Series Solution of Non-similarity Boundary-Layer Flows Over a Porous Wedge

Solution of non-similarity boundary-layer flows over a porous wedge is studied. The free stream velocity U _w (x) ~ a x ^m and the injection velocity V _w (x) ~ b x ⁿ at the surface are considered, which result in the corresponding non-similarity boundary-layer flows governed by a nonlinear partial differential equation. An analytic technique for strongly nonlinear problems, namely, the homotopy analysis method (HAM), is employed to obtain the series solutions of the non-similarity boundary-layer flows over a porous wedge. An auxiliary parameter is introduced to ensure the convergence of solution series. As a result, series solutions valid for all physical parameters in the whole domain are given. Then, the effects of the physical parameters on the skin friction coefficient and displacement thickness are investigated. To the best of our knowledge, it is the first time that the series solutions of this kind of non-similarity boundary-layer flows are reported.     

An investigation of the viscoelastic properties of molten polypropylene

Two series of polypropylene samples of different molecular weight, the first obtained directly from polymerization reactions and the second from controlled thermal degradation, were studied by dynamic testing in the melt state. Several viscoelastic parameters were determined, and correlated with weight-average molecular weightM _w . It is found that theM _w -dependence of the two series is rather different.     

Determination of the Boundaries between the Domains of Stability and Instability for the Hill Equation

The stability problem for the Hill equation containing two parameters is analyzed using the Mathematica computer algebra system. The characteristic constant is found as a series expansion in powers of a small parameter e. It is shown that the domains of instability are located only between the curves a = a(e) on the a-e plane crossing the axis e = 0 at the points a = (2k – 1)² / 4, k = 1, 2, 3, ....The corresponding curves are found as power series in e with accuracy O(e ⁶).     

Combustion process in a spark ignition engine: analysis of cyclic peak pressure and peak pressure angle oscillations

In this paper we analyze the cycle-to-cycle variations of peak pressure p _max and peak pressure angle α _pmax in a four-cylinder spark ignition engine. We examine the experimental time series of p _max and α _pmax for three different spark advance angles. Using standard statistical techniques such as return maps and histograms we show that depending on the spark advance angle, there are significant differences in the fluctuations of p _max and α _pmax . We also calculate the multiscale entropy of the various time series to estimate the effect of randomness in these fluctuations. Finally, we explain how the information on both p _max and α _pmax can be used to develop optimal strategies for controlling the combustion process and improving engine performance.     

Dynamic Model for LES Without Test Filtering: Quantifying the Accuracy of Taylor Series Approximations

The dynamic model for large-eddy simulation (LES) of turbulent flows requires test filtering the resolved velocity fields in order to determine model coefficients. However, test filtering is costly to perform in LES of complex geometry flows, especially on unstructured grids. The objective of this work is to develop and test an approximate but less costly dynamic procedure which does not require test filtering. The proposed method is based on Taylor series expansions of the resolved velocity fields. Accuracy is governed by the derivative schemes used in the calculation and the number of terms considered in the approximation to the test filtering operator. The expansion is developed up to fourth order, and results are tested a priori based on direct numerical simulation data of forced isotropic turbulence in the context of the dynamic Smagorinsky model. The tests compare the dynamic Smagorinsky coefficient obtained from filtering with those obtained from application of the Taylor series expansion. They show that the expansion up to second order provides a reasonable approximation to the true dynamic coefficient (with errors on the order of about 5% for c _s ²), but that including higher-order terms does not necessarily lead to improvements in the results due to inherent limitations in accurately evaluating high-order derivatives. A posteriori tests using the Taylor series approximation in LES of forced isotropic turbulence and channel flow confirm that the Taylor series approximation yields accurate results for the dynamic coefficient. Moreover, the simulations are stable and yield accurate resolved velocity statistics. Received 20 February 2001 and accepted 24 July 2001     

Variable-coefficient harmonic balance for periodically forced nonlinear oscillators

This paper explores the application of the method of variable-coefficient harmonic balance to nonautonomous nonlinear equations of the form XsF(X, t:), and in particular, a one-degree-of-freedom nonlinear oscillator equation describing escape from a cubic potential well. Each component of the solution, X(t), is expressed as a truncated Fourier series of superharmonics, subharmonics and ultrasubharmonics. Use is then made of symbolic manipulation in order to arrange the oscillator equation as a Fourier series and its coefficient are evaluated in the traditional way. The time-dependent coefficients permit the construction of a set of amplitude evolution equations with corresponding stability criteria. The technique enables detection of local bifurcations, such as saddle-node folds, period doubling flips, and parts of the Feigenbaum cascade. This representation of the periodic solution leads to local bifurcations being associated with a term in the Fourier series and, in particular, the onset of a period doubled solution can be detected by a series of superharmonics only. Its validity is such that control space bifurcation diagrams can be obtained with reasonable accuracy and large reductions in computational expense.     

