Decision Support Model for Production Disturbance Estimation

A current modeling framework for disturbance in manufacturing systems (MS) is given by concepts like discrete-event systems, stochastic fluid models and infinitesimal disturbance analysis. The goal of modeling is to achieve control and structural and functional optimization of MS. Objective functions of these optimization models are focused on quantities which reflect the level of reliability, the level of manufactured products, the quality of products or the impact on the environment of MS with disturbances. These models do not allow a dynamic evaluation of consequences of the disturbances which appears in the operation of MS machines and also do not allow an evaluation of the evolution in time of disturbance consequence indicators. Disturbances in technological lines of MS represent local bottlenecks of production with severe economic consequences in what regards production time losses. Good estimation of disturbances dynamics can be very helpful to both technological line designers, who can optimize their projects and production managers who can minimize their losses. Our model allows a dynamic evaluation of consequences of some disturbance of machine operation in MS, using indicators based on time, energy and costs. A MATLAB software package was developed for tests.

Linear water waves: the horizontal motion of a structure in the time domain

The framework of the linearized theory of water waves in the time domain is used to examine the horizontal motion of an unrestrained floating structure. One of the principal assumptions of the theory is that an infinitesimal disturbance of the rest state will lead to an infinitesimal motion of the fluid and structure. It has been known for some time that for some initial conditions the theory predicts an unbounded horizontal motion of the structure that violates this assumption, but the possibility does not appear to have been examined in detail. Here some circumstances that lead to predictions of large motions are identified and, in addition, it is shown that not all non-trivial initial conditions lead to violations of the assumptions. In particular, it is shown that the horizontal motion of a floating structure remains bounded when it is initiated by the start up of a separate wave maker. The general discussion is supported by specific calculations for a vertical circular cylinder.

The Viscosity Coefficients of Biaxial-Nematic Liquid Crystals. Phenomenology and Affine Transformation Model

The viscosity tensor of biaxial-nematic liquid crystals contains 16 independent elements. A complete set of these viscosity coefficients is introduced and related to experimentally accessible quantities. Furthermore, the flow alignment and its stability against an arbitrary infinitesimal disturbance are discussed. By the affine transformation model, formerly established for the uniaxial symmetric case, one can express the viscous anisotropy of perfectly ordered biaxial ellipsoids in terms of the two viscosities of an isotropic reference system and the axes ratios of the nonspherical particles.

The stability of stationary waves in a wavy-walled channel

Experiments by Binnie showed that unsteady waves were produced by flow through a channel with symmetric, wavy sidewalls, with waves propagating both upstream and downstream. However, the first-order solution to this problem that was obtained by Yih is a set of steady waves. The steady solution is shown to be unstable to a pair of infinitesimal disturbance waves which satisfy the resonance conditions of Phillips. For the Froude-number range used by Binnie, a pair of disturbances has been found such that one wave propagates upstream, one propagates downstream, and the amplitudes have an exponential growth. The Froude numbers outside the range of Binnie are also shown to be unstable. The steady waves produced by flow through an antisymmetric channel are shown to be unstable in the same manner.

Stability of the Stewartson layer in a rapidly rotating gas

The linear stability of the Stewartson layer in a compressible fluid is studied. The viscosity and the heat conductivity are shown to be negligible for a special kind of infinitesimal disturbance. The basic equations of the disturbance are shown to reduce to those for a Boussinesq fluid subject to a virtual radial stratification. A Miles-type sufficient condition for stability and a Howard-type semicircle theorem are derived. The growth rates of unstable modes with wavenumber and shear strength are summarized in stability diagrams for typical cases. The results clarify the situation in which the stability of the Stewartson layer is governed by a balance between the shear strength and the temperature stratification in the layer.

Nonlinear stability of parallel flows with subcritical Reynolds numbers. Part 1. An asymptotic theory valid for small amplitude disturbances

A strict distinction is made between the two fundamental assumptions in the Stuart-Watson theory of nonlinear stability, one of which is that the amplitude of disturbance is sufficiently small, while the other is that the damping or amplification rate for an infinitesimal disturbance is small. This distinction leads to classification of the nonlinear stability theory into two asymptotic theories: the theory based on the first assumption can be applied to subcritical flows with Reynolds numbers away from the neutral curve, even to flows with no neutral curve, such as plane Couette flow or pipe Poiseuille flow, while the theory based on the second assumption is available only for Reynolds numbers and wavenumbers in the neighbourhood of the neutral curve. In the theory based on the first assumption the concept of trajectories in phase space, together with the method of eigenfunction expansion, is introduced in order to display nonlinear behaviour of the disturbance amplitude and to provide the most rational definition of the Landau constant available for classification of the behaviour patterns.

Pressure-wave propagation through a separated gas-liquid layer in a duct

Pressure-wave propagation through a separated gas-liquid layer at rest in a duct of constant rectangular cross-section and infinite length is considered. Such a system is dispersive, possessing an infinite number of modes which depend on the ratios of the densities, thicknesses and sound speeds of the two phases. The transitional variation of an infinitesimal disturbance initially having a step profile is investigated analytically and numerically. In addition, it is shown that a weak but finite disturbance is described asymptotically by the solution of the Korteweg-de Vries equation.

On the nonlinear response of a marginally unstable plane parallel flow to a two-dimensional disturbance

The differential equation governing the nonlinear evolution of an initial centred infinitesimal disturbance to a marginally unstable plane parallel flow was obtained by Stewartson & Stuart (1971) and some of its properties elucidated by Hocking, Stewartson & Stuart (1972). Of especial interest is the final localized burst of the solution which occurs when all the coefficients of the equation are real and the first Landau constant is positive. In plane Poiseuille flow, however, the standard example of plane parallel flow, these coefficients are complex and in the present paper an analytic and numerical study is made of the evolution of the solution when they are permitted to take general values. It is found that if the real part 8r of the first Landau constant is positive it is possible to have either a burst or a solution which remains finite for all time depending on the values of the other coefficients. In addition when a burst occurs it can take on two different structures. If δ r < 0 all solutions remain finite but the amplitude of the oscillation does not tend to a limit if the imaginary part δ of the first Landau constant is large enough. For the particular example of plane Poiseuille flow, skewed disturbances burst only if they are inclined to the main stream at an angle exceeding about 56°.

Finite-amplitude stability of pipe flow

In this paper we present some results concerning the stability of flow in a circular pipe to small but finite axisymmetric disturbances. The flow is unstable if the amplitude of a disturbance exceeds a critical value, the equilibrium amplitude, which we have calculated for a wide range of wave-numbers and Reynolds numbers. For large values of the Reynolds number, R, and for a real value of the wave-number, α, we indicate that the energy density of a critical disturbance is of order c2i, where −ααci is the damping rate of the associated infinitesimal disturbance. The energy, per unit length of the pipe, of a critical disturbance which is concentrated near the axis of the pipe is of order R−2, and the wave-number α is of order R1/3 For a critical disturbance which is concentrated near the wall of the pipe the energy is of order $R^{-\frac{3}{2}}$ and α is of order R½. This suggests that non-linear instability is most likely to be caused by a ‘centre’ mode rather than by a ‘wall’ mode. The wall mode solution is also essentially the solution for the problem of plane Couette flow when αR is large. We compare it with the true solution.In an appendix Dr A. E. Gill indicates how some of the results of this paper may be inferred from a simple scale analysis.

Finite-amplitude stability of a parallel flow with a free surface

The linearized problem of the instability of a layer of liquid flowing down an inclined plane was formulated by Yih (1954) and was solved by Benjamin (1957). It was found that the instability of such a film flow is initially due to long surface waves of infinitesimally small amplitudes. In the present study, a closed-form expression for the non-linear development of these long surface waves is obtained. It is shown that in the neighbourhood of the neutral curve an exponentially growing infinitesimal disturbance may develop into supercritically stable wave motion of small but finite amplitude if the surface tension of the liquid is sufficiently large. Theoretically obtained amplitudes of such waves are consistent with Kapitza's (1949) observation. The approach used in this analysis is a modification of the method used by Reynolds & Potter (1967), who extended the method of Stuart (1960) and Watson (1960) in their study of the non-linear instability of plane Poiseuille and Poiseuille-Couette flow.