Symmetry groups of attractors

For maps equivariant under the action of a finite group on ⁿ, the possible symmetries of fixed points are known and correspond to the isotropy subgroups. This paper investigates the possible symmetries of arbitrary, possibly chaotic, attractors and finds that the necessary conditions of Melbourne, Dellnitz & Golubitsky [15] are sufficient, at least for continuous maps.This result shows that the reflection hyperplanes are important in determining those groups which are admissible; more precisely, a subgroup of is admissible as the symmetry group of an attractor if there exists a with / cyclic such that fixes a connected component of the complement of the set of reflection hyperplanes of reflections in but not in . For finite reflection groups this condition on reduces to the condition that is an isotropy subgroup. Our results are illustrated for finite subgroups of O(3).     

Asymptotic behavior of solutions of nonlinear elliptic equations

We study and obtain formulas for the asymptotic behavior as ¦x¦ of C ² solutions of the semilinear equation u=f(x, u), x (*) where is the complement of some ball in ⁿ and f is continuous and nonlinear in u. If, for large x, f is nearly radially symmetric in x, we give conditions under which each positive solution of (*) is asymptotic, as ¦x¦, to some radially symmetric function. Our results can also be useful when f is only bounded above or below by a function which is radially symmetric in x or when the solution oscillates in sign. Examples when f has power-like growth or exponential growth in the variables x and u usefully illustrate our results.     

A visualization study of the interaction of a free vortex with the wake behind an airfoil

The two-dimensional interaction of a single vortex with a thin symmetrical airfoil and its vortex wake has been investigated in a low turbulence wind tunnel having velocity of about 2 m/s in the measuring section. The flow Reynolds number based on the airfoil chord length was 4.5 × 10³. The investigation was carried out using a smoke-wire visualization technique with some support of standard hot-wire measurements. The experiment has proved that under certain conditions the vortex-airfoil-wake interaction leads to the formation of new vortices from the part of the wake positioned closely to the vortex. After the formation, the vortices rotate in the direction opposite to that of the incident vortex.List of symbols c test airfoil chord - C vortex generator airfoil chord - TA test airfoil - TE test airfoil trailing edge - TE _G vortex generator airfoil trailing edge - t dimensionless time-interval measured from the vortex passage by the test airfoil trailing edge: gDt=(T-T- _TE)·U/c - T time-interval measured from the start of VGA rotation - U free stream velocity - U vortex induced velocity fluctuation - VGA vortex generator airfoil - y distance in which the vortex passes the test airfoil - Z vortex circulation coefficient: Z=/(U · c/2) - vortex generator airfoil inclination angle - vortex circulation - vortex strength: =/2     

Numerische Untersuchung des Lax-Verfahrens

Übersicht Das von P. D. Lax angegebene Verfahren zur numerischen Lösung nichtlinearer hyperbolischer Systeme partieller Differentialgleichungen wird auf seine Brauchbarkeit zur Berechnung entsprechender Gasströmungen untersucht, besonders in bezug auf die Genauigkeit der Wiedergabe auch bei komplexeren Problemen. Außerdem werden zwei aufeinander aufbauende Modifikationen dieses Verfahrens angegeben, die die Konvergenzgeschwindigkeit beträchtlich erhöhen und (zweite Modifikation) näherungsweise eine Optimierung bezüglich des in realen Problemen u. U. stark wechselnden Verhältnisses t/x gestatten.

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Summary The accuracy of P. D. Lux's scheme for numerically solving nonlinear hyperbolic systems of partial differential equations has been tested with regard to instationary compressible fluid flow. Two modifications of the original scheme are given, which considerably increase the speed of convergence, one of them optimizing the calculation with regard to the eventually wide range of values of the ratio t/x.

     

Die Eigenschaften von Wasser und Wasserdampf nach „The 1968 IFC Formulation“

Zusammenfassung Diese Arbeit enthält Druck-Temperatur-Diagramme für 6 spezifische Zustandsgrößen und 16 erste Ableitungen und zusammengesetzte Größen von Wasser und Wasserdampf, die nach einem Gleichungssystem berechnet wurden, das unter dem Namen The 1968 IFC Formulation for Scientific and General Use von der 6. Internationalen Konferenz für die Eigenschaften des Wasserdampfes angenommen wurde. Einige Konsequenzen der thermodynamischen Konsistenz, das Verhalten im kritischen Gebiet und bei sehr kleinen Drücken werden diskutiert. Ferner werden die kinematische Viskosität und die Temperaturleitfähigkeit, sowie eine Beziehung zwischen dynamischer Viskosität und isenthalpem Drosselkoeffizienten angegeben.

