Does Surface Tension Stabilize Liquid Films Inside a Rotating Horizontal Cylinder? Part 2: Multidimensional Disturbances

We examine the stability of a thin film of viscous fluid on the inside surface of a cylinder with horizontal axis, rotating about this axis. Depending on the parameters involved, the dynamics of the film can be described by several asymptotic models, one of which was examined by Benilov [ J. Fluid Mech. 501:105–124 (2004)]. It turned out that the linearized stability problem for this model admits infinitely many neutrally stable eigenmodes, which form a complete set. Despite that, the film is unstable with respect to exploding disturbances, which grow infinitely in a finite time. The present paper examines the effect of surface tension on the stability of the film. Two cases are considered: short-scale disturbances (such that the axial wavelength λ is much smaller than the radius R of the cylinder) and long-scale disturbances (for which  λ≳ R   ). In the former case, surface tension is a stabilizing influence, because it regularizes the exploding solutions and makes all eigenmodes asymptotically (not just neutrally) stable. The latter case was previously examined by Acrivos and Jin [ J. Eng. Math. 50:99–120 (2004)], who showed that surface tension destabilizes some of the eigenmodes. We argue, however, that the corresponding growth rate is much smaller than that of the so-called inertial instability.     

Local Instability in Curved Flow

Expressions are obtained for the rates of change of the vorticitycomponents about the tangent, principal normal, and binormalof inviscid flow of arbitrary configuration with curvature andtorsion of the particle paths, and non-uniform density. If a steady flow pattern exists the vorticity changes are exactlythose required to carry the fluid particles through it. A smallrotational displacement of a fluid element about each of thesedirections is then considered separately and if the additionalrate of change of vorticity has the same sign and order of magnitudeas the displacement the motion is unstable locally regardlessof the motion elsewhere. The equations depend on four quantities: the curvature, torsion,velocity and vorticity at the point where stability is beinginvestigated. These four quantities define a helical vortex,with the same stability properties, in which the flow is equivalent.There is therefore special merit in studying this case. It is found that a vortex with helical particle paths is unstablefor rotation of the fluid elements about directions which liebetween the vorticity vector and the direction of the axis ofthe motion, when the density is uniform. More generally, themotion is unstable for these disturbances if the stagnationpressure decreases radially outwards. A gradient of axial velocitycomponent always causes some local instability, and the mostlikely (fastest growing) disturbance is one in which rotationsoccur around a line everywhere bisecting the angle between thevorticity vector and the direction of the axis. The analysis shows that in two dimensional circular flow themost unstable disturbances are toroidal and in general curvedflow in two dimensions (without torsion) the most likely disturbancesare rotations around the tangent, i.e. longitudinal rolls. The criteria obtained for local instability do not agree witha Richardson-type criterion for local stability, but both showthe destabilizing effect of a radiating gradient of axial velocity.     

A Note on the Linear Stability of Burgers Vortex

A two-parameter family of analytical solutions of the linearized equations for axially dependent disturbances to the three-dimensional base strain field associated with the well-known axisymmetric Burgers vortex is presented. The solutions are valid asymptotically at large axial distances from the stagnation point. By a formal perturbation analysis, perturbative solutions are also found for disturbances to the Burgers vortex for small Reynolds numbers. The solutions are believed to provide important insights into the nature of the as-yet-unsolved problem of the linear stability of Burgers vortex to axially varying disturbances.     

Transition to turbulence, small disturbances, and sensitivity analysis II: The Navier-Stokes equations

Recent research has shown that small disturbances in the linearized Navier-Stokes equations cause large energy growth in solutions. Although many researchers believe that this interaction triggers transition to turbulence in flow systems, the role of the nonlinearity in this process has not been thoroughly investigated. This paper is the second of a two part work in which sensitivity analysis is used to study the effects of small disturbances on the transition process. In the first part, sensitivity analysis was used to predict the effects of a small disturbance on solutions of a motivating problem, a highly sensitive one-dimensional Burgers' equation. In this paper, we extend the analysis to study the effects of small disturbances on transition to turbulence in the three-dimensional Navier-Stokes equations. We show that the change in a laminar flow with respect to small variations in the initial flow or small forcing acting on the system is large when the linearized operator is stable yet nonnormal. In this case, the solution of the disturbed problem can be very large (and potentially turbulent) even if the disturbances are extremely small. We also give bounds on the disturbed flow in terms of certain constants associated with the linearized operator.     

Three-Dimensional Stability and Bifurcation of Capillary and Gravity Waves on Deep Water

The stability to three-dimensional disturbances of finite-amplitude capillary waves on deep water is studied. Two families of steady finite-amplitude three-dimensional capillary waves, which result from the bifurcation associated with stationary disturbances, are calculated. Earlier work on three-dimensional gravity waves is extended, and a new family of solutions if found.     

The fracture of brittle and ductile crystals. Force and deformation criteria

The growth and branching of sharp cracks in ideal single crystals are investigated. Neuber-Novozhilov force and deformation criteria are proposed for the branching of sharp cracks; these criteria describe the brittle, quasibrittle, quasiductile and ductile behaviour of materials on fracture. For internal cracks, simple relations are obtained that describe the branching of cracks when the Coulomb-Mohr single-crystal theoretical strength curves are known for a generalized stress state. The possibility of multiple branching of cracks is found, which is linked to the multiplicity of the eigenvalues on loss of stability of the system. It is established that, for ideal single crystals, the principle of local symmetry is satisfied in the vicinity of the crack tip if the axis of symmetry of the crystal coincides with the axis of the crack. When there are asymmetrical disturbances of the atomic lattice in the vicinity of the crack tip, or when the axis of symmetry of the single crystal does not coincide with the crack axis, the principle of local symmetry is not satisfied.     

Numerical Approximation of the Spectra of Phan-Thien Tanner Liquids

The linear stability of the linear Phan-Thien Tanner (PTT) fluid model is investigated for plane Poiseuille flow. The PTT model involves parameters that can be used to fit shear and extensional data, which makes it suitable for describing both polymer solutions and melts. The base flow is determined using a Chebyshev-tau method. The linear stability equations are also discretized using Chebyshev approximations to furnish a generalized eigenvalue problem. The spectrum is shown to comprise a continuous part and a discrete part. The theoretical and numerical results are validated for the UCM and Oldroyd-B models, which are special cases of the PTT model, by comparing with results in the literature. It is demonstrated that the linear PTT fluid is stable to infinitesimal disturbances with respect to the range of shear-thinning, extensional and elasticity parameters considered. The computational efficiency and accuracy of the numerical method are also investigated.     

On Truncated Matricial Stieltjes Type Moment Problems

We study two slightly different versions of truncated generalized matricial moment problems of Stieltjes type. A central topic is the construction and investigation of distinguished solutions of both moment problems under consideration. These solutions turn out to be nonnegative Hermitian Borel measures on the real axis which are concentrated on a finite number of points. Our approach is mainly algebraic. It is based on the use of particular matrix polynomials constructed from a nonnegative Hermitian block Hankel matrix.     

旋转流体中圆球沿旋转轴运动产生的扰动

本文用线化理论分析了整体旋转的理想流体中有一个圆球沿旋转轴作匀速运动时流体的扰动,基于旋轴对称流动的假设导出了决定运动稳定性的扰动压力方程和扰动流函数方程.用简正模法分析了扰动流函数方程,得出了非平凡中性扰动的波数与波速必须满足的约束条件,并求出了扰动的精确表达式.文中得出结论,中性扰动共有三种可能的形式.     

