On the stability of solutions of a non-linear, non-autonomous equation

Sufficient conditions for the stability of solutions of the equation

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are deduced. These conditions depend on coefficient ratios a/c and b/c as well as on initial conditions. As an example the pendulum of variable length is discussed.     

A study of two alternate tangent modulus formulations and attendant implicit algorithms for creep as well as high-strain-rate plasticity

Two alternate tangent modulus formulations based on the use of material characteristics expressed as

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, respectively, are prese for the analysis of material response under conditions such as high temperature creep and high strainrate dynamic plasticity. In each formulation, implicit algorithms of generalized midpoint radial mapping are presented to compute stress histories at a material point. Several examples to illustrate the stability, accuracy, and convergence of the presented computational methodologies are included. In each instance, the two alternate tangent modulus approaches lead to results of comparable accuracy and compare excellently with each other as well as other available independent solutions.     

Flow in a cylinder driven by rotating split-disk endwalls

Flow of an incompressible viscous fluid contained in a cylindrical vessel (radius R, height H) is considered. Each of the cylinder endwalls is split into two parts which rotate steadily about the central axis with different rotation rates: the inner disk (r < r₁) rotating at Ω₁, and the outer annulus (r₁ < r < R) rotating at Ω₂. Numerical solutions to the axisymmetric Navier-Stokes equations are secured for small system Ekman numbers E ( v/(ΩH²)). In the linear regime, when the Rossby number Ro

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, the numerical results are shown to be compatible with the theoretical prediction as well as the available experimental measurements. Emphasis is placed on the results in the nonlinear regime in which Ro is finite. Details of the structures of azimuthai and meridional flows are presented by the numerical results. For a fixed Ekman number, the gross features of the flow remain qualitatively unchanged as Ro increases. The meridional flows are characterized by two circulation cells. The shear layer is a region of intense axial flow toward the endwall and of vanishing radial velocity. The thicknesses of the shear layer near r = r₁ and the Ekman layer on the endwall scale with E and E , respectively. The numerical results are consistent with these scalings.     

Numerical prediction of locally forced turbulent separated and reattaching flow

An unsteady numerical simulation was performed for locally forced separated and reattaching flow over a backward-facing step. The local forcing was given to the separated and reattaching flow by means of a sinusoidally oscillating jet from a separation line. A version of the k––f_μ model was employed, in which the near-wall behavior without reference to distance and the nonequilibrium effect in the recirculation region were incorporated. The Reynolds number based on the step height (H) was fixed at Re_H=33 000, and the forcing frequency was varied in the range 0St_H2. The predicted results were compared and validated with the experimental data of Chun and Chun. It was shown that the unsteady locally forced separated and reattaching flows are predicted reasonably well with the k––f_μ model. To characterize the large-scale vortex evolution due to the local forcing, numerical flow visualizations were carried out.     

A general correlation for the local loss coefficient in Newtonian axisymmetric sudden expansions

Results from numerical simulations and guidance from an approximated corrected-theory, developed by Oliveira and Pinho (1997), (Oliveira, P.J. and Pinho, F.T. 1997. Pressure drop coefficient of laminar Newtonian flow in axisymmetric sudden expansions. Int. J. Heat and Fluid flow 18, 518–529) have been used to arrive at a correlation expressing the irreversible loss coefficient for laminar Newtonian flow in axisymmetric sudden expansions. The correlation is valid for the ranges 1.5 < D₂/D₁ < 4 and 0.5 < Re < 200 with errors of less than 5%, except for 25 < Re < 100 where the error could be as much as 7%. The recirculation bubble length is also presented for the same range of conditions and the pressure recovery coefficient was calculated for Reynolds numbers above 15.     

The scattering of a plane wave by two parallel semi-infinite overlapping screens with dielectric loading

The diffraction of a plane electromagnetic wave by two parallel, overlapping, perfectly conducting half-planes with a dielectric loading is considered by using the Fourier transform technique in conjunction with the mode-matching method. This mixed method of formulation enables one to obtain directly two uncoupled modified Wiener–Hopf equations instead of a matrix one. When the overlap length is set to zero and the dielectric loading is removed, the results are shown to coincide exactly with those obtained previously by Abrahams [1]. Numerical results showing the effects of the dielectric loading on the transmitted field are also presented.     

On the flow in a uniformly porous pipe

This article considers fully laminar flow of an incompressible viscous fluid in a uniformly porous pipe with suction and injection. An exact solution of the Navier–Stokes equations is given. The velocity filed can be expressed in a series form in terms of the modified Bessel function of the first kind of order n. The volume flux across a plane normal to the flow, the vorticity and the stress on the boundary are presented. The flow properties depend on the cross-Reynolds number, Ua/ν, where U is the suction velocity, a is the radius of the pipe and ν is the kinematic viscosity of the fluid. It is found that for large values of the cross-Reynolds number, the flow near the region of the suction shows a boundary layer character. In this region the velocity and the vorticity vary sharply. Outside the boundary layer, the velocity and the vorticity do not show an appreciable change.     

