Linear Stability Problem Research Articles

A parallel adaptive unstructured finite volume method for linear stability (normal mode) analysis of viscoelastic fluid flows

A parallel unstructured finite volume method is presented for analysis of the stability of two-dimensional steady Oldroyd-B fluid flows to small amplitude three-dimensional perturbations. A semi-staggered dilation-free finite volume discretization with Newton’s method is used to compute steady base flows. The linear stability problem is treated as a generalized eigenvalue problem (GEVP) in which the rightmost eigenvalue determines the stability of the base flow. The rightmost eigenvalues associated with the most dangerous eigenfunctions are computed through the use of the shift-invert Arnoldi method. To avoid fine meshing in the regions where the flow variables are changing slowly, a local mesh refinement technique is used in order to increase numerical accuracy with a lower computational cost. The CUBIT mesh generation environment has been used to refine the quadrilateral meshes locally. In order to achieve higher performance on parallel machines the algebraic systems of equations resulting from the steady problem and the GEVP have been solved by implementing the MUltifrontal Massively Parallel Solver ( MUMPS). The proposed method is applied to the linear stability analysis of the flow of an Oldroyd-B fluid past a linear periodic array of circular cylinders in a channel and a linear array of circular half cylinders placed on channel walls. Two different leading eigenfunctions are identified for close and wide cylinder spacing for the periodic array of cylinders. The numerical results indicate good agreement with the numerical and experimental results available in the literature.

Effect of an electric field on the stability of contaminated film flow down an inclined plane

The stability of a liquid film flowing down an inclined plane is considered when the film is contaminated by an insoluble surfactant and subjected to a uniform normal electric field. The liquid is treated as a perfect conductor and the air above the film is treated as a perfect dielectric. Previous studies have shown that, when acting in isolation, surfactant has a stabilizing influence on the flow while an electric field has a destabilizing influence. The competition between these two effects is the focus of the present study. The linear stability problem is formulated and solved at arbitrary parameter values. An extended form of Squire's theorem is presented to argue that attention may be confined to two-dimensional disturbances. The stability characteristics for Stokes flow are described exactly; the growth rates of the normal modes at finite Reynolds number are computed numerically. We plot the neutral curves dividing regions of stability and instability, and trace how the topology of the curves changes as the intensity of the electric field varies both for a clean and for a contaminated film. With a sufficiently strong electric field, the neutral curve for a clean film consists of a lower branch trapping an area of stable modes around the origin, and an upper branch above which the flow is stable. With surfactant present, a similar situation obtains, but with an additional island of stable modes disjoint from the upper and lower branches.

Exponential Dissipativity of Discrete-time Stochastic Systems and Robust Simultaneous Stabilization via Output Feedback

The paper studies robust simultaneous stabilization problem via output feedback for a set of nonlinear discrete-time uncertain systems. New version of dissipativity called exponential dissipativity is defined for discrete-time stochastic system with Markovian switching. It is proved that if this stochastic system with special choice of its parameters is exponentially dissipative then the original set of uncertain system is simultaneously stabilizable and the stabilizing control has robustness properties in the sense that it admits some feedback uncertainties. The linear robust simultaneous stabilization problem is considered as a particular case. In this case the results are obtained in the form of convergent algorithm for computing of output feedback gain, based on iterative solution of LMI's. This algorithm is applied to the problem of angular stabilization of longitudinal multi-regime aircraft motion.

Towards the problem of hydrodynamic stability of plane‐parallel dusty‐gas flows

AbstractSeveral modifications of the classical Saffman formulation of the dusty‐gas flow linear stability problem are considered. Dispersed flows are described by a two‐fluid model. Linear stability problems are reduced to the solution of modified Orr‐Sommerfeld equations which are solved by the orthogonalization method. It is shown, that the additional factors taken into account in the problem formulation affect significantly the flow stability limits. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Pseudospectral integration matrix and boundary value problems

This paper presents a new spectral successive integration matrix. This matrix is used to construct a Chebyshev expansion method for the solution of boundary value problems. The method employs the pseudospectral approximation of the highest-order derivative to generate an approximation to the lower-order derivatives. Application to the linear stability problem for plane Poiseuille flow is presented. The present numerical results are in satisfactory agreement with the exact solutions.

