Fixed pansystems theorems and pansystems logic conservation of chaos

The study of nonlinear problems was developed in works ^[1,2] by means of the pansystems methodology (PM) which does not need the condition related to differential manifolds and linear space. In work [2], within the framework of PM, we proved that study of panchaos, panattractor and strange panattractor can be transformed conditionally into some forms of fixed subsets. As a continuation of work [2], we now research the pansystems logic conservation of panchaos, panattractor, strange panattractor and some other fixed subsets.     

An approach on fixed pansystems theorems: Panchaos and strange panattractor

The investigations about chaos, attractor and strange attractor are main subjects in nonlinear analysis. Under the framework of pansystems methodology, reference [1] discussed these problems and introduced the concepts of panchaos, panattractor and strange panattractor. These concepts omitted the condition of continuity, compactness, etc. and put stress on the properties of binary relations on a set. A certain obtained result indicates that panchaos, panattractor and strange panattractor correspond respectively to fixed subsets of certain pansystems operators. This paper continues the investigation of [1,2], discusses the existence of these pansystems fixed subsets, their construction and interrelations.The authors appreciate Associate Research Fellow Qin Guo-guang for his discussion on related problems.     

Fixed pansystems theorems and eigen-fuzzy sets

In this paper, we apply fixed pansystems theorems to study when a fuzzy transformation has a nonzero eigen-fuzzy set. Many necessary conditions and sufficient conditions are obtained. Furthermore, the properties of nonzero eigen-fuzzy sets are investigated, and the results presented in Ref. [4] are extended.     

INVESTIGATION AND APPLICATIONS OF PANSYSTEMS RECOGNITION THEORY AND PANSYSTEMS-OPERATIONS RESEARCH OF LARGE-SCALE SYSTEMS(Ⅰ)

In this work, a sort of recognition theory and operations research of large scale systems are developed within the framework of pansystems methodology. We establish a series of theorems concerning pansystems relations, which discuss some fundamental problems of interdisciplines from the viewpoint of generalized system-transformation-symmetry in things mechanism, and are connected closely with mathematical physics systems thinking science bioecological sciences and the methodological investigation of mechanical foundation. This paper offers 100 pansystems theorems.     

Pansystems clustering analysis of complex systems

In this paper, by use of equivalence operators δ_i and semi-equivalence operators ε_i we study the clustering problems of complex systems, present δ_i(1,3) disconnection principle,dual transformation principle and large-scale systems decomposition principle for analizing and operating complex systems, discuss interconnectivity and disconnectivity of complex systems in detail and present some related theorems. Finally, we discuss the levels of systems according to pansystems, clustering approach proposed in this paper.     

The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations

We prove that viscosity solutions in W ^1, of the second order, fully nonlinear, equation F(D ² u, Du, u) = 0 are unique when (i) F is degenerate elliptic and decreasing in u or (ii) F is uniformly elliptic and nonincreasing in u. We do not assume that F is convex. The method of proof involves constructing nonlinear approximation operators which map viscosity subsolutions and supersolutions onto viscosity subsolutions and supersolutions, respectively. This method is completely different from that used in Lions [8, 9] for second order problems with F convex in D ² u and from that used by Crandall & Lions [3] and Crandall, Evans & Lions [2] for fully nonlinear first order problems.The research reported here was supported in part by grants from the Alfred P. Sloan Foundation and the National Science Foundation.     

The description of the characteristics of the structure and the quantity in fixed pansystems theorems

Pansymmetry is the abstract of symetry, stability and other concepts in physics and so on. Fixed pansystems theorems portray a typical pansymmetry of systemic structure. The present paper complements and extends the Work in concerned in fixed pansystems theorems. It gives in finite case the structural character of fixed subsets; the criterion of existence of the least fixed subset, and the numbering formula of fixed subsets and minimal fixed subsets.     

PANSYSTEMS METHODOLOGY AND CONSTRUCTION OF MAGIC SQUARES

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Variational principles in elasticity with nonlinear stress-strain relation

Since 1979, a series of papers have been published concerning the variational principles and generalized variational principles in elasticity such as [1] (1979), [6] (1980), [2,3] (1983) and [4,5] (1984). All these papers deal with the elastic body with linear stress-strain relations. In 1985, a book was published on generalized variational principles dealing with some nonlinear elastic body, but never going into detailed discussion. This paper discusses particularly variational principles and generalized variational principles for elastic body with nonlinear stress-strain relations. In these discussions, we find many interesting problems worth while to pay some attention. At the same time, these discussions are also instructive for linear elastic problems. When the strain is small, the high order terms may be neglected, the results of this paper may be simplified to the well-known principles in ordinary elasticity problems.     