Variational formulation of ideal fluid flows according to gauge principle

On the basis of the gauge principle of field theory, a new variational formulation is presented for flows of an ideal fluid. The fluid is defined thermodynamically by mass density and entropy density, and its flow fields are characterized by symmetries of translation and rotation. The rotational transformations are regarded as gauge transformations as well as the translational ones. In addition to the Lagrangians representing the translation symmetry, a structure of rotation symmetry is equipped with a Lagrangian Λ_A including the vorticity and a vector potential bilinearly. Euler's equation of motion is derived from variations according to the action principle. In addition, the equations of continuity and entropy are derived from the variations. Equations of conserved currents are deduced as the Noether theorem in the space of Lagrangian coordinate a. Without Λ_A, the action principle results in the Clebsch solution with vanishing helicity. The Lagrangian Λ_A yields non-vanishing vorticity and provides a source term of non-vanishing helicity. The vorticity equation is derived as an equation of the gauge field, and the Λ_A characterizes topology of the field. The present formulation is comprehensive and provides a consistent basis for a unique transformation between the Lagrangian a space and the Eulerian x space. In contrast, with translation symmetry alone, there is an arbitrariness in the transformation between these spaces.     

Messung der Viskosität von Kohlendioxid und Propylen

Zusammenfassung Die Viskosität von Kohlendioxid und Propylen wurde bei Temperaturen zwischen 298 K und 473 K und Drücken zwischen 1 bar und dem zweifachen kritischen Druck mit einem Schwingscheibenviskosimeter gemessen. Dieses unterscheidet sich von den bisher bekannten Schwingscheibenviskosimetern insbesondere durch die optische Einrichtung zur Bestimmung der Winkelamplituden. Die mittleren relativen Fehler der Meßwerte sind vom thermodynamischen Zustandsbereich abhängig und liegen zwischen 0,9 % und 1,6%. Für Interpolationsrechnungen werden einfache Gleichungen angegeben.

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Measurements on the viscosity of carbon dioxide and propylene

Measurements on the viscosity of carbon dioxide and propylene are reported. The experimental investigations have been performed with an oscillating disk viscosimeter at temperatures between 298 K and 473K and pressures from 1 bar up to the twofold critical pressure of each gas. The optical system for reproducing the oscillations of the disk on a scale is modified to the yet known oscillating disk viscosimeters. With respect to the thermodynamic state of the gas an accuracy between 98,4% and 99,1% could be reached. For correlating the measured values two polynomial approximations are reported.

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Formelzeichen b mittlere Spaltbreite; - C_N,C Gerätekonstante - D Scheibenabstand - d Scheibendicke - J Trägheitsmoment - p Druck - R Scheibenradius - T Temperatur - T_kr kritische Temperatur - T_N normierte Temperatur, T_N=T/T_kr - t Zeit - Zeitverhältnis; =t/t_O - logarithmisches Dekrement - dynamische Viskosität - kinematische Viskosität - Kreisfrequenz - Dichte - _kr kritische Dichte - _N normierte Dichte; _N= /_kr Index 0 Vakuum Der Verfasser dankt Herrn Prof. Dr.-Ing. K. Stephan für die grozügige Förderung dieser Arbeit.     

STEADY STRATIFIED CIRCULATION INDUCED BY CONSTANT COASTAL WIND

SUMMARY

The nearshore zone is the area where the action of waves and currents on the sea bed is most intense, and where thermal stratification plays an important role in the transport of momentum, heat and pollutants. Therefore, in recent years this topic has drawn the attention of various investigations using analytical, experimental or numerical methods. The coastal water surface can be thought to be agitated by the uniform wind shear stress under flow field with thermal stratified density if the on-shore and off-shore wind blowing over the coastal water surface is assumed to be uniform. The physical properties that enter into the governing equations, which both describe the problem and are also essential to the solution of the phenomena of interest, are the eddy viscosity and the eddy diffusivity. The nonlinear governing equation is simplified in this present work by using the Boussinesq Approximation. A quadratic upwind finite volume method employing contravariant velocities as dependent variables is applied to the simulation of a three-dimensional steady state laminar incompressible flow. The flux terms in the basic equations were discretized via the QUICK ¹ Leonard , B. P. ( 1979 ) “ A stable and accurate convective modelling procedure based on quadratic upstream interpolation ”, J. Comput. Meth. Appl. Mech. Engrg. , 19 , 59 – 98 .[Crossref], [Web of Science ®] , [Google Scholar] scheme for internal nodes and power-law scheme for the boundary nodes. The resulting algebraic equations were solved in a decoupled manner by the SIMPLEC ² Patankar , S. V. ( 1980 ) “ Numerical Heat Transfer and Fluid Flow ”, McGrawHill , New York . [Google Scholar] solution algorithm. A numerical method was then utilized in calculating the flow conditions for ratios of shear stress region and depth, and for different obstacles near the shoreline. Two circulation forms would be formed when the distance of the obstacle (Ex. breakwater) from the coast is equal to the depth. The cellular circulation in the lower layer would become gradually weakened with an increasing distance of the obstacle from the coast. If the eddy viscosity and eddy diffusivity are considered, the cellular circulation in the lower layer would also become gradually weakened. The cellular circulation would arise both behind and in front of the coastal structures, if the terrains with sedimentary obstacles are considered. The cellular circulation in the low layer would also tend to gradually disappear; however, the tendency of weakening is more apparent than the former. The effects of the cellular circulation in the lower layer would make it difficult to transport oxygen, heat and pollutants. Finally, the Coriolis force effect would be taken into consideration and therefore, its order of influence could be indicated.     

