On the unsteady motion of two-dimensional sails

An equation is derived to describe the motion of a two-dimensionalinextensible sail at a small, time-dependent, angle of incidenceto a uniform two-dimensional flow. The equation derived isa singular partial integro-differential equation, which in the steady case reduces to the sail equation of Voelz. A numberof limiting versions of the equation are derived and analysedfor cases where the relative mass of the sail is large or small.For general unsteady sail motions the governing equation mustbe solved numerically. A scheme is proposed that employs Chebyshevpolynomials to approximate the position of the sail; ordinarydifferential equations are derived to determine the relevantChebyshev coefficients and a number of examples are illustratedand discussed. It is found that in some cases where the angleof attack changes sign the tension may become large; in theseinstances the underlying physical assumptions of the modelmay be violated. Received 12 November, 1998.     

Blowup of solutions of the unsteady Prandtl's equation

We prove that for certain classes of compactly supported C^∞ initial data, smooth solutions of the unsteady Prandtl's equation blow up in finite time. © 1997 John Wiley & Sons, Inc.     

Ultraparabolic equations and unsteady heat transfer

The present paper is concerned with the boundary value problem for the equation _t + u· = k_yy + f. Existence and uniqueness of the generalized solution are proved.     

Some unsteady diffusion and absorption problems

Specific problems of the diffusion type, focusing on the absorptionof heat (or matter) at one or more boundary surfaces after itsrelease within the adjacent medium at a localized source, areconsidered, and the measures of absorption or outflow duringarbitrary intervals of time are directly expressed in termsof auxiliary temperature/matter distributions with given sourceson the pertinent surfaces. Detailed applications of these reciprocalrelations are presented, as is a brief analysis that includessome nonlinear or convective aspects in the basic diffusionequations.     

Exact solutions of the unsteady two-dimensional Navier-Stokes equations

This paper studies the two-dimensional incompressible viscous flow in which the local vorticity is proportional to the stream function perturbed by a uniform stream. It was known by Taylor and Kovasznay that the Navier-Stokes equations for flow of this kind become linear. From the general solution to the linear equations for steady flow, we show that there exist only two types of steady flow of this kind: Kovasznay downstream flow of a two-dimensional grid and Lin and Tobak reversed flow about a flat plate with suction. In the unsteady flow case, new classes of exact analytical solutions are found which include Taylor vortex array solution as a special case. It is shown that these unsteady flows are, as viewed from a frame of reference moving with the undisturbed uniform stream, pseudo-steady in the sense that the flow pattern is steady but the magnitude of motion decays, or grows, exponentially in time. All these solutions are valid for any Reynolds number.

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Résumé Dans ce travail nous étudions l'écoulement plan d'un fluide visqueux incompressible dans lequel la rotation locale est proportioneile à la fonction de courant perturbée par un courant uniforme. Conformément aux travaux de Taylor et Kovasznay les équations de Navier-Stokes pour cet écoulement deviennent linéaires. Par conséquent nous utilisons la solution générale pour démontrer que seulement deux catégories d'écoulement stationnaire peuvent exister: l'écoulement de Kovasznay en aval d'une grille plane, et l'écoulement inversé de Lin et Tobak pour une plaque plane avec aspiration. Nous étudions aussi l'écoulement non stationnaire et nous découvrons des classes nouvelles de solutions exactes qui contiennent, en particulier, le réseau de tourbillons de Taylor. Enfin nous démontrons que ces écoulements sont pseudo-stationnaires dans un système de coordonnées en mouvement avec le courant uniforme non perturbé; ce qui signifie que l'amplitude de l'écoulement stationnaire croit ou décroit exponentiellment dans le temps. Toutes ces solutions sont valides pour tous les nombres de Reynolds.

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On leave from University of Waterloo, Ontario, Canada.     

Asymptotic solution of the unsteady boundary layer equations

Summary An extension of the Meksyn asymptotic method to unsteady boundary layers in laminar, incompressible flow is investigated. The results indicate that unsteady boundary layers can be calculated by the Meksyn asymptotic method with comparable accuracy to that obtained for steady flows. Several differences from the well developed steady-flow application exist and require further work before general problems can be treated. The calculation technique is more straight-forward for cases involving acceleration because three or four terms in the expansions may then yield sufficient accuracy. The form of the governing equation required by the Meksyn method indicates that it is most useful for unsteady stagnation boundary layers since some basic unsteady flows are not directly accessible in their simplest form from that equation. The effect of unsteadiness on the rate of asymptotic convergence is assessed by detailed comparison of a similar solution for unsteady, stagnation flow with analogous results from the Falkner-Skan equation and of reliable numerical results for both cases.

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Résumé On étudie une extension de la méthode asymptotique de Meksyn aux couches limites instables des écoulements laminaires de fluides incompressibles. Les résultats montrent que les couches limites instables peuvent être calculées à l'aide de la méthode asymptotique de Meksyn avec une précision comparable à celle obtenue pour les écoulements stables. Plusieurs différences existent par rapport à l'application, bien mise au point, aux écoulements stables; elles demandent encore du travail avant que les problèmes généraux puissent être traités. La méthode de calcul est plus directe dans les cas impliquant une accélération, car 3 ou 4 termes dans les développements assurent alors une précision suffisante. La forme de l'équation principale nécessaire à la méthode de Meksyn indique qu'elle est très utile pour les couches limites instables au repos; en effet, certains écoulements instables de base ne peuvent être atteints directement dans leur forme la plus simple à partir de cette équation. L'effet de l'instabilité sur la vitesse de convergence asymptotique est établi grâce à une comparaison détaillée d'une solution analogue pour un écoulement instable stagnant avec les résultats semblables obtenus par l'équation Falkner-Skan, et des résultats numériques sûrs obtenus dans les deux cas.