Inflow conditions for inhomogeneous turbulent flows

A cost‐effective method to generate inflow conditions for direct numerical simulations of wall‐bounded flows is presented. The method recycles a finite‐length time series of instantaneous velocity planes extracted from a precursor simulation and has earlier proved efficient for free shear layers. Now a spatially developing plane channel flow is considered. Different durations t_s of the time series are tested and compared. Excellent agreement with fully developed channel flow statistics is observed when t_s equals or exceeds the large‐eddy turnover time scale. The present results are more realistic than those obtained with synthetic turbulence generation and at the same time substantially cheaper than running an auxiliary simulation in parallel. Copyright © 2008 John Wiley & Sons, Ltd.     

MHD boundary-layer flow of an upper-convected Maxwell fluid in a porous channel

Two-dimensional magnetohydrodynamic (MHD) boundary layer flow of an upper-convected Maxwell fluid is investigated in a channel. The walls of the channel are taken as porous. Using the similarity transformations and boundary layer approximations, the nonlinear partial differential equations are reduced to an ordinary differential equation. The developed nonlinear equation is solved analytically using the homotopy analysis method. An expression for the analytic solution is derived in the form of a series. The convergence of the obtained series is shown. The effects of the Reynolds number Re, Deborah number De and Hartman number M are shown through graphs and discussed for both the suction and injection cases.     

A singular perturbation problem of laminar flow in a uniformly porous channel with large injection and with an applied transverse magnetic field

Summary The problem of the steady flow of an electrically conducting viscous fluid through porous walls of a channel in the presence of an applied transverse magnetic field is considered. A solution for the case of small M ²/R (where M = Hartmann number, R = suction Reynolds number) with large blowing at the walls has been given by Terrill and Shrestha [3]. Their solution, on differentiating three times, is found to become infinite at the centre of the channel. Physically this means that there must be a viscous layer at the centre of the channel and Terrill and Shrestha are neglecting the shear layer. In this paper the solution given by Terrill and Shrestha is extended by obtaining an extra term of the series of expansion and the method of inner and outer expansion is used to obtain the complete solution which includes the viscous layer. The resulting series solutions are confirmed by numerical results.     

Convergence of biorthogonal series of biharmonic eigenfunctions by the method of titchmarsh

Canonical edge problems for the biharmonic equation can be solved by separating variables. The eigenvalues and eigenvectors arising in this separation are derived from a reduced system of ordinary differential equations along lines suggested in the excellent work of R. C. Smith (1952). We study the reduced system which is governed by a vector ordinary differential equation. A solution of the biharmonic problem, governed by a partial differential equation, can be found only if the prescribed data is restricted to a subspace of the space spanned by the eigenfunctions of the reduced problem. The theory leads to problems in generalized harmonic analysis which seek conditions under which arbitrary vector fields f(y) with values in ² can be represented in terms of eigenvectors of the reduced problem. This paper adds new theorems and conjectures to the theory. We extend Smith's generalization to fourth-order problems of the methods introduced by Titchmarsh (1946) to study eigenfunction expansions associated with second-order problems. We use this method to prove that, if f(y)=[(f ₁(y), f ₂y)], -1y1, f(y) C¹[-1, 1], f L₂[-1, 1], then the series expressing f(y) converges uniformly to f(y) in the open interval (-1, 1), uniformly in [-1, 1] if f ₁(±1)=0 and, in any case, to [0, f ₂(±1)-f ₁(±1)] at y=±1. This is unlike Fourier series, which converge to the mean value of the periodic extension of a function. The series exhibits a Gibbs phenomenon near the end points of discontinuity when f ₁(±1) 0.The Gibbs undershoot and overshoot for the step function vector [1, 0] and ramp function vector [y, 0] are computed numerically. The undershoot and overshoot are much larger than in the case of Fourier series and, unlike Fourier series, the Gibbs oscillations do not appear to be entirely suppressed by Féjer's method of summing Cesaro sums. We show that, when f(y) has interior points of discontinuity, the series for f(y) diverges and we present numerical results which indicate that, in this divergent case, the Cesaro sums converge to f(y) apparently with Gibbs oscillations near the point of discontinuity.     

A Cusp Catastrophe Approach to the Prediction of Temporal Patterns in the Kill Dates of Individual Serial Murderers

A cusp catastrophe model was fitted to the time series of kill dates of eleven serial murderers, using no additional external information. As predicted, this model provided a superior fit (Median R² =0.67) to the data than competing linear models (Median R² =0.43 and 0.22). Moreover, a single equation could be identified which was moderately effective (R² =0.53) in predicting the changes in the intervals between successive killings across individual killers. All time series showed evidence for the existence of a cyclic underlying process, although distinctions could be made depending on whether the phase portrait showed an attractor, a repellor, or a pulse type pattern. Finally, using a neural net approach it proved possible to distinguish pairs of serial killers solely based on properties of their time series. These findings are best explained by biological theories which assume that the cyclic nature of serial murder is the result of the successive domination of opposing forces.     

On the solution of certain field problems by the schwarz-christoffel method

In many practical cases the usefulness of the Schwarz-Christoffel method to solve two-dimensional field problems (Laplace equation with Dirichlet boundary conditions) is limited by the presence of transcendental functions of complex variables. We demonstrate here a new technique whereby, in lieu of qualitative plots of equipotential surfaces and flux lines, field components and potential can be expressed as real power series of the coordinates (x, y). The convergence of these series is only limited by the proximity of singular points corresponding to the physical convex corners. By choosing suitable points on the boundary around which the series of expansion are developed, fringing field components in the regions of interest between the boundaries can be computed directly. In some cases the series converges rapidly and assumes a remarkably simple form.     

Molecular weight polydispersity effects on the melt viscoelasticity of styrene-acrylonitrile random copolymers

A series of polydisperse SAN (styrene-co-acrylonitrile) random copolymers was studied by means of oscillatory rheometry in the rubbery plateau zone and in the terminal zone. The plateau modulus, the Newtonian viscosity, and the critical frequencies for the onset of non-Newtonian behavior were extracted from the experimental data. All these viscoelastic quantities consistently indicate that the tail of molecular weights below approximately M _e (the entanglement spacing) acts as a solvent for the rest of the polymer with M>M _e .     

Effects of relaxation time on start-up time for starting flow of Maxwell fluid in a pipe

Although the analytical solution of the starting flow of Maxwell fluid in a pipe has been derived for a long time, the effect of relaxation time λ on start-up time ts of this flow is still not well understood. Especially, there exist a series of jumps on the ts-λ. curve. In this paper we introduce a normalized mechanical energy by mode decomposition and mathematical analogy to describe the start-up process. An improved definition of start-up time is presented based on the normalized mechanical energy. It is proved that the ts-λ. curve contains a series of jumps if λ is larger than a critical value. The exact positions of the jumps are determined and the physical reason of the jumps is discussed.     

Mixed boundary value problem of heat conduction for infinite slab

Summary An axisymmetric steady state heat conduction boundary value problem having mixed boundary conditions on both faces of an infinite slab, is reduced to a pair of Fredholm integral equations of the second kind. For large values of h, the slab thickness, a solution correct to O(h ^–6) is obtained by expanding the kernels in power series.Presently at Imperial College, London.     

The effect of cold-work on the electrical resistivity of alloys and the law of recovery

Summary The change in resistivity due to cold-work has been determined for a number of binary alloy series having copper, silver or gold as basic metal. The effects observed are very different in the different alloy systems. A change as large as 42% of the total resistivity is found in one case. Alloys from the systems gold-chromium and gold-iron show a decrease of resistivity when cold-worked. In addition to the binary series, some alloys containing three or four components have also been investigated, and on the basis of the results obtained the question of the influence of the different solutes upon the change of resistivity is discussed. Results are given regarding the recovery of the resistivity changes as a function of time for different annealing temperatures. The resistivity change Δϱ as a function of temperatureT and time τ is found to obey the law Δϱ=cτ ⁿ exp (−nE/RT), wheren is a number of the order 0.1–0.4 andE is the activation energy of the process;R is the gas constant, andc a quantity of proportionality. Alloys belonging to the following binary systems have been investigated: Cu-Al, Cu-Si, Ag-Al, Ag-Mn, Ag-Sn, Au-Al, Au-Cr, Au-Mn, Au-Fe and Au-Sn, and the following multi-component systems: Au-Mn-Cr, Au-Mn-Cr-Fe and Au-Al-Cr.     

Definition and Empirical Characterization of Creative Processes

Creative processes exhibit a new, thus far unrecognized, form of dynamical behavior distinct from the known classes of mechanical and chaotic dynamics. We present quantitative methods of time series analysis that distinguish creative processes from random and chaotic systems. Creative processes exhibit diversification, indicating an expanding phase space volume, which contrasts with processes that converge to equilibrium, or to periodic or chaotic attractors. Creative processes exhibit novelty, that is, they produce less recurrence than obtains from random series. Creative processes exhibit arrangement, a measure of patterned recurrences that indicates nonrandom complexity. These three measures, diversification, novelty, and arrangement, reliably identify creative dynamics and distinguish creativity from chaos and from randomness.     

Rheologische Messungen an Polypropylenschmelzen

Zusammenfassung Es wurden Fließkurven von Polypropylenschmelzen bei 190 °C in einem Schergefällebereich von etwa 10^–1 bis 10² sec^–1 aufgenommen, wobei drei Serien von Polymerproben herangezogen wurden: eine Serie PP 1–7 von isotaktischen Propylenpolymerisaten mit einer Molekulargewichtsverteilung, die durch¯M _w /¯M _n 4–6 gekennzeichnet war; eine zweite Serie PP 1d–5d von abgebauten Polymerproben mit¯M _w /¯M _n 2–3; eine dritte Serie PP 1m–3m, die aus Mischungen von Polypropylenen bestand und die durch¯M _w /¯M _n >10 gekennzeichnet war. Es wurde der Einfluß des Molekulargewichts und der Molekulargewichtsverteilung auf die Fließkurven untersucht, insbesondere der Einfluß auf den Exponentenn in derOstwald-de Waele-Gleichung=KD ⁿ . Es wurde versucht, die Ergebnisse durch die universaleVinogradov-Viskositätsfunktion darzustellen, wobei sich aber sehr wesentliche Abweichungen infolge Variation des Molekulargewichts und der Molekulargewichtsverteilung ergaben.

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Summary Flow curves of molten polypropylene were determined at 190 °C over a shear gradient range of about l0^–1 to 10² sec^–1. Three series of polymer samples have been used: one series PP 1–7 of isotactic propylene polymerization products with a molecular weight distribution characterized by¯M _w /¯M _n 4–6, another series of degraded polymer samples PP 1d–5d with¯M _w /¯M _n 2–3, and still another series of polymer mixtures PP 1m–3m with¯M _w /¯M _n >10. The influence of molecular weight and molecular weight distribution on the flow curves was investigated, particularly on the exponentn in theOstwald-de Waele-power law=KD ⁿ . An attempt to plot the results by the universalVinogradov-viscosity function produced marked deviations due to variation of molecular weight and molecular weight distribution.

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Vorgetragen auf der Arbeitstagung der Sektion Rheologie des Vereins Österreichischer Chemiker am 17. November 1967 in Graz. Mitteilung I, II: Rheol. Acta5, 89 (1966); ibid. im Druck [Messung und Interpretation von Filmkurven].     

Photoelastic determination ofK I andK II : A numerical study on experimental data

The most efficient photoelastic methods to obtain stress intensity factors are those based on stress functions series expansions. The coefficients of these expansions are fitted to the experimental isochromatic pattern using an overdeterministic Newton-Raphson least squares method. In this paper, a study has been carried out to analyze the influence on the results of several numerical and experimental factors. It is shown that accurate values of the stress intensity factorsK _I andK _II can be obtained by following some recommendations given in the text and summarized in the conclusions at the end of the paper.     

Dynamic measurements of initiation toughness at high loading rates

An experimental method is described for measuring the dynamic initiation toughness of a sharp stationary crack. A plane specimen is utilized which consists of a central region 50-mm wide and 200-mm long with integral dog-bone ends. The loading is accomplished by the detonation of four small explosive charges which produce two tensile stress waves upon reflection from the dog-bone ends. The stress waves meet at the midpoint of the specimen and reinforce to produce a relatively large, uniformly stressed region with a very high loading rate. The crack is positioned at the midpoint of the specimen at the location where the reinforcing tensile stress waves meet. A series of photoelastic experiments were conducted using Homalite 100 as the model material to observe, in a full-field view, the arrival of the dilatational waves, the subsequent development of the stress field at the tip of the stationary crack and the initiation of the crack. The isochromatic fringe pattern was also used to determine the instantaneous value of the stress-intensity factorK(t) after the characteristic fringe loops developed in the region near the crack tip. Finally,K(t) was measured using a single strain gage positioned and oriented so that its signal output was proportional toK(t) and independent of the next two higher order terms in the series representation of the strain field. A method was developed to determine the instant of initiation from the strain-time trace. Results obtained from the photoelastic and strain measurements of the dynamic-initiation toughnessK _ID were consistently higher than the static value ofK _IC . Paper was presented at the 1987 SEM Fall Conference on Experimental Mechanics held in Savannah, GA on October 25–28.