Viscous Approximations and Decay Rate of Maximal Vorticity Function for 3-D Axisymmetric Euler Equations

In this paper, we are first concerned with viscous approximations for the three-dimensional axisymmetric incompressible Euler equations. It is proved that the viscous approximations, which are the solutions of the corresponding Navier-Stokes equations, converge strongly in L^2([0, T]; Lloc^2(R^3)) provided that they have strong convergence in the region away from the symmetry axis. This result has been proved by the authors for the approximate solutions generated by smoothing the initial data,with no restriction of the sign of the initial data. Then we discuss the decay rate for maximal vorticity function, which is established for both approximate solutions generated by smoothing the initial data and viscous approximations respectively. One sufficient condition to guarantee the strong convergence in the region away from the symmetry axis is given, and a decay rate for maximal vorticity function in the region away from the symmetry axis is obtained for non-negative initial vorticity.     

On solutions defined on an axis for differential equations with shifts of the argument

We consider linear first-order differential equations with shifts of the argument for functions with values in a Banach space. Sufficient conditions for the existence of nontrivial solutions of homogeneous equations are obtained. Ordinary differential equations are constructed for which all solutions defined on the entire axis are solutions of a given equation with shifts of the argument.     

Short-Term Capacity Adjustment with Offline Production for a Flexible Manufacturing System under Abnormal Disturbances

Large production variations caused by abnormal disturbances can significantly reduce the production capacity of a flexible manufacturing system (FMS). To prevent production delays, short-term capacity adjustment strategies can be used to augment the capacity of the FMS, such as working overtime, using alternative tools that are suited for faster processing, and producing parts outside of the FMS. We propose a mixed integer programming (MIP) model to obtain an optimal production plan for a multi-machine FMS. Our model evaluates both the FMS loading decision and the effective use of short-term capacity adjustment strategies to minimize the total part production cost. We develop an iterative procedure to solve the model that uses the Lagrangian relaxation method for finding lower bounds and a Lagrangian heuristic for obtaining feasible solutions. The procedure exploits certain special structures found in the Lagrangian multipliers which enable us to obtain good solutions to reasonably large test problems quickly.     

The Evolution of Large-Horizontal-Scale Disturbances in Marginally Stable,Inviscid, Shear Flows. II. Solutions of the Boussinesq Equation

In part I the Boussinesq equation was found to describe large-horizontal-scale disturbances in certain marginally stable flows. In this paper the properties of the solutions of this equation are explored. Solutions that are bounded for all time and those that blow up in finite time are observed analytically and numerically. Sufficient conditions for both boundedness and blowup are given.     

A NONAUTONOMOUS STAGE-STRUCTURED SINGLE SPECIES MODEL WITH DIFFUSION

1IntroductionRecent1ytheso-calledsinglespeciesm0delwasconsidered,andmanyrcsultshavebeen.bt.in.d[1-7J.In[l],thesinglespeciesmodelwithstage-structtlredinthefOrmofisstudied,whereIi(t)andx.(t)representtheimmatureandmaturepopu-lati0nsdensitiesrespectively;ofisthe0bservedorassumedbirthrateofxi(t)attimet(-T5t5O),andTrepresentsac0nstanttime1engthfortheiITl-maturetobecomematurity.aisthebirthrateofmaturepopulation.7isthedeathrateofimmaturepopulationandfiistheLogisticconstantofthemature.Somesufficien…     

Existence of solutions of two generalized riccati operator equations and their representation

The solutions of two generalized Riccati operator equations are discussed in terms of two critical parameter values, which are related to the application of optimal control under unknown disturbances. Explicit formulas for calculating these two critical parameters as well as the closedform solutions of these two generalized Riccati operator equations are given. The connection between these two parameters and a zero-sum differential game is also investigated.     

Transient Nonlinear Disturbances and Shelves in a Stratified Fluid Layer

We revisit the classical problem of internal wave propagation in a stratified fluid layer bounded by rigid walls and point out a mechanism by which unsteady locally confined disturbances generate far-field shelves. Carrying the standard expansion procedure to fourth order in the wave amplitude reveals that weakly nonlinear long waves of a certain mode shed, in general, lower- and higher-mode shelves, which propagate upstream and downstream with the corresponding long-wave speeds. This phenomenon is brought about by the combined effect of nonlinear interactions and the presence of transience in the main disturbance. While the shelves accompanying small-amplitude waves are relatively weak, numerical solutions of the full Euler equations indicate that shelves induced by unsteady disturbances of finite amplitude close to breaking can be quite significant.     