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This work was supported by the Energy Research and Development Administration.     

Study of unsteady oligopoly markets

Under study is some simulation model of an oligopolisticmarket in continuous time. For the suppliers (oligopolists) the two rules of behavior are defined by a system of ordinary differential equations. The scenario determines the strategy that is used by the supplier, including the possibility of changing it during the interaction. It is shown, that there are possible stationary states in the model when a supplier profitability can be tools to start using a new rule of his behavior. In result of changing strategy his profit rises as compared to the other participants.     

On the unsteady Poiseuille flow in a pipe

We show that in an unsteady Poiseuille flow of a Navier–Stokes fluid in an infinite straight pipe of constant cross-section, σ, the flow rate, F(t), and the axial pressure drop, q(t), are related, at each time t, by a linear Volterra integral equation of the second type, where the kernel depends only upon t and σ. One significant consequence of this result is that it allows us to prove that the inverse parabolic problem of finding a Poiseuille flow corresponding to a given F(t) is equivalent to the resolution of the classical initial-boundary value problem for the heat equation.     

On the unsteady Poiseuille flow in a pipe

We show that in an unsteady Poiseuille flow of a Navier–Stokes fluid in an infinite straight pipe of constant cross-section, σ, the flow rate, F(t), and the axial pressure drop, q(t), are related, at each time t, by a linear Volterra integral equation of the second type, where the kernel depends only upon t and σ. One significant consequence of this result is that it allows us to prove that the inverse parabolic problem of finding a Poiseuille flow corresponding to a given F(t) is equivalent to the resolution of the classical initial-boundary value problem for the heat equation. G. P. Galdi: Partially supported by the NSF grant DMS–0404834. K. Pileckas: Supported by EC FP6 MCToK program SPADE2, MTKD–CT–2004–014508 A. L. Silvestre: Supported by FCT-Project POCI/MAT/61792/2004     

Performance evaluation of purchasing and supply management using value chain DEA approach

Purchasing and Supply Management (PSM) today is increasingly becoming more important to senior management due to its potential to strategically influence both operational performance as well as financial performance outcomes. However the cross-functional nature of many PSM activities has led to inadequate data collection and performance measurement resulting in weak performance evaluation methodologies and mixed results. We address this gap in the current study, firstly by using an external assessment survey methodology that complements the internal perceptional measures of PSM performance, to collect data for a sample of over 120 firms across the globe with more than 3 billion US dollar turnover, representing seven industry sectors. Next, we develop a comprehensive performance measurement framework using the classical and two-stage Value Chain Data Envelopment Analysis models, which make use of multiple PSM measures at various stages and provide a single efficiency measure that estimates the all-round performance of a PSM function and its contribution to the long term corporate performance in each of these seven industry sectors. The relevance of this measurement methodology is demonstrated through an in-depth analysis of the distribution of efficiencies within and across industry sectors and through the estimation of target PSM performance levels.     

Performance analysis and design of supply chains: a Petri net approach

In this paper, we investigated a dynamic modelling technique for analysing supply chain networks using generalised stochastic Petri nets (GSPNs). The customer order arrival process is assumed to be Poisson and the service processes at the various facilities of the supply chain are assumed to be exponential. Our model takes into account both the procurement process and delivery logistics that exist between any two members of the supply chain. We compare the performance of two production planning and control policies, the make-to-stock and the assemble-to-order systems in terms of total cost which is the sum of inventory carrying cost and cost incurred due to delayed deliveries. We formulate and solve the decoupling point location problem in supply chains as a total relevant cost (sum of inventory carrying cost and the delay costs) minimisation problem. We use the framework of integrated GSPN-queuing network modelling—with the GSPN at the higher level and a generalised queuing network at the lower level—to solve the decoupling point location problem.     

Polynomial-time universality and limitations of deep learning

The goal of this paper is to characterize function distributions that general neural networks trained by descent algorithms (GD/SGD), can or cannot learn in polytime. The results are: (1) The paradigm of general neural networks trained by SGD is poly-time universal: any function distribution that can be learned from samples in polytime can also be learned by a poly-size neural net trained by SGD with polynomial parameters. In particular, this can be achieved despite polynomial noise on the gradients, implying a separation result between SGD-based deep learning and statistical query algorithms, as the latter are not comparably universal due to cases like parities. This also shows that deep learning does not suffer from the limitations of shallow networks. (2) The paper further gives a lower-bound on the generalization error of descent algorithms, which relies on two quantities: the cross-predictability, an average-case quantity related to the statistical dimension, and the null-flow, a quantity specific to descent algorithms. The lower-bound implies in particular that for functions of low enough cross-predictability, the above robust universality breaks down once the gradients are averaged over too many samples (as in perfect GD) rather than fewer (as in SGD). (3) Finally, it is shown that if larger amounts of noise are added on the initialization and on the gradients, then SGD is no longer comparably universal due again to distributions having low enough cross-predictability.     

Some mathematical limitations of the general-purpose analog computer

We prove that the Dirichlet problem on the disc cannot be solved by the general-purpose analog computer, by constructing, on the boundary, a function u₀ that does satisfy an algebraic differential equation, but whose Poisson integral u satisfies no algebraic differential equation on some line segment inside the disc.