to a model reduction method with an a priori stability guarantee. The approach is applied to two non‐linear problems: the Allen–Cahn (or ‘bistable’) equation and a convection‐diffusion‐reaction system representing a tubular reactor. For each of these problems, bounds on the penalty parameters that ensure asymptotic stability of the ROM solutions are derived. The non‐linear terms in the equations are handled efficiently using the ‘best points’ interpolation method proposed recently by Peraire, Nguyen et al. Numerical experiments reveal that the POD/Galerkin ROMs with stability‐preserving penalty boundary treatment for the two problems considered, both without as well as with interpolation, remain stable in a way that is consistent with the solutions to the governing continuous equations and capture the correct non‐linear dynamics exhibited by the exact solutions to these problems. Published 2012. This article is a US Government work and is in the public domain in the USA.

On higher order effects in marginally separated flows

AbstractIf the angle of attack α of a slender airfoil reaches a critical value αs flow separation is known to occur at the upper s surface. Further increase of α initially leads to the formation of a short laminar separation bubble which has an extremely weak influence on the external flow field – a phenomenon known as marginal separation – but then rather rapidly causes a severe change of the flow behaviour, leading to leading edge stall. According to the asymptotic theory of marginal separation holding in the limit of large Reynolds numbers Re, the flow in the neighbourhood of the separation bubble is governed by an integro‐differential equation. This so‐called interaction equation contains a single controlling parameter which relates the angle of attack to the Reynolds number, with a value Γs corresponding to αs. Some recent results concerning higher order s s corrections to this theory and their effect on the stability of steady solutions will be presented. (© 2011 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

A new algorithm for analysis of SVC’s impact on bifurcations, chaos and voltage collapse in power systems

In this paper, a numerical algorithm, based on initial value problem, using local parameterisation continuation technique is proposed for tracing stable and unstable steady state periodic solution branches of power systems. Bifurcation diagrams of steady state solutions are constructed by the application of the proposed algorithm. From the bifurcation diagrams, the existence of various bifurcation points such as, unstable Hopf bifurcation (UHB), stable Hopf bifurcation (SHB), cyclic fold bifurcation (CFB), saddle node bifurcation (SNB) and period doubling bifurcation (PDB) are identified. With the use of tools of nonlinear dynamics, voltage collapse points, and chaotic solutions due to period doublings are unearthed. Simulations have been carried out to analyse the sensitivity of the system with respect to load reactive power and compensating capacitor. The impact of SVC on Hopf bifurcations and occurrence of SNB are investigated. The algorithm is validated by applying it to a standard power system reported in literature and the results obtained are presented.

Rapid granular flows down inclined planar chutes. Part 2. Linear stability analysis of steady flow solutions

The linear stability of steady solutions for a rapid granular flow down an inclined chute, modelled using a kinetic theory continuum model, is analysed. The previous studies of Forterre &amp; Pouliquen (J. Fluid Mech., vol. 467, 2002, p. 361) and Mitarai &amp; Nakanishi (J. Fluid Mech., vol. 507, 2004, p. 309) are extended by considering fully three-dimensional perturbations, allowing variations in both the cross-slope and downslope directions, as well as normal to the base. Our results demonstrate the existence of three qualitatively different unstable perturbations, each of which can be the most rapidly growing instability for different steady flows. By considering the linear stability of many steady solutions along macroscopic flow curves, we show that linear stability occurs in only a small part of parameter space, and furthermore the regions of linear instability do not correlate with density inversion of the underlying steady solutions. Our results suggest that inelastic clustering is the dominant instability mechanism.

Time-dependent regimes of a Bursian diode I: Stability of steady solutions

A study is made of physical phenomena characteristic of the Bursian diode—a plane vacuum diode with a monoenergetic electron beam. The paper consists of two parts. The first part is devoted to analyzing the stability of steady solutions against small perturbations. It is shown that there exists a critical current density below which all steady solutions with a virtual cathode that lie on branch B are stable. The second part is aimed at investigating the features of oscillatory solutions.

Convection in a rapidly rotating spherical shell with an imposed laterally varying thermal boundary condition

We investigate thermally driven convection in a rotating spherical shell subject to inhomogeneous heating on the outer boundary, extending previous results to more rapid rotation rates and larger amplitudes of the boundary heating. The analysis explores the conditions under which steady flows can be obtained, and the stability of these solutions, for two boundary heating modes: first, when the scale of the boundary heating corresponds to the most unstable mode of the homogeneous problem; second, when the scale is larger. In the former case stable steady solutions exhibit a two-layer flow pattern at moderate rotation rates, but at very rapid rotation rates no steady solutions exist. In the latter case, stable steady solutions are always possible, and unstable solutions show convection rolls that cluster into nests that are out of phase with the boundary anomalies and remain trapped for many thermal diffusion times.

Stability of steady thermal convection in a tilted rectangular cavity in low-mode approximation

The model system of ordinary differential equations [1, 2] governing the behavior of a non-uniformly heated fluid in a tilted cavity is used for studying the stability of steady regimes of thermal convection at arbitrary (not small) tilting of the rectangular cavity. The bifurcation curve is constructed, which separates the region of parameters (the Rayleigh number — the cavity tilting angle) into two regions — the internal and external ones. In the external region, the system has one stable steady solution, and in the internal region, it has three steady solutions. One of them is always unstable in a monotone way, and two others may be both stable and unstable. The neutral curves are constructed, which determine the boundaries of the incipience of the oscillatory and monotone instabilities.