Asymptotic axial symmetry of solutions of parabolic equations in bounded radial domains

We consider solutions of some nonlinear parabolic boundary value problems in radial bounded domains whose initial profile satisfies a reflection inequality with respect to a hyperplane containing the origin. We show that, under rather general assumptions, these solutions are asymptotically (in time) foliated Schwarz symmetric, that is, all elements in the associated omega limit set are axially symmetric with respect to a common axis passing through the origin and nonincreasing in the polar angle from this axis. In this form, the result is new even for equilibria (i.e., solutions of the corresponding elliptic problem) and time periodic solutions.     

Minmax robustness for multi-objective optimization problems

In real-world applications of optimization, optimal solutions are often of limited value, because disturbances of or changes to input data may diminish the quality of an optimal solution or even render it infeasible. One way to deal with uncertain input data is robust optimization, the aim of which is to find solutions which remain feasible and of good quality for all possible scenarios, i.e., realizations of the uncertain data. For single objective optimization, several definitions of robustness have been thoroughly analyzed and robust optimization methods have been developed. In this paper, we extend the concept of minmax robustness (Ben-Tal, Ghaoui, & Nemirovski, 2009) to multi-objective optimization and call this extension robust efficiency for uncertain multi-objective optimization problems. We use ingredients from robust (single objective) and (deterministic) multi-objective optimization to gain insight into the new area of robust multi-objective optimization. We analyze the new concept and discuss how robust solutions of multi-objective optimization problems may be computed. To this end, we use techniques from both robust (single objective) and (deterministic) multi-objective optimization. The new concepts are illustrated with some linear and quadratic programming instances.     

Existence, Uniqueness, and Construction of the Solution of a System of Ordinary Functional Differential Equations, with Application to the Design of Perfectly Focusing Symmetric Lenses

In the design of perfectly focusing symmetric lenses, one isled, in a natural way, to a set offunctional differential equations;that is, differential equations involving composites of unknownfunctions, with initial conditions prescribed on the lens axis.This paper concentrateson those features of the equations whichmake them uniquely solvable. They are: (i) a contractivenessproperty of the equations near the axis; (ii) a uniform retardationin the arguments of thecomposite functions away from the axis.The second and third sections of this paper generalize and formalizethese properties and provide proofs of existence, uniqueness,and continuous dependence on the data for solutions of suchgeneralized systems of functional differential equations. Becauseof the lens context which motivates our study, the problem inwhich the contractiveness property (i) above holds is calledthe ‘local’ problem, and the problem in which thearguments of composite functions are uniformly retarded is calledthe ‘global’ problem. In the final section of thepaper we apply the general results of the preceding sectionsto prove existence and uniqueness of perfectly focusing symmetriclenses up to distances from the lens axis at which various typesof breakdown, discussed in the text, may occur.     

Comparison principle for equations of the Hamilton-Jacobi type in control theory

This paper deals with the comparison principle for the first-order ODEs of the Hamilton-Jacobi-Bellman and Hamilton-Jacobi-Bellman-Isaacs type which describe solutions to the problems of reachability and control synthesis under complete as well as under limited information on the system disturbances. Since the exact solutions require fairly complicated calculation, this paper presents the upper and lower bounds to these solutions, which in some cases may suffice for solving such problems as the investigation of safety zones in motion planning, verification of control strategies or of conditions for the nonintersection of reachability tubes, etc. For systems with original linear structure it is indicated that present among the suggested estimates are those of ellipsoidal type, which ensure tight approximations of the convex reachability sets as well as of the solvability sets for the problem of control synthesis.