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This paper contains pressure-temperature diagrams for 6 properties and 16 first derivatives and combined terms for water and steam. These were calculated from a system of equations accepted by the 6^th International Conference on the Properties of Steam, and called The 1968 IFC Formulation For Scientific and General Use. Some consequences of thermodynamic consistency, and the behaviour in the critical region and at very small pressures are discussed. Further, the kinematic viscosity and the thermal diffusivity and a relation between the dynamic viscosity and the throttling coefficient at constant enthalpy are given.

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Bezeichnungen (s. auch Tabelle 1) k Temperaturleitfähigkeit:k=/c _p - p Druck - r spezifische Verdampfungsenthalpie:r=h –h - T thermodynamische oder Kelvin-Temperatur - t Celsius-Temperatur - dynamische Viskosität - Wärmeleitfähigkeit - v kinematische Viskosität:=/ - Dichte:=1/v Indices und Sättigungswerte des Dampfes und der Flüssigkeit Differenz der Sättigungswerte, z. B. h=h –h     

Piezo-electric foils as a means of sensing unsteady surface forces

The experimental determination of steady and unsteady surface forces is an elementary problem in experimental fluid dynamics, e.g., in experimental aerodynamics. Up to now, unsteady forces such as pressure or shear fluctuations have been detected by means of special plug-in probes (e.g., miniature pressure transducers). An alternative and attractive technique of monitoring unsteady surface forces has become possible through the development of piezoelectric foils. With this novel type of sensor, which simply can be glued onto a surface, the piezoelectric effect of polarized plastic foils is used to register time-dependent pressure or shear loads. First of all, the paper concentrates on the fundamentals of this new measuring technique. Furthermore, some practical applications in experimental aerodynamics are outlined.List of symbols A area - c chord - C capacity - D pipe diameter - d _3i piezoelectric constant, i = 1–3 - f frequency - h foil thickness - p pressure - p pressure fluctuation - Q electric charge - R impedance - t time - u flow velocity - U voltage - x streamwise coordinate - Re Reynolds number - angle of attack - friction factor - aspect ratio - sweep angle - _w wall shear stress - wall shear stress fluctuation Indices i direction of mechanical stress 1, 2: in direction of foil plane 3: normal to foil plane - ll lower limit - RMS root-mean-square - w wall condition - free stream condition A version of this paper was presented at the 6th Symposium on Turbulent Shear Flows, Toulouse, France, September 1987     

Mean and turbulent velocity measurements of supersonic mixing layers

The behavior of supersonic mixing layers under three conditions has been examined by schlieren photography and laser Doppler velocimetry. In the schlieren photographs, some large-scale, repetitive patterns were observed within the mixing layer; however, these structures do not appear to dominate the mixing layer character under the present flow conditions. It was found that higher levels of secondary freestream turbulence did not increase the peak turbulence intensity observed within the mixing layer, but slightly increased the growth rate. Higher levels of freestream turbulence also reduced the axial distance required for development of the mean velocity. At higher convective Mach numbers, the mixing layer growth rate was found to be smaller than that of an incompressible mixing layer at the same velocity and freestream density ratio. The increase in convective Mach number also caused a decrease in the turbulence intensity ( _u/U).List of symbols a speed of sound - b total mixing layer thickness between U ₁ – 0.1 U and U ₂ + 0.1 U - f normalized third moment of u-velocity, f u³/(U)³ - g normalized triple product of u² , g u²/(U)³ - h normalized triple product of u ², h u ²/(U)³ - l _u axial distance for similarity in the mean velocity - l _u axial distance for similarity in the turbulence intensity - M Mach number - M _c convective Mach number (for ₁ = ₂), M _c (U ₁ – U ₂)/(a ₁ + a ₂) - P static pressure - r freestream velocity ratio, r U ₂/U ₁ - Re unit Reynolds number, Re U/ - s freestream density ratio, s ₂/₁ - T _t total temperature - u instantaneous streamwise velocity - u deviation of u-velocity, uu – U - U local mean streamwise velocity - U ₁ primary freestream velocity - U ₂ secondary freestream velocity - average of freestream velocities, (U ₁ + U ₂)/2 - U freestream velocity difference, U U ₁ – U ₂ - instantaneous transverse velocity - v deviation of -velocity, – V - V local mean transverse velocity - x streamwise coordinate - y transverse coordinate - y ₀ transverse location of the mixing layer centerline - ensemble average - ratio of specific heats - boundary layer thickness (y-location at 99.5% of free-stream velocity) - similarity coordinate, (y – y ₀)/b - compressible boundary layer momentum thickness - viscosity - density - standard deviation - dimensionless velocity, (U – U ₂)/U - 1 primary stream - 2 secondary stream A version of this paper was presented at the 11th Symposium on Turbulence, October 17–19, 1988, University of Missouri-Rolla     

A solitary wave in a porous medium

The one-phase Darcy continuity equation, including the quadratic gradient term, is considered. The exact linearization of the equation is found by a functional transformation for an arbitrary spatial dimension in the limit case where the constant fluid compressibility is much more dominant than the constant compressibilities of the reservoir parameters.The equation permits a solution representing a localized wave travelling through a one-dimensional reservoir without changing its form. This is the actual long-time limit of the transient solution for a constant sandface-rate injection of a compressible fluid with a constant compressibility if the fluid is much more compressible than the matrix. A solitary wave solution is not possible for production.A fully developed solitary wave would appear only for very high pressure increases, but the first signs of the emerging solitary wave are detectable at the sandface for moderate pressure increases which can appear under physical reservoir conditions.Latin symbols a Dimensionless wave propagation velocity - A _N Sandface area (N = 0, 1, 2) - c ₁, c ₂ Sums of compressibilities - c _x Generic (generalized) compressibility - c Fluid compressibility - c _h Reservoir height (i.e. bulk volume) compressibility (N = 0, 1) - c _k , c , c Generalized compressibilities - D Spatial reservoir dimensionality (D = 1, 2, 3) - f Fractional change of p _n1 due to nonlinear effects - h Reservoir height (proportional to bulk volume for N = 0, 1) - Horizontal reservoir width (N = 0) - k Reservoir permeability - K _N Constant with dimension of pressure (N = 0, 1, 2) - n Sum index - N Integer variable (N = D – 1) - p Reservoir pressure - p^* Overburden pressure - p _D Dimensionless (scaled) version of p - p ₀ Initial pressure - q Volumetric flow rate referred to sandface - r Radial (or linear) spatial distance from center of well - r _w Well radius - r _e External reservoir radius (or length) from center of well - t Time variable - t _f Injection/production time corresponding to fraction f - T Cole-Hopf-transformed version of dimensionless pressure y - u Rescaled (dimensionless) version of v _D - v Darcy velocity - v _d Dimensionless (scaled) version of v - x Generic symbol in compressibility expression (also used for auxiliary function and for auxiliary variable) - y Rescaled (dimensionless) version of p _D - z Dimensionless (scaled) version of r Greek symbols Coefficient of inertial resistance - Variable in wave solution for y - p _n1 Absolute change in physical sandface pressure due to production or injection - _p Pressure change over (dimensionless) distance behind and far away from front - _r Physical distance at constant time corresponding to - Characteristic (dimensionless) width of solitary wave - Formation porosity - ₁, ₂ Integration constants - Dimensionless (scaled) length of finite reservoir - Fluid viscosity - Fluid density - Dimensionless (scaled) version of t - Wave solution for dimensionless pressure y - Integer variable (±1) distinguishing between production and injection     

Über Differentialgleichungen für Wärme- und Stoffaustausch von hygroskopischen Stoffen

Zusammenfassung Die beiden Differentialgleichungssysteme vonKrischer undLykow werden miteinander verglichen. Dabei ergibt sich, daß die in der deutschen und russischen Literatur angewandten mathematischen Modelle der Trocknung von kapillarporösen Körpern praktisch übereinstimmen. Es werden die Transformationsgleichungen der dimensionslosen Kenngrößen angegeben, die die Beziehungen zwischen den beiden Systemen herstellen.

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The differential equations ofKrischer andLuikow for unsteady internal heat and mass transfer in the porous medium are compared. It is shown, that the mathematical models for drying in the German and Russian literature are equivalent. The transform relations of the non-dimensional parameters between the two models are given.

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Formelzeichen nach Krischer z laufende Koordinate in Strömungsrichtung in m - R kennzeichnende Abmessung des Körpers in m - t Zeit in h - Raumgewicht bei mittlerer Feuchtigkeit in kg/m³ - _w Teilgewicht des Wassers in 1 m³ Trockengut in kg/m³ - _wa Anfangsfeuchtigkeit in kg/m³ - _D Dampfdichte in kg/m³ - _L Luftvolumen je m³ Trocknungsgut in m³/m³ - Temperatur in °C - _u Umgebungstemperatur in °C - _a Anfangstemperatur in °C - r Verdampfungswärme in kcal/kg - q _E Wärmeentwicklung in kcal/m³ h - c spezifische Wärmekapazität des Trockengutes in kcal/kg grd - Wärmeleitfähigkeit in kcal/m h grd - Feuchtigkeitsleitzahl des Trockengutes in m²/h - wirksame Diffusionszahl von Wasserdampf in Luft in m²/h - Diffusionswiderstandszahl des Trockengutes — - Konstante — - Konstante in kg/m³ grd Formelzeichen nach Lykow X=r/R dimensionslose Koordinate des Körpers;r laufende Koordinate in m;R kennzeichnende Abmessung in m; - Fo=a/R ² Fourier-Zahl;a Temperaturleitzahl in m²/h; Zeit in h - T(X, Fo)=t(r, )– 0/t dimensionslose Temperatur des Körpers im Punkt mit KoordinateX für den ZeitpunktFo;t(r, ) Temperatur in °C; 0 mittlere Anfangstemperatur in °C; t ein vorher angenommener Temperaturunterschied in grd - (X, Fo)= ₀–u(r, )/u dimensionsloses Potential des Stoffübergangs im Punkt mit KoordinateX für den ZeitpunktFo;u(r, ) Feuchtigkeitsgehalt des Trockengutes in kg/kg; ₀ mittlerer Anfangsfeuchtigkeitsgehalt in kg/kg; u ein vorher angenommener Unterschied des Feuchtigkeitsgehalts in kg/kg - Ko= u/c t Kosowitsch-Zahl; Verdampfungswärme in kcal/kg;c spezifische Wärmekapazität des Trockengutes in kcal/kg - Ko*=Ko modifizierte Kosowitsch Zhal; Kenngröße der Zustandsänderung - Pn= t/u Posnowsche Zahl;=a _T ^m /a _m Thermogradientkoeffizient in 1/grd;a _T ^m thermische Stoffübergangszahl (charakterisiert den Stoffstrom unter der Einwirkung von Temperaturgradienten) in m²/h grd;a _m Stoffübergangszahl (charakterisiert den Stoffstrom unter der Einwirkung von Feuchtigkeitsgradienten) in m²/h - Lu=a _m/a Lykowsche Zahl - Ki _q=q _q ()·R/ _q t dimensionsloser Wärmestrom (Kirpitschew-Zahl);q _q() Wärmestrom durch die Körperoberfläche in kcal/m²; _q Wärmeleitfähigkeit in kcal/m² h grd - Ki _m=q _m ()·R/a _{m 0} u dimensionsloser Stoffstrom;q _m() Stoffstrom durch die Körperoberfläche in kg/m² h; ₀ Wichte des Trockengutes in kg/m³     

On slip effect in free coating of non-Newtonian fluids

The failure of the current theories to predict the coating thickness of non-Newtonian fluids in free coating operations is shown to be a result of the effective slip at the moving rigid surface being coated. This slip phenomenon is a consequence of stress induced diffusion occurring in flow of structured liquids in non-homogeneous flow fields. Literature data have been analysed to substantiate the slip hypothesis proposed in this work. The experimentally observed coating thickness is shown to lie between an upper bound, which is estimated by a no-slip condition for homogeneous solution and a lower bound, which is estimated by using solvent properties. Some design considerations have been provided, which will serve as useful guidelines for estimating coating thickness in industrial practice.fa exponent in eq. (15) - b n/(4 –n)(n + 1) - Ca Capillary number - D diffusivity - De Deborah number - g acceleration due to gravity - G Goucher number - h thickness profile - h ₀ final coating thickness - K consistency index - L length available for diffusion - L _t tube length - n power-law index - P pressure drop - Q flow rate - R cylinder radius - R _t tube radius - t time available for diffusion - T ₀ dimensionless thickness without slip - T _s dimensionless thickness with slip - U _c theoretically calculated withdrawal velocity to match the film thickness - u _s slip velocity - U withdrawal velocity - U _w theoretically calculated withdrawal velocity based on solvent properties - U ^* effective withdrawal velocity - x distance in the direction of flow - y distance transverse to the flow direction - curvature coefficient - slip coefficient - curvature coefficient - rate of deformation tensor - u _s /U - relaxation time - density - surface tension - shear stress in tube flow - _w wall shear stress in tube flow - stress tensor - _w wall shear stress - T _s /T ₀ NCL-Communication No. 2818     

Lower semicontinuity of the attractor for a singularly perturbed hyperbolic equation

For a smooth, bounded domain R, n 3, and a real, positive parameter, we consider the hyperbolic equationu _tt +u _t –u=–f(u) –g in with Dirichlet boundary conditions. Under certain conditions onf, this equation has a global attractorA inH ₀ ¹ () ×L ²(). For=0, the parabolic equation also has a global attractor which can be naturally embedded into a compact setA ₀ inH ₀ ¹ () ×L ²(). If all of the equilibrium points of the parabolic equation are hyperbolic, it is shown that the setsA are lower semicontinuous at=0. Moreover, we give an estimate of the symmetric distance betweenA ₀ andA .     

Rheologie pathologischer Gelenkflüssigkeiten

Zusammenfassung Die eingehende Analyse des viskoelastischen Verhaltens von 193 Kniegelenkspunktaten verschiedenster entzündlicher und nichtentzündlicher Gelenkerkrankungen ließ keine wesentlichen diagnostischen Hilfen für klinische Problemfälle erkennen. Untersucht wurden im einzelnen Fließkurven einschließlich der Anfangsviskosität ₀ und durch eine Normierungsmethode ermittelte master-curves, sowie Normalspannungen und in 3 Fällen gleichzeitig auch der SpeichermodulG und der VerlustmodulG mit Hilfe von dynamischen Messungen.Durch Vergleich der pathologischen Gelenkpunktate mit normaler, post mortal gewonnener gepoolter Synovia ließ sich ein Eindruck vom Grad der gestörten Viskoelastizität gewinnen. Dabei lassen die erniedrigten Hyaluronsäure-Konzentrationen, die Veränderungen der konzentrationsunabhängigen Knickzeitt _k> und die master-curve erkennen, daß hierfür sowohl eine verringerte Konzentration als auch ein geringeres Molekulargewicht der Hyaluronsäure verantwortlich ist. Konzentrierungsversuche pathologischer Synovia ergaben den Hinweis auf die Entstehung von Mikrogelen und ließen in Fällen zuvor fehlender Normalspannungen auch nach der Eindickung keine Normalspannungen erkennen. Es wird deshalb auch die Möglichkeit gestörter intermolekularer Interaktionen in der pathologischen Synovia diskutiert.

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Summary A thorough analysis of the viscoelastic behaviour of 193 synovial fluid samples of knee joints concerning different joint diseases (inflammatory and non-inflammatory) gives no essential diagnostic help in case of clinical problems.Investigations were done particularly on flow curves including the Newtonian viscosity ₀ and normal forces, and with the help of a standardization-method we got master curves. In three cases we also got dynamic properties i.e. the elastic modulusG and the loss modulusG. By comparison of the pathological synovial fluid samples with normal, post-mortem pooled synovial fluid one gets an idea of the degree of disturbance on viscoelasticity. It was found that the reduced concentration and the lower molecular weight of the hyaluronic acid are responsible for the pathological variation of the concentration independent bending timet _k as well as the shape of the master curves.Tests on concentrated pathological synovial fluids indicate the beginning formation of micro-gels. In cases of absence of normal forces even after concentration no normal forces could be detected. Therefore the possibility of disturbed intermolecular interactions in pathological synovial fluids will be discussed, too.

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D Schergeschwindigkeit - D _K D-Koordinate des Tangentenschnittpunktes - t _k Knickzeit - s Steigung des geradlinigen Anteils der Fließkurve - scheinbare Viskosität - _m mittlere Viskosität beiD = 10² s^–1 - ₀ Anfangsviskosität, Nullviskosität - Endviskosität - _{N 11} – ₂₂ 1. Normalspannungsdifferenz - G ₀ Ruheschermodul - G ^* komplexer (dynamischer) Schermodul - G Speichermodul - G Verlustmodul - Winkelgeschwindigkeit - Winkel der Phasendifferenz - Kegelwinkel - d Durchmesser von Kegel und Platte des Meßsystems - f Frequenz der vorgegebenen Oszillation - f _n Eigenfrequenz des Torsionskopfes - _IA Amplitude der Eingangsschwingung - _TA Amplitude der Ausgangsschwingung - _I axiale Bewegung der Schneckenwelle - _T Bewegung des Torsionskopfverminderers - [] Grenzviskositätszahl (Staudinger-Index) - v Verhängungszahl - r Korrelationskoeffizient - m Mittelwert - s Standardabweichung - p Signifikanzniveau - n.s. nicht signifikant p > 0,05 Auszugsweise vorgetragen auf der Jahrestagung der Deutschen Rheologischen Gesellschaft in Berlin vom 8.–10. Mai 1978.Mit 9 Abbildungen und 8 Tabellen     

Reservoir transport equations by compositional approach

The objective of this paper is to present an overview of the fundamental equations governing transport phenomena in compressible reservoirs. A general mathematical model is presented for important thermo-mechanical processes operative in a reservoir. Such a formulation includes equations governing multiphase fluid (gas-water-hydrocarbon) flow, energy transport, and reservoir skeleton deformation. The model allows phase changes due to gas solubility. Furthermore, Terzaghi's concept of effective stress and stress-strain relations are incorporated into the general model. The functional relations among various model parameters which cause the nonlinearity of the system of equations are explained within the context of reservoir engineering principles. Simplified equations and appropriate boundary conditions have also been presented for various cases. It has been demonstrated that various well-known equations such as Jacob, Terzaghi, Buckley-Leverett, Richards, solute transport, black-oil, and Biot equations are simplifications of the compositional model.Notation List B reservoir thickness - B formation volume factor of phase - Cⁱ mass of component i dissolved per total volume of solution - C ⁱ mass fraction of component i in phase - C heat capacity of phase at constant volume - C_p heat capacity of phase at constant pressure - D ⁱ hydrodynamic dispersion coefficient of component i in phase - D_MTf thermal liquid diffusivity for fluid f - F = F(x, y, z, t) defines the boundary surface - f_p fractional flow of phase - g gravitational acceleration - H_p enthalpy per unit mass of phase - J_p volumetric flux of phase - k_rf relative permeability to fluid f - k₀ absolute permeability of the medium - M_p ⁱ mass of component i in phase - n porosity - N rate of accretion - P_f pressure in fluid f - p_ca capillary pressure between phases and =p-p - Rⁱ rate of mass transfer of component i from phase to phase - Rⁱ _source source rate of component i within phase - S saturation of phase - s gas solubility - T temperature - t time - U displacement vector - u velocity in the x-direction - v velocity in the y-direction - V volume of phase - V_s velocity of soil solids - W_i body force in coordinate direction i - x horizontal coordinate - z vertical coordinate Greek Letters _p volumetric coefficient of compressibility - _T volumetric coefficient of thermal expansion - _ij Kronecker delta - volumetric strain - _m thermal conductivity of the whole matrix - internal energy per unit mass of phase - gf suction head - density of phase - _ij tensor of total stresses - _ij tensor of effective stresses - volumetric content of phase - f viscosity of fluid f     

Thermal free convection from a solid sphere

Summary Thermal free convection from a sphere has been studied by melting solid benzene spheres in excess liquid benzene (Pr=8,3; 10⁸<Gr<10⁹). Overall heat transfer as well as local heat transfer were investigated. For the effect of cold liquid produced by the melting a correction has been applied. Results are compared with those obtained by other workers who used alternative experimental methods.Nomenclature coefficient of heat transfer - d characteristic length, here diameter of sphere - thermal conductivity - g acceleration of free fall - cubic expansion coefficient - T temperature difference between wall and fluid at infinity - kinematic viscosity - density - c specific heat capacity - a thermal diffusivity (=/c) - D diffusion coefficient - Nu dimensionless Nusselt number (=d/) - Nu* the analogous number for mass transfer (=kd/D) - mean value of Nusselt number - Gr dimensionless Grashof number (=gd ³T/ ²) - Gr* the analogous number for mass transfer (=gd ³x/ ²) - Pr dimensionless Prandtl number (=/a) - Sc dimensionless Schmidt number (=/D)     

Elongational flow opto-rheometry for polymer melts

Polymer melt elongation is one of the most important procedures in polymer processing. To understand its molecular mechanisms, we constructed an elongational flow opto-rheometer (EFOR) in which a high precision birefringence apparatus of reflection-double path type was installed into a Meissner's new elongational rheometer of a gas cushion type (commercialized as RME from Rheometric Scientific) just by mounting a small reflecting mirror at the center of the RME's sample supporting table. The EFOR enabled us to achieve simultaneous measurements of tensile stress (t) and birefringence n(t) as a function of time t under a given constant strain rate within the range of 0.001 to 1.0s^–1. (t) can be monitored upto the maximum Hencky strain (t) of 7 as attained, in principle, with RME, while the measurable range of the phase difference in the birefringence was 0 to 250 (0 to 79 100 nm for He-Ne laser light) within the accuracy of ±0.1 (±31.6 nm) up to (t) 4. The performance was tested on an anionically polymerized polystyrene (PS) and a low density polyethylene (LDPE). For both polymers (t) first followed the linear viscoelasticity rule in that the elongational viscosity, , is three times the steady shear viscosity, 3 _o(t), at low shear rate , but the _E (t) tended to deviate upward after a certain Hencky strain was attained. The birefringence n(t) was a function of both Hencky strain and strain rate in such a way that the stress-optical law holds with the stress-optical coefficient C(t) = n(t)/(t) being equal to the ones reported from shear flow experiments. Interestingly, however, for PS elongated at low strain rates the C(t) vs (t) relation exhibited a strong nonlinearity as soon as (t) reached steady state. This implies that the tensile stress reaches the steady state but the birefringence continues to increase in the low strain-rate elongation. For the PS melt elongated at high strain rates, on the other hand, C(t) was nearly a constant in the entire range observed. For LDPE with long-chain branchings, (t) exhibited tendency of strain-induced hardening after certain critical strain, but C(t) was nearly a constant in the entire range of (t) observed.     

New calculations on the Faraday effect in wave guides

Summary The propagation of electromagnetic waves is investigated theoretically for a round wave guide, containing a gyroelectrie-gyromagnetic medium with gyroaxis parallel to the guide in the form of a cylindrical shell of thickness, adjacent to the wall of the guide. An equation is set up, permitting to compute the change in the propagation constant due to the presence of the shell, including terms proportional to ². Assuming only the presence of gyromagnetism, the change ₁ of first order in for TE-waves is determined and is found to be the same fpr right- and left-circular polarization. The second order difference ₂ ⁺ — ₂ ^- for the two senses of polarization, however, appears to have a non-vanishing value which, just like ₁ can be expressed in terms of the radius of the guide, the frequency, the dielectric constant and the elements of the gyromagnetic permeability tensor which characterize the medium of the shell.     

Fatigue crack propagaion under mixed mode loading

Mixed model fatigue crack propagation is analyzed in this paper, using a centre cracked plate geometry, loaded under un-iaxial cyclic tension. Based on maximum principal stress criterion, a modified Paris expression of fatigue crack growth rate is derived in terms of ΔK and crack angle β_α for an inclined crack. It is also shown that it is more convenient to express the Paris equation by means of crack length projected on the x -axis, α_x rather than the actual length, α itself. The crack trajectory due to cyclic loading is predicted, β is varied from 29° to 90°. Experimental data on Type L3 aluminium agree fairly well with predicted values when β_α exceeds 30°.     

Two-color particle-imaging velocimetry using a single argon-ion laser

A swept-beam, two-color particle-imaging velocimetry (PIV) technique has been developed which utilizes a single argon-ion laser for illuminating the seed particles in a flowfield. In previous two-color PIV techniques two pulsed lasers were employed as the different-color light sources. In the present experiment the particles in a two-dimensional shear-layer flow were illuminated using arotating mirror to sweep the 488.0-nm (blue) and 514·5-nm (green) lines of the argon-ion laser through a test section. The blue- and greenparticle positions were recorded on color film with a 35-mm camera. The unique color coding eliminates the directional ambiguities associated with single-color techniques because the order in which the particle images are produced is known. Analysis of these two-color PIV images involved digitizing the exposed film to obtain the blue and green-particle image fields and processing the digitized images with velocity-displacement software. Argon-ion lasers are available in many laboratories; with the addition of a rotating mirror and a few optical components, it is possible to conduct flow-visualization experiments and make quantitative velocity measurements in many flow facilities.List of symbols d length of displacement vector - d _m distance between rotating mirror and concave mirror - n _f number of facets on rotating mirror - R seed-particle radius - v velocity in x, y plane - v _s sweep velocity of laser beams, assumed to be in y direction from top to bottom of field of view - v _x, v _y, v _z x, y, and z components of velocity - x ₁, y ₁ color-1 particle coordinates - x ₂, y ₂ color-2 particle coordinates - y _max y dimension of field of view, assumed to be the long dimension - s spatial separation of beams as they approach rotating mirror - t time separation of laser sheets or of swept beams passing fixed point - t _b time between successive sweeps through test section by same beam - t _s time required for both beams to sweep through test section - angular separation of beams reflecting from rotating mirror - fluid viscosity - v angular velocity of rotating mirror in cycles per second - seed-particle density - seed-particle response time - _v, _d, _t standard deviation of velocity, displacement, and time - vorticity This work was supported, in part, by the Aero Propulsion and Power Directorate of Wright Laboratory under Contract No. F33615-90-C-2033.     

Effects of MHD free convection and mass transfer on the flow past an oscillating infinite coaxial vertical circular cylinder

The effects of MHD free convection and mass transfer are taken into account on the flow past oscillating infinite coaxial vertical circular cylinder. The analytical expressions for velocity, temperature and concentration of the fluid are obtained by using perturbation technique.

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Einwirkungen von freier MHD-Konvektion und Stoffübertragung auf eine Strömung nach einem schwingenden unendlichen koaxialen vertikalen Zylinder

Zusammenfassung Die Einwirkungen der freien MHD-Konvektion und Stoffübertragung auf eine Strömung nach einem schwingenden, unendlichen, koaxialen, vertikalen Zylinder wurden untersucht. Die analytischen Ausdrücke der Geschwindigkeit, Temperatur und Fluidkonzentration sind durch die Perturbationstechnik erhalten worden.

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Nomenclature C _p Specific heat at constant temperature - C the species concentration near the circular cylinder - C _w the species concentration of the circular cylinder - C the species concentration of the fluid at infinite - * dimensionless species concentration - D chemical molecular diffusivity - g acceleration due to gravity - Gr Grashof number - Gm modified Grashof number - K thermal conductivity - Pr Prandtl number - r _a ,r _b radius of inner and outer cylinder - a, b dimensionless inner and outer radius - r,r coordinate and dimensionless coordinate normal to the circular cylinder - Sc Schmidt number - t time - t dimensionless time - T temperature of the fluid near the circular cylinder - T _w temperature of the circular cylinder - T temperature of the fluid at infinite - u velocity of the fluid - u dimensionless velocity of the fluid - U ₀ reference velocity - z,z coordinate and dimensionless coordinate along the circular cylinder - coefficient of volume expansion - * coefficient of thermal expansion with concentration - dimensionless temperature - H ₀ magnetic field intensity - coefficient of viscosity - _e permeability (magnetic) - kinematic viscosity - electric conductivity - density - M Hartmann number - dimensionless skin-friction - frequency - dimensionless frequency     

Certain yawed magnetogasdynamic flows

Certain steady yawed magnetogasdynamic flows, in which the magnetic field is everywhere parallel to the velocity field, are related to certain reduced three-dimensional compressible gas flows having zero magnetic field. Under a restriction, the reduced flows are linked, by certain reciprocal relations, to a four parameter class of plane gas flows. In the instance of constant entropy an approximation method is suggested for obtaining magnetogasdynamic flows from the corresponding plane, irrotational gasdynamic flows and examples are given.

Nomenclature

magnetogasdynamic flow variables H magnetic intensity - q fluid velocity - fluid density - p pressure - s entropy - Q _t, H _t component of q, H in the x–y plane - w , h component of q, H perpendicular to the x–y plane reduced gasdynamic flow factor of proportionality - q* fluid velocity - * fluid density - p* pressure - Q _t ^* =u*î+v*, w* components of q* - l arbitrary constant - A _v Alfvén speed - Q _t, , p fluid velocity, density, pressure of the reciprocal gas dynamic flow - L, n, k, arbitrary constants - , velocity potential, stream function - angle made by Q _t, Q _t ^* , and V with the x-axis - adiabatic gas constant - a ²=(–1)/2 constant - M Mach number - W constant value of w* - E approximate constant value of g(p) - * modified potential function - modified velocity coordinate - +i - complex potential of the irrotational flow - B arbitrary constant - V incompressible flow velocity - V modified fluid velocity - X _p, Y _p points on the profile