Stability analysis of a class of unsteady nonparallel incompressible flows via separation of variables

Stability of some unsteady three-dimensional flows (exact solutions of the viscous incompressible Navier–Stokes equations in cylindrical coordinates) is studied via separation of variables in the linearized equations for the flow perturbations. The flows in an expanding rotating porous cylinder and in a gap between two coaxial rotating cylinders are considered. Converting the stability equations to the new variables allows perturbation forms (counterparts of normal modes of the steady state parallel flow stability problem) such that the linear stability problems are exactly reduced to eigenvalue problems of ordinary differential equations. The eigenvalue problems are solved numerically with the help of the spectral collocation method based on Chebyshev polynomials. The results showing dependence of the stability threshold on the parameters of the problems and a spatial structure of the unstable perturbation modes are presented. For some classes of perturbations, exact analytical solutions of the eigenvalue problems are available. A combination of analytical and numerical solutions can provide useful testing for numerical methods used in the hydrodynamic stability studies. It may also provide a basis for a well-grounded discussion of some problematic points of (numerical) stability analysis. In particular, in the present paper, a problem of formulation of the boundary conditions for perturbations at the axis r=0 is discussed on the basis of the solutions obtained.

Disturbance growth in two-fluid channel flow: The role of capillarity

This work examines the role of capillarity in the non-modal linear stability properties of three-dimensional disturbances in sheared two-layer flow of immiscible fluids of similar density. Capillarity reduces the transient growth of energy that occurs due to the non-normality of the linear stability problem according to a scaling of peak energy with We 1/2 over a wide range of Weber number, viscosity ratio and wavenumber. More importantly, the participation of capillary modes in non-modal growth leads to oscillatory energy growth and to larger disturbance growth rates, features that are confirmed by computing the numerical range and numerical abscissa of the non-normal disturbance evolution operator. Examination of energy components and disturbance structure reveals that early rapid growth and subsequent oscillations are due to the coupling of streamwise vortices – the two-fluid analog of lift-up – to the displaced interface.

Nonlinear high-frequency flutter of a plate

In recent studies of the problem of linear stability of a plate in a supersonic gas flow a new (“high-frequency”) type of flutter, which cannot be obtained by means of the piston theory usually employed in these problems, was found to exist together with the classical (“low-frequency”) type. In the present study a new method of calculating the pressure acting on a high-frequency vibrating plate is proposed and, using this method, high-frequency flutter is investigated in the nonlinear formulation and the flutter vibration amplitudes are determined.

On the uniqueness and stability of solutions to some continuum mechanics problems

One-dimensional shear flows of a perfect fluid, a perfect rigid-plastic body, a Newtonian viscous fluid, and a viscoplastic medium in a plane layer and the compression problem for a Euler beam are considered as examples to discuss a relation between the nonuniqueness of a solution to the basic continuum mechanics problem and the instability of the unique solution to this problem. When the basic solution is not unique, the question is raised on the mechanical interpretation of the linearized stability problem for this solution.

Lyapunov–Schmidt reduction algorithm for three-dimensional discrete vortices

We address the persistence and stability of three-dimensional vortex configurations in the discrete nonlinear Schrödinger equation and develop a symbolic package based on Wolfram’s MATHEMATICA for computations of the Lyapunov–Schmidt reduction method. The Lyapunov–Schmidt reduction method is a theoretical tool which enables us to study continuations and terminations of the discrete vortices for small coupling between lattice nodes as well as the spectral stability of the persistent configurations. The method was developed earlier in the context of the two-dimensional lattice and applied to the onsite and offsite configurations (called the vortex cross and the vortex cell) by using semianalytical computations [D.E. Pelinovsky, P.G. Kevrekidis, D. Frantzeskakis, Physica D 212 (2005) 20–53; P.G. Kevrekidis, D.E. Pelinovsky, Proc. R. Soc. A 462 (2006) 2671–2694]. The present treatment develops a full symbolic computational package which takes a desired waveform at the anticontinuum limit of uncoupled sites, performs a required number of Lyapunov–Schmidt reductions and outputs the predictions on whether the configuration persists, for finite coupling, in the three-dimensional lattice and whether it is stable or unstable. It also provides approximations for the eigenvalues of the linearized stability problem. We report a number of applications of the algorithm to important multisite three-dimensional configurations, such as the simple cube, the double cross and the diamond. For each configuration, we identify exactly one solution, which is stable for small coupling between lattice nodes.

Spontaneous swirling in axisymmetric MHD flows of an ideally conducting fluid with closed streamlines

The region of instability of the Hill-Shafranov viscous MHD vortex with respect to azimuthal axisymmetric perturbations of the velocity field is determined numerically as a function of the Reynolds number and magnetization in a linear formulation. An approximate formulation of the linear stability problem for MHD flows with circular streamlines is considered. The further evolution of the perturbations in the supercritical region is studied using a nonlinear analog model (a simplified initial system of equations that takes into account some important properties of the basic equations). For this model, the secondary flows resulting from the instability are determined.

On ice-induced instability in free-surface flows

The problem of stability of a water-coated ice layer is investigated for a free-surface flow of a thin water film down an inclined plane. An asymptotic (double-deck) theory is developed for a flow with large Reynolds and Froude numbers which is then used to investigate linear two-dimensional, three-dimensional and nonlinear two-dimensional stability characteristics. A new mode of upstream-propagating instability arising from the interaction of the ice surface with the flow is discovered and its properties are investigated. In the linear limit, closed-form expressions for the dispersion relation and neutral curves are obtained for the case ofPr= 1. For the general case, the linear stability problem is solved numerically and the applicability of the solution withPr= 1 is analysed. Nonlinear double-deck equations are solved with a novel global-marching-type scheme and the effects of nonlinearity are investigated. An explanation of the physical mechanism leading to the upstream propagation of instability waves is provided.

Critical stability of solutions to linear ordinary differential equations with large high-frequency terms

The stability problem is considered for certain classes of systems of linear ordinary differential equations with almost periodic coefficients. These systems are characterized by the presence of rapidly oscillating terms with large amplitudes. For each class of equations, a procedure for analyzing the critical stability of solutions is constructed on the basis of the Shtokalo-Kolesov method. A verification scheme is described. The theory proposed is illustrated by using a linearized stability problem for the upper equilibrium of a pendulum with a vibrating suspension point.

Stability of liquid films down an inclined plane

In this paper, the problem of linear stability of viscous liquid films down an inclined plane is solved by finite difference method. It is applicable for moderate values of Reynolds and wave numbers. The obtained results by this method is compared with ones of some papers and with experimental data.

Nonlinear three-dimensional flow in the lid-driven square cavity

The three-dimensional flow in a lid-driven cuboid is investigated numerically. The geometry is an extension to three dimensions of the lid-driven square cavity by translating the two-dimensional lid-driven cavity parallel to the third orthogonal direction. The incompressible Navier–Stokes equations are discretized by a pseudospectral Chebyshev-collocation method. The singularities caused by the discontinuous velocity boundary conditions are reduced by including asymptotic analytical solutions in the solution ansatz. The flow is computed for Reynolds numbers above the critical onset of Taylor–Görtler vortices. Nonlinear Taylor–Görtler cells are calculated for periodic and for realistic no-slip endwall conditions. For periodic boundary conditions the bifurcation is either sub- or supercritical, depending on the wavenumber. The limiting tricritical case arises near the critical wavenumber of the linear-stability problem. On the other hand, no-slip endwall conditions have a significant effect on the supercritical three-dimensional flow. In agreement with recent experimental results we find that Taylor–Görtler vortices are suppressed near no-slip endwalls.

A resonant instability of steady mountain waves

A new mechanism for the instability of steady mountain waves is found through analysis of the linear stability problem. Steady flow of a hydrostatic stratified fluid is known to be unstable when the streamlines are at, or very close to, overturning. When the topography has multiple peaks, it is shown that this criterion can be superseded by an instability owing to a resonant triad interaction. For flow over two peaks, the threshold heights for instability are roughly half those which produce overturning streamlines. The mechanism behind the instability is the parametric amplification of counter-propagating gravity waves. The resonant nature of the instability is further illustrated by the existence of discrete peak-to-peak separation distances where the growth rate is a maximum.

Effects of load height application and pre-buckling deflections on lateral buckling of thin-walled beams

Based on a non-linear model taking into account flexural-torsional couplings, analytical solutions are derived for lateral buckling of simply supported I beams under some representative load cases. A closed form is established for lateral buckling moments. It accounts for bending distribution, load height application and pre-buckling deflections. Coefficients C1 and C2 affected to these parameters are then derived. Regard to well known linear stability solutions, these coefficients are not constant but depend on another coefficient k1 that represents the pre-buckling deflection effects. In numerical simulations, shell elements are used in mesh process. The buckling loads are achieved from solutions of eigenvalue problem and by bifurcations observed on non linear equilibrium paths. It is proved that both the buckling loads derived from linear stability and eigenvalue problem lead to poor results, especially for I sections with large flanges for which the behaviour is predominated by pre-buckling deflection and the coefficient k1 is large. The proposed solutions are in good agreement with numerical bifurcations observed on non linear equilibrium paths.

Linear stability and instability of relativistic Vlasov‐Maxwell systems

AbstractWe consider the linear stability problem for a symmetric equilibrium of the relativistic Vlasov‐Maxwell (RVM) system. For an equilibrium whose distribution function depends monotonically on the particle energy, we obtain a sharp linear stability criterion. The growing mode is proved to be purely growing, and we get a sharp estimate of the maximal growth rate. In this paper we specifically treat the periodic 1½D case and the 3D whole‐space case with cylindrical symmetry. We explicitly illustrate, using the linear stability criterion in the 1½D case, several stable and unstable examples. © 2006 Wiley Periodicals, Inc.

Stability and instability of Taylor–Couette flows of a Bingham fluid

We consider in detail the Taylor-Couette problem for a Bingham fluid, presenting a range of analytical and computational results. First, for co-rotating cylinders it is known that the critical inner cylinder Reynolds number Re 1,c , does not increase monotonically with the Bingham number B, over a range of small to moderate B. It is the only situation that we know of where a yield stress fluid flow is less stable that the corresponding Newtonian fluid flow. This effect was discovered independently by Landry (2003, MSc thesis) and Peng & Zhu (J. Fluid Mech. vol. 512, 2004, p. 21), but the mechanism has not been explained. Here we show that the decrease in critical Reynolds number is due to an increase (at small B) in the rate of strain of the basic flow, which amplifies the transfer of energy from the basic flow to the perturbation, via the inertial terms in the energy equation. At larger B, the yielded region contracts and the inertial energy transfer is bounded by the yield stress dissipation. We next consider the effects of large B. For fixed radius and Reynolds number ratios, we show that for sufficiently large B all basic flows have an unyielded fluid layer attached to the outer wall. For these flows we show that there is a similarity mapping that maps both the basic solution and the linear stability problem onto the stability problem for an outer cylinder of radius equal to the yield surface radius. The Reynolds and Bingham numbers of the transformed problems are smaller than that of the original problem, as is the wavenumber k. As B → ∞, the yield surface approaches the inner cylinder, defining a narrow gap limiting problem that differs from the classical narrow gap limit. Via the transformed problem we derive an energy estimate for stability: Re 1,c k c ∼ B 1.5 as B → ∞, which compares well with our computed results for a stationary outer cylinder: Re 1,c ∼B 1.25 and k c ∼ B 0.375 . We also show how Re 1,c ∼ B 1.25 can be deduced from a simple order of magnitude analysis, for a stationary outer cylinder. Finally, we consider the second (classical) narrow gap limit in which the radius ratio η, approaches unity, for fixed B and Reynolds number ratio. We show that Re 1,c ≥ (k 2 [1 + O(B)] + n 2 )/(1-η) 1/2 in this limit.

Linear instability of planar shear banded flow of both diffusive and non-diffusive Johnson–Segalman fluids

We consider the linear stability of shear banded planar Couette flow of the Johnson–Segalman fluid, with and without the addition of stress diffusion to regularise the equations. In particular, we investigate the linear stability of an initially one-dimensional “base” flow, with a flat interface between the bands, to two-dimensional perturbations representing undulations along the interface. We demonstrate analytically that, for the linear stability problem, the limit in which diffusion tends to zero is mathematically equivalent to a pure (non-diffusive) Johnson–Segalman model with a material interface between the shear bands, provided the wavelength of perturbations being considered is long relative to the (short) diffusion lengthscale. For no diffusion, we find that the flow is unstable to long waves for almost all arrangements of the two shear bands. In particular, for any set of fluid parameters and shear stress there is some arrangement of shear bands that shows this instability. Typically the stable arrangements of bands are those in which one of the two bands is very thin. Weak diffusion provides a small stabilising effect, rendering extremely long waves marginally stable. However, the basic long-wave instability mechanism is not affected by this, and where there would be instability as wavenumber k → 0 in the absence of diffusion, we observe instability for moderate to long waves even with diffusion. This paper is the first full analytical investigation into an instability first documented in the numerical study of Fielding [S.M. Fielding, Linear instability of planar shear banded flow, Phys. Rev. Lett. 95 (2005) 134501]. Authors prior to that work have either happened to choose parameters where long waves are stable or used slightly different constitutive equations and Poiseuille flow, for which the parameters for instability appear to be much more restricted. We identify two driving terms that can cause instability: one, a jump in N 1 , as reported previously by Hinch et al. [E.J. Hinch, O.J. Harris, J.M. Rallison, The instability mechanism for two elastic liquids being coextruded, J. Non-Newtonian Fluid Mech. 43 (1992) 311–324]; the second, a discontinuity in shear rate. The mechanism for instability from the second of these is not thoroughly understood. We discuss the relevance of this work to recent experimental observations of complex dynamics seen in shear-banded flows.