Conservation laws and constitutive relations for density-gradient-dependent viscous fluids

Conservation laws and constitutive relations for a density-gradient-dependent viscous fluid as a multipolar continuum obeying an entropy inequality with generalized entropy flux and supply density are considered in this paper. A decomposition of the rate of work of dipolar stress, which reveals the contribution of various parts of this stress to the energy equation, is used to discuss the inconsistencies between the results obtained here and those obtained by Bluestein and Green [1] on the basis of the pioneering work of Green and Rivlin [8]. Furthermore, we discuss the connection between the model presented here and the materials of Korteweg type considered by Dunn and Serrin [6]. In particular, we relate the rate of work of dipolar stress and the interstitial working introduced by Dunn and Serrin [6].Received: 3 April 2004, Accepted: 6 December 2004, Published online: 4 March 2005 Correspondence to: M.M. Mehrabadi     

The effects of span position of winglet vortex generator on local heat/mass transfer over a three-row flat tube bank fin

The naphthalene sublimation method was used to study the effects of span position of vortex generators (VGs) on local heat transfer on three-row flat tube bank fin. A dimensionless factor of the larger the better characteristics, JF, is used to screen the optimum span position of VGs. In order to get JF, the local heat transfer coefficient obtained in experiments and numerical method are used to obtain the heat transferred from the fin. A new parameter, named as staggered ratio, is introduced to consider the interactions of vortices generated by partial or full periodically staggered arrangement of VGs. The present results reveal that: VGs should be mounted as near as possible to the tube wall; the vortices generated by the upstream VGs converge at wake region of flat tube; the interactions of vortices with counter rotating direction do not effect Nusselt number (Nu) greatly on fin surface mounted with VGs, but reduce Nu greatly on the other fin surface; the real staggered ratio should include the effect of flow convergence; with increasing real staggered ratio, these interactions are intensified, and heat transfer performance decreases; for average Nu and friction factor (f), the effects of interactions of vortices are not significant, f has slightly smaller value when real staggered ratio is about 0.6 than that when VGs are in no staggered arrangement. A cross section area of flow passage [m²] - A _mim minimum cross section area of flow passage [m²] - a width of flat tube [m] - b length of flat tube [m] - B _pT lateral pitch of flat tube: B _pT = S ₁/T _p - d _h hydraulic diameter of flow channel [m] - D _naph diffusion of naphthalene [m²/s] - f friction factor: f = pd _h/(Lu ² _max/2) - h mass transfer coefficient [m/s] - H height of winglet type vortex generators [m] - j Colburn factor [–] - JF a dimensionless ratio, defined in Eq. (23) [–] - L streamwise length of fin [m] - L _PVG longitudinal pitch of vortex generators divided by fin spacing: L _pVG = l _VG/T _p - l _VG pitch of in-line vortex generators [m] - m mass [kg] - m mass sublimation rate of naphthalene [kg/m²·s] - Nu Nusselt number: Nu = d _h/ - P pressure of naphthalene vapor [Pa] - p non-dimensional pitch of in-line vortex generators: p = l _VG/S ₂ - Pr Prandtl number [–] - Q heat transfer rate [W] - R universal gas constant [m²/s²·K] - Re Reynolds number: Re = ·u _max·d _h/ - S ₁ transversal pitch between flat tubes [m] - S ₂ longitudinal pitch between flat tubes [m] - Sc Schmidt number [–] - Sh Sherwood number [–]: Sh = hd _h/D _naph - Sr staggered ratio [–]: Sr = (2Hsin – C)/(2Hsin) - T _p fin spacing [m] - T temperature [K] - u _max maximum velocity [m/s] - u average velocity of air [m/s] - V volume flow rate of air [m³/s] - x,y,z coordinates [m] - z sublimation depth[m] - heat transfer coefficient [W/m²·K] - heat conductivity [W/m·K] - viscosity [kg/m²·s] - density [kg/m³] - attack angle of vortex generator [°] - time interval for naphthalene sublimation [s] - fin thickness, distance between two VGs around the tube [m] - small interval - C distance between the stream direction centerlines of VGs - p pressure drop [Pa] - 0 without VG enhancement - 1, 2, I, II fin surface I, fin surface II, respectively - atm atmosphere - f fluid - fin fin - local local value - m average - naph naphthalene - n,b naphthalene at bulk flow - n,w naphthalene at wall - VG with VG enhancement - w wall or fin surface     

Fracture of composite laminates containing cracks due to shear loading

The responses of graphite/epoxy [0/90/±45] _s , [±45]_2s , [0/90]_2s and [0/±45]_2s composite laminates with and without center cracks were studied under shear loading using the three-rail shear test. The shear stress/strain relationship, the failure mechanisms and the notched strength were analyzed. Substantial amounts of local buckling were observed in some of the laminates. The present paper shows that shear modulus can be determined accurately using the three-rail shear test with proper interpretation of data. Using the minimum strength model, only one characteristic length was needed to predict accurately the notched strength of a composite laminate under shear and tensile loadings.Paper was presented at the 1987 SEM Spring Conference on Experimental Mechanics held in Houston, TX on June 14–19.     

On the use of crack-tip-opening displacement to predict the fracture strength of notched graphite/epoxy laminates

The fracture strength and crack-opening displacement of notched graphite/epoxy laminates were measured experimentally using the center-cracked tension-specimen geometry. Four replicate tests were conducted for a variety of laminate stacking sequences, thicknesses, and notch lengths. Most laminates exhibited extensive notch-tip damage prior to fracture. Values of crack-tip-opening displacement (CTOD) at fracture were estimated from values of crack-opening displacement measured at the crack center line. CTOD was independent of specimen crack length for the [0/±45/90] _s , [0/±45/90]_15s , [0/±45] _s , [0/±45/]_15s , and [0/90]_24s laminates. In addition, notched laminate strength was accurately predicted using a Dugdale-type model along with the estimated CTOD.Paper was presented at V International Congress on Experimental Mechanics held in Montreal, Quebec, Canada on June 10–15, 1984.     

Pansystems logic conservation of bifurcation,catastrophe, chaos and stability

It is in references[4,5]that the combination of the relative researches of pansystems methodology and the researches of bifurcation,catastrophe,chaos and stability in nonlinear mechanics was put forward and the concepts were redefined from the point of view of pansystems methodology.The present paper studies the logic conservation law of these nonlinear mechanics phenomena under the framework of pansystems methodology.     

Three-segment electrodiffusion probe for velocity measurements

A cylindrical electrodiffusion probe for the measurement of liquid velocity vectors in the plane perpendicular to its axis was developed as an analogue to the triple-split film thermoanemometer. The geometry of the probe enables high directional resolution in the whole range of 360°. The total mass transfer of the probe was well correlated by the relation Sh = 0.76 Sc ^0.33 Re ^0.47.List of symbols A _kj , B _kj Fourier coefficient - c [mol/m³] depolarizer concentration - te]D [m²/s] diffusion coefficient of species - d [m] diameter of probe - f [1/s] frequency of vortex formation - h [mol/m²s] coefficient of mass transfer - I _k normalized current of K-th segment - i [A] total current - i _k [A] current of K-th segment - Re Reynolds number, u d/v - Sc Schmidt number, v/D - Sh Sherwood number, h d/c D - Sr Strouhal number, f d/v - v [m/s] free stream velocity - [°] flow angle, i.e. angle between approaching stream and reference direction of probe - v [m²/s] kinematic viscosity     

Pansymmetry and fixed pansystems theorems

In this paper, we continue the research of fixed pansystems theorems presented in ref [1], and discuss the existence, the structure and the number of fixed subsets.     

Strömungsmodell der Gas-Flüssigreaktoren nach der Filmtheorie

Zusammenfassung Es wird ein mathematisches Strömungsmodell für Gas-Flüssigreaktoren aufgestellt, das auf der Filmtheorie basiert. Für den Fall einer chemischen Reaktion erster Ordnung läßt sich eine geschlossene analytische Lösung finden, mit deren Hilfe man den Stoffaustauschgrad, den Reaktionsumsatz und die Reaktorkapazität leicht ermitteln kann. Das Modell eignet sich also unmittelbar als Auslegungsbasis für Gas-Flüssigreaktoren.

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A flowingmodel for gas-liquid reactors based on the film theory

A design model for gas-liquid reactors is developed based on the film theory and under condition that the gas and liquid phase are in plug flow. An analytical solution of this system has been achieved. The mass transfer degree, the reaction conversion and the reactor capacity can be easily calculated by means of the analytical solutions. Therefore, this model can be used directly to design the gas-liquid reactors.

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Formelzeichen a _i [m ²/m ³] spezifische Phasengrenzfläche - C [kmol/m³] Konzentration - D [m²/s] Diffusionskoeffizient - F [kmol/s] Masseneinströmung der Gasphase - H [J/kmol] Henry'sche Konstante - Ha [J/kmol] Hatta-Zahl definiert in Gl. (3) - L [m] charakteristische Länge des Reaktors - Q _L [m ³/s] Volumenströmung der Flüssigphase - N [kmol/m²·s] Stoffübergangsgeschwindigkeit - p [N/m²] Partialdruck einer Komponente - p [N/m²] Gesamtdruck des Systems - r [N/m²] Strömungsstatus - x [m] Ortskoordinate längs der Diffusionsrichtung - x _A [m] Reaktionsumsatz des EduktesA - V [m³] Reaktorvolumen Griechische Buchstaben [m] Diffusionsgrenzschichtdicke - _L Flüssig-Holdup - [m] Austauschgrad - [m] Abkürzung definiert in Gl. (13) - 0 bezogen auf Anfangsstelle des Reaktors - A bezogen auf KomponenteA - b bezogen auf Bulkphase - L bezogen auf Flüssigphase - bezogen auf Einströmung - bezogen auf Ausströmung     

PIV measurements of velocities and concentrations of wood fibres in pneumatic transport

A method for simultaneous measurement of the concentration and velocity of wood fibres suspended in air was developed. The velocity of the wood fibres was measured by the use of particle image velocimetry (PIV). The concentration of wood fibres was measured using the raw images from the PIV equipment as input data. An image processing procedure was used to determine the volume fraction of the fibre particles in the images. The method gave good qualitative and quantitative results for low volume fractions of fibres; for higher volume fractions the quantitative results were unsatisfactory.Latin letters C Concentration of fibres [g/m³] - d Diameter of fibre [m] - M_w Water mass [kg] - M_f Fibre mass [kg] - m Calibration mass flow [kg/s] - m₂₅ Calibration mass flow at C=25 g/m³ [kg/s] - n Fan rpm [-] - t Thickness of light sheet [m] - t Time between laser pulses [s] - U_i Velocity component in i-direction [m/s] - v Streamwise velocity [m/s] - v_average Average streamwise velocity [m/s] - x_i Particle displacement in i-direction [m] Greek letters _f Volume fraction of fibres [-] - _average Average volume fraction of fibres [-] - Area fraction of fibres in image [-] - Density of fibre particle [kg/m³]     

Study of thermal flows from two-dimensional,upward-facing isothermal surfaces using a laser speckle photography technique

A laser specklegram or speckle photography technique allows a direct measurement of surface temperature gradients and provides a full field interrogation with an extremely high resolution from a single data taking. The specklegram technique has been successfully applied to investigate the natural convection heat transfer from an upward-facing isothermal plate. For a plate with a large aspect ratio of 15, both local and global Nusselt numbers have been determined from the direct measurement of local temperature gradients. The Rayleigh number, based on the length scale equivalent to the ratio of the surface area to the perimeter, has been varied from 9.0 × 10³ to 4.0 × 10⁴. The present result for the global heat transfer has shown that a 1/5-power law, i.e., Nu = C₁ Ra ^1/5, correlates the data more properly whilst previously published results showed a large scatter in the exponent, ranging from 1/8-power to 1/4-power. The proportional constant, C₁ has been determined to be 0.56 which shows a fairly good agreement with previously published theoretical results. The laser specklegram technique has shown a strong potential as a powerful and convenient method for an experimental assessment of natural convection heat transfer problems. The specklegram technique at the same time has eliminated the deficiencies of both the mass transfer analogy technique and the classical heat transfer measurement technique.List of symbols a characteristic length scale defined as a = A/P where A is the surface area and P is the perimeter of the plate edge [mm] - AR aspect ratio [L/H] - c defocusing distance [mm] - d image distance of Young's fringes from speckle negative - h thermal convection coefficient [W/m² · K] - average thermal convection coefficient [W/m² · °C] - H width of the test section measured perpendicular to the optic axis [mm] - k thermal conductivity [W/m · K] - L length of the test section measured parallel to the optical axis [mm] - n index of refraction - Nu local Nusselt number [ha/k] - global Nusselt number - Pr Prandtl number [v/] - q heat flux per unit area [W/m² · s] - Ra Rayleigh number - s fringe spacing [mm] - Sc Schmidt number [v/D] - T temperature [K] Greek symbols thermal diffusivity [m²/s] - volumetric coefficient of expansion (1/T) - v kinematic viscosity of air [m²/s] - wavelength of helium-neon laser [632.8 nm] - amount of speckle dislocation     

A bar impact tester for dynamic fracture testing of ceramics and ceramic composites

A bar impact test was developed to study the dynamic fracture responses of precracked ceramic bars, Al₂O₃ and 15/29-percent volume SiC_w/Al₂O₃. Crack-opening displacement was measured with a laser-interferometric displacement gage and was used to determine the crack velocity and the dynamic stress-intensity factorK _I ^dyn . The crack velocity andK _I ^dyn increased with increasing impact velocity while the dynamic-initiation fracture toughness,K _Id, did not vary consistently with increasing impact velocities.Paper was presented at the 1992 SEM Spring Conference on Experimental Mechanics held in Las Vegas on June 8–11.