Der Einfluß variabler Stoffwerte auf natürliche laminare Konvektionsströmungen

Zusammenfassung Es wird zunächst die laminare natürliche Konvektionsströmung in der Nähe eines ebenen Staupunktes und für die senkrechte Platte betrachtet. Die Stoffgesetze werden in der Umgebung des Bezugszustandes T (Umgebungstemperatur) in Taylor-Reihen entwickelt, deren Koeffizienten dimensionslose Stoffkennzahlen — wie die Prandtl-Zahl — sind, die als freie Parameter in die Rechnung eingehen. Wandschubspannung und Wärmeübergang lassen sich für beliebige Stoffgesetze als Potenzreihe eines Parameters universell angeben. Der Entwicklungsparameter ist dabei ein Maß für die Stärke der Wärmeübertragung. Ein Vergleich mit der Methode der Stoffwertverhältnisse ermöglicht die Bestimmung der dort vorkommenden Exponenten für alle Stoffe, ohne daß auf empirische Daten zurückgegriffen werden muß. Aus den Ergebnissen wird dann eine nicht-rationale Näherungsbeziehung für beliebige zylindrische Körper gewonnen.

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The influence of variable fluid properties to free convection laminar flows

First the free convection laminar flow near a plane stagnation point and at the vertical flat plate is investigated. The functions describing the temperature dependence of the fluid properties are expanded as Taylor series at the reference state T (ambient temperature) whose coefficients are dimensionless fluid properties like the Prandtl number, but are not specified for particular fluids. Shear stress and heat flux at the wall are given for arbitrary temperature dependence of the fluid properties as universal power series of a parameter. This perturbation parameter describes the strength of heat transfer. Comparison with the property-ratio method shows how the exponents in that method depend on the fluid properties without any need of empirical information. From these results a non-rational approximation for arbitrary cylindrical bodies is developed.

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Formelzeichen c _a integrierter Reibungsbeiwert, Gl. (64) - c _f Reibungsbeiwert, Gl. (49) - c _p spez. Wärmekapazität bei konstantem Druck - d Transformationsparameter, Gl. (6) - e Exponent bei der Verteilung der Wandtemperatur, Kap. 2 - f( _s ) dimensionslose Stromfunktion, Gl. (7) - f ₀ f(_s) für konstante Stoffwerte - f _1i dimensionslose Stromfunktionen, Gl. (25) i=1,2, 3, 4 - g Erdbeschleunigung - Gr Grashof-Zahl, Kap. 4 - K _a Kombination aus dimensionslosen Stoffwerten, Gl. (24) - k _ij dimensionslose Stoffwerte, Gln. (13) bis (17) i=1, 2; j=,,,c - k _ij dimensionslose Stoffwerte, Gln. (20) bis (23) i=1,2; j=, - L Bezugslänge, Tabelle 1 - L _i Linear-Operatoren, Gln. (37) bis (40) i=1,2,3,4 - m _i Exponenten, Gl. (59), i=1, 2, 3, 4 - Hilfsfunktionen, Gl. (53), i=1,2, 3, 4 - n _i Exponenten, Gl. (60), i=1,2,3,4 - Hilfsfunktionen, Gl. (55), i=1, 2, 3, 4 - N _u Nusselt-Zahl, Kap. 6 - Pr ^*,Pr Prandtl-Zahl, Tabelle 1 - q _w Wärmefluß an der Wand, Gl. (50) - Q _w Gesamt-Wärmefluß an der Wand, Gl. (63) - T absolute Temperatur - u _b Bezugsgeschwindigkeit, Kap. 4 - u, v Geschwindigkeitskomponenten - x, y kartesische Koordinaten - Kontur-Neigungswinkel, Bild 1 - Volumenausdehnungskoeffizient, Gl. (13) - Entwicklungsparameter, Gl. (15) - Viskosität - _s Ähnlichkeitsvariable, Gl. (6) - ( _S ) dