Stability Of Steady Solutions Research Articles

Supercritical surface gravity waves generated by a positive forcing

Forced surface waves on an incompressible, inviscid fluid in a two-dimensional channel with a small bump on a horizontal rigid flat bottom are studied. The wave motion on the free surface is determined by a nondimensional wave speed F, called Froude number, and F = 1 is a critical value of F. If F = 1 + λ ϵ with ϵ > 0 a small parameter, then a time-dependent forced Korteweg–de Vries (FKdV) equation can be derived to model the wave motion on the free surface. Here, the case λ ⩾ 0 (or F ⩾ 1 , called supercritical case) is considered. The steady FKdV equation is first studied both theoretically and numerically. It is shown that there exists a cut-off value λ 0 of λ. For λ ⩾ λ 0 there are steady solutions, while for 0 ⩽ λ < λ 0 no steady solution of FKdV exists. For the unsteady FKdV equation, it is found that for λ > λ 0 , the solution of FKdV with zero initial condition tends to a stable steady solution, whilst for 0 < λ < λ 0 a succession of solitary waves are periodically generated and continuously propagating upstream as time evolves. Moreover, the solutions of FKdV equation with nonzero initial conditions are studied.

Effects of Curvature and Convective Heat Transfer in Curved Square Duct Flows

Non-isothermal flows with convective heat transfer through a curved duct of square cross section are numerically studied by using a spectral method, and covering a wide range of curvature, δ, 0&lt;δ≤0.5 and the Dean number, Dn, 0≤Dn≤6000. A temperature difference is applied across the vertical sidewalls for the Grashof number Gr=100, where the outer wall is heated and the inner one cooled. Steady solutions are obtained by the Newton-Raphson iteration method and their linear stability is investigated. It is found that the stability characteristics drastically change due to an increase of curvature from δ = 0.23 to 0.24. When there is no stable steady solution, time evolution calculations as well as their spectral analyses show that the steady flow turns into chaos through periodic or multi-periodic flows if Dn is increased no matter what δ is. The transition to a periodic or chaotic state is retarded with an increase of δ. Nusselt numbers are calculated as an index of horizontal heat transfer and it is found that the convection due to the secondary flow, enhanced by the centrifugal force, increases heat transfer significantly from the heated wall to the fluid. If the flow becomes periodic and then chaotic, as Dn increases, the rate of heat transfer increases remarkably.

Stability of the Global Ocean Circulation: Basic Bifurcation Diagrams

AbstractA study of the stability of the global ocean circulation is performed within a coarse-resolution general circulation model. Using techniques of numerical bifurcation theory, steady states of the global ocean circulation are explicitly calculated as parameters are varied. Under a freshwater flux forcing that is diagnosed from a reference circulation with Levitus surface salinity fields, the global ocean circulation has no multiple equilibria. It is shown how this unique-state regime transforms into a regime with multiple equilibria as the pattern of the freshwater flux is changed in the northern North Atlantic Ocean. In the multiple-equilibria regime, there are two branches of stable steady solutions: one with a strong northern overturning in the Atlantic and one with hardly any northern overturning. Along the unstable branch that connects both stable solution branches (here for the first time computed for a global ocean model), the strength of the southern sinking in the South Atlantic changes substantially. The existence of the multiple-equilibria regime critically depends on the spatial pattern of the freshwater flux field and explains the hysteresis behavior as found in many previous modeling studies.

Transition from Steady to Chaotic States of Isothermal and Non-isothermal Flows through a Curved Rectangular Duct

Flows through a curved rectangular duct for the aspect ratio l =2 are numerically studied by use of the spectral method with and without a temperature difference between the vertical outer and inner sidewalls. We find three branches of steady solutions for the case without the temperature difference (isothermal case). Then we investigate linear stability of the steady solutions and find that only a portion of one steady solution branch is linearly stable but other branches are unstable. We obtain five branches of steady solutions for the case with the temperature difference (non-isothermal case) for the Grashof number Gr =100. Linear stability shows that only a portion of one of them is linearly stable while other branches are unstable like the isothermal case. The change of the flow state, as the Dean number Dn is increased, obtained by time evolution calculations, is found to be similar for both the isothermal and non-isothermal cases. When there is no stable steady solution, the time evolution calculations show that typical transition occurs from a steady flow to a chaotic state through a periodic flow when Dn is increased whether the system is isothermal or non-isothermal.

The properties of solutions of the fundamental problem of dynamics in systems with non-ideal constraints

The problem of determining the accelerations and reactions of constraints in systems with Coulomb friction is discussed. It is shown that a realistic solution of the problem is possible provided only that allowance is made for deformations in the bodies comprising the system. For this purpose, the initial system is expanded by including local deformations among the generalized coordinates. Asymptotic methods are used to divide the expanded system into a “slow” initial subsystem and a “fast” subsystem that serves to determine the reactions. Analysis of the fast subsystem is the key to understanding the dynamics on the whole. The results obtained for the number of steady solutions and their stability are invariant with respect to the viscoelastic characteristics of the contact pairs. For every stable steady solution there is a realizable motion of the initial system, but in order to select the true motion one has to know the initial deformations and their derivatives with respect to time. Along with steady solutions, the “fast” subsystem may have stable oscillatory solutions. It is proved that to the stable limit cycle of the “fast” subsystem there correspond motions of the initial system in which the reactions oscillate at a high frequency about their mean values. The Painlevé-Klein system and the problem of braking a wheel with two brake shoes are considered as examples.

AN EXPLICIT SUBPARAMETRIC SPECTRAL ELEMENT METHOD OF LINES APPLIED TO A TUMOUR ANGIOGENESIS SYSTEM OF PARTIAL DIFFERENTIAL EQUATIONS

In this paper we consider a numerical solution to Anderson and Chaplain's tumour angiogenesis model1 over two-dimensional complex geometry. The numerical solution of the governing system of non-linear evolutionary partial differential equations is obtained using the method of lines: after a spatial semi-discretisation based on the subparametric Legendre spectral element method is performed, the original system of partial differential equations is replaced by an augmented system of stiff ordinary differential equations in autonomous form, which is then advanced forward in time using an explicit time integrator based on the fourth-order Chebyshev polynomial. Numerical simulations show the convergence of the steady state numerical solution towards the linearly stable steady state analytical solution.

Direct Computational Simulations for Internal Condensing Flows and Results on Attainability/Stability of Steady Solutions, Their Intrinsic Waviness, and Their Noise Sensitivity

The paper presents a new two-dimensional computational approach and results for laminar/laminar internal condensing flows. Accurate numerical solutions of the full governing equations are presented for steady and unsteady film condensation flows on a sidewall inside a vertical channel. It is found that exit conditions and noise sensitivity are important. Even for stable steady solutions obtained for nearly incompressible vapor phase flows associated with unconstrained exit conditions, the noise sensitivity to the condensing surface’s minuscule transverse vibrations is high. The structure of waves, the underlying characteristics, and the “growth/damping rates” for the disturbances are discussed. A resonance condition for high “growth rates” is proposed and its efficacy in significantly enhancing wave motion and heat transfer rates is computationally demonstrated. For the unconstrained exit cases, the results make possible a separately reported study of the effects of shear, gravity, and surface tension on noise sensitive stable solutions.

The structure of flame filaments in chaotic flows

The structure of flame filaments resulting from chaotic mixing within a combustion reaction is considered. The transverse profile of the filaments is investigated numerically and analytically based on a one-dimensional model that represents the effect of stirring as a convergent flow. The dependence of the steady solutions on the Damköhler number and Lewis number is treated in detail. It is found that, below a critical Damköhler number Da crit, the flame is quenched by the flow. The quenching transition appears as a result of a saddle-node bifurcation where the stable steady filament solution collides with an unstable one. The shape of the steady solutions for the concentration and temperature profiles changes with the Lewis number and the value of Da crit increases monotonically with the Lewis number. Properties of the solutions are studied analytically in the limit of large Damköhler number and for small and large Lewis number.

Bifurcation and stability of combined free and forced convection in rotating curved ducts of square cross-section

A numerical study is made on fully developed bifurcation structure and stability of combined free and forced convection in a rotating curved duct of square cross-section. The solution structure is determined as the variation of a parameter indicating the magnitude of buoyancy force. Steady solution structure is very complicated. Flow and temperature fields on various solution branches are identified to be symmetric/asymmetric multi-cell patterns. Dynamic responses of multiple solutions to finite random disturbances are examined by direct transient computation. Five types of physically realizable solutions are identified numerically. They are stable steady 2-cell solution, stable steady multi-cell solution, periodic oscillation, chaotic oscillation and symmetry-breaking oscillation led by sub-harmonic bifurcation (period doubling). Among them, three kinds of stable steady solutions are found to co-exist within a range of parameters. In addition, temporal periodic and chaotic oscillations can also co-exist in another range of parameters. Furthermore, sub-harmonic bifurcation is identified to be another route to chaos. Spectral analysis is used to demonstrate the presence of additional frequencies for the case of sub-harmonic bifurcations. Results show that symmetry-breaking oscillation driven by sub-harmonic bifurcations appear to be identical with the mode observed in Lipps [J. Fluid Mech. 75 (1976) 113], McLaughlin and Orszag [J. Fluid Mech. 122 (1982) 123], and Gollub and Benson [J. Fluid Mech. 100 (1980) 449] for problem of free convection between flat horizontal plates.

Laminar flows through a curved rectangular duct over a wide range of the aspect ratio

The laminar flow in a curved rectangular duct for a range of the aspect ratio 1 ≤ l ≤ 12 is investigated by use of the spectral method. The steady solutions are obtained using the Newton–Raphson method with the symmetry condition. As a result, five branches of steady solutions are found. Linear stability characteristics are also investigated for all the steady solutions. It is found that one steady solution is linearly stable for most of l, but two linearly stable steady solutions exist for a region of small l and there are several intervals of l where there is no linearly stable steady solution. We performed time-evolution calculations with and without the symmetry condition, and observed periodic oscillations with the symmetry condition and aperiodic time evolutions without the symmetric condition. Finally, the present results numerically suggest that what determines which solution is realizable may be the maximum of the momentum transfer in the cross section.

Solution Structure and Stability of Viscous Flow in Curved Square Ducts

The bifurcation structure of viscous flow in curved square ducts is studied numerically and the stability of solutions on various solution branches is examined extensively. The solution structure of the flow is determined using the Euler-Newton continuation, the arc-length continuation, and the local parameterization continuation scheme. Test function and branch switch technique are used to monitor the bifurcation points in each continuation step and to switch branches. Up to 6 solution branches are found for the case of a flow in the curved square channel within the parameter range under consideration. Among them, three are new. The flow patterns on various bifurcation branches are also examined. A direct transient calculation is made to determine the stability of various solution branches. The results indicate that, within the scope of the present work, at given set of parameter values, the arbitrary initial disturbances lead all solutions to the same state. In addition to stable steady two-vortex solutions and temporally periodic solutions, intermittent and chaotic oscillations are discovered within a certain region of the parameter space. Temporal intermittency that is periodic for certain time intervals manifests itself by bursts of aperiodic oscillations of finite duration. After the burst, a new periodic phase starts, and so on. The intermittency serves as one of the routes for the onset of chaos. The results show that the chaotic flow in the curved channel develops through the intermittency. The chaotic oscillations appear when the number of bursts becomes large. The calculations also show that transient solutions on various bifurcation branches oscillate chaotically about the common equilibrium states at a high value of the dynamic parameter.

A Mathematical Model for Quorum Sensing in Pseudomonas aeruginosa

The bacteria Pseudomonas aeruginosa use the size and density of their colonies to regulate the production of a large variety of substances, including toxins. This phenomenon, called quorum sensing, apparently enables colonies to grow to sufficient size undetected by the immune system of the host organism. In this paper, we present a mathematical model of quorum sensing in P. aeruginosa that is based on the known biochemistry of regulation of the autoinducer that is crucial to this signalling mechanism. Using this model we show that quorum sensing works because of a biochemical switch between two stable steady solutions, one with low levels of autoinducer and one with high levels of autoinducer.

The unsteady motion of two-dimensional flags with bending stiffness

The motion of a two-dimensional flag at a time-dependent angle of incidence to an irrotational flow of an inviscid, incompressible fluid is examined. The flag is modelled as a thin, flexible, impermeable membrane of finite mass with bending stiffness. The flag is fixed at the leading edge where it is assumed to be either freely hinged or clamped with zero gradient. The angle of incidence to the outer flow is assumed to be small and thin aerofoil theory and simple beam theory are employed to obtain a partial singular integro-differential equation for the flag shape. Steady solutions to the problem are calculated analytically for various limiting cases and numerically for order one values of a non-dimensional parameter that measures the relative importance of outer flow momentum flux and flexural rigidity. For the unsteady problem, the stability of steady solutions depends only upon two non-dimensional parameters. Stability analysis is performed in order to identify the regions of instability. The resulting quadratic eigenvalue problem is solved numerically and the marginal stability curves for both the hinged and the clamped flags are constructed. These curves show that both stable and unstable solutions may exist for various values of the mass and flexural rigidity of the membrane and for both methods of attachment at the leading edge. In order to confirm the results of the linear stability analysis, the full unsteady flag equation is solved numerically using an explicit method. The numerical solutions agree with the predictions of the linear stability analysis and also identify the shapes that the flag adopts according to the magnitude of the flexural rigidity and mass.

Transitions of natural convection in a horizontal annulus

Transitions of natural convection in an annulus between horizontal concentric cylinders are investigated theoretically by assuming two-dimensional and incompressible flow fields. It is assumed that the inner cylinder is kept at a higher temperature than the outer cylinder. It is confirmed by numerical simulations that dual stable steady solutions exist for Rayleigh numbers larger than a critical value. Bifurcation diagrams of the steady solutions are obtained by Newton–Raphson's method for various values of ratio of the inner cylinder diameter to the gap width and their linear stability is investigated. The origin of the dual solutions is clarified from the bifurcation analysis.

Matched asymptotic solutions for the steady banded flow of the diffusive Johnson–Segalman model in various geometries

We present analytic solutions for steady flow of the Johnson–Segalman (JS) model with a diffusion term in various geometries and under controlled strain rate conditions, using matched asymptotic expansions. The diffusion term represents a singular perturbation that lifts the continuous degeneracy of stable, banded, steady states present in the absence of diffusion. We show that the stable steady flow solutions in Poiseuille and cylindrical Couette geometries always have two bands. For Couette flow and small curvature, two different banded solutions are possible, differing by the spatial sequence of the two bands.

Stability of Pole Solutions for Planar Propagating Flames: II. Properties of Eigenvalues/Eigenfunctions and Implications to Stability

In a previous paper (Part I) we focused our attention on pole solutions that arise in the context of flame propagation. The nonlinear development that follows after a planar flame front becomes unstable is described by a single nonlinear PDE which admits pole solutions as equilibrium states. Specifically, we were concerned with coalescent steady states, which correspond to steadily propagating single-peak structures extended periodically over the infinite domain. This pattern is one that is commonly observed in experiments. In order to examine the linear stability of these equilibrium solutions, we formulated in Part I the corresponding eigenvalue problem and derived exact analytical expressions for the spectrum and the corresponding eigenfunctions. In this paper, we examine their properties as they relate to the stability issue. Being based on analytical expressions, our results resolve earlier controversies that resulted from numerical investigations of the stability problem. We show that, for any period 2L, there always exists one and only one stable steady coalescent pole solution. We also examine the dependence of the eigenvalues and eigenfunctions on L which provides insight into the behavior of the nonlinear PDE and, consequently, on the nonlinear dynamics of the flame front.

Steady states of the surface of a nonadiabatic flame near the limits

Within the framework of a weakly linear model, which describes a nonadiabatic flame near the limit of its propagation caused by heat losses, steady states of the combustion-wave from are studied. Three-dimensional structures of the wave front are formed because of diffusion-thermal instability of the planar flame. The limits of propagation of a curved flame front are shown to expand if the diffusion-thermal instability is taken into account: a cellular flame can exist at heat losses higher than the critical value for the two-dimensional flame. The stability of steady solutions, which describe the cellular flame near the limits of its propagation, is studied. For sufficiently high heat losses, steady solutions for a nonadiabatic flame with front discontinities are obtained.

Computations of a laminar backward-facing step flow atRe=800 with a spectral domain decomposition method

The two-dimensional laminar incompressible flow over a backward-facing step is computed using a spectral domain decomposition approach. A minimum number of subdomains (two) is used; high resolution being achieved by increasing the order of the basis Chebyshev polynomial. Results for the case of a Reynolds number of 800 are presented and compared in detail with benchmark computations. Stable accurate steady flow solutions were obtained using substantially fewer nodes than in previously reported simulations. In addition, the problem of outflow boundary conditions was examined on a shortened domain. Because of their more global nature, spectral methods are particularly sensitive to imposed boundary conditions, which may be exploited in examining the effect of artificial (non-physical) outflow boundary conditions. Two widely used set of conditions were tested: pseudo stress-free conditions and zero normal gradient conditions. Contrary to previous results using the finite volume approach, the latter is found to yield a qualitatively erroneous yet stable flow-field. Copyright © 1999 John Wiley & Sons, Ltd.

Stability of Steady Solutions of Burgers's Equation

We develop tools to discuss the stability of viscous shock profiles for conservation laws. The technique is based on replacing the far field regions by boundary conditions that describe the effect of the outer solution on the solution in the shock layer region. We apply the technique to Burgers's equation and prove stability for perturbations that are not localized in space.

Energy constraints in forced recirculating MHD flows

We are concerned here with forced steady recirculating flows which are laminar, two-dimensional and have a high Reynolds number. The body force is considered to be prescribed and independent of the flow, a situation which arises frequently in magnetohydrodynamics. Such flows are subject to a strong constraint. Specifically, the body force generates kinetic energy throughout the flow field, yet dissipation is confined to narrow singular regions such as boundary layers. If the flow is to achieve a steady state, then the kinetic energy which is continually generated within the bulk of the flow must find its way to the dissipative regions. Now the distribution of u2/2 is governed by a transport equation, in which the only cross-stream transport of energy is diffusion, v∇2 (u2/2). It follows that there are only two possible candidates for the transport of energy to the dissipative regions: the energy could be diffused to the shear layers, or else it could be convected to the shear layers through entrainment of the streamlines. We investigate both options and show that neither is a likely candidate at high Reynolds number. We then describe numerical experiments for a model problem designed to resolve these issues. We show that, at least for our model problem, no stable steady solution exists at high Reynolds number. Rather, as soon as the Reynolds number exceeds a modest value of around 10, the flow becomes unstable via a supercritical Hopf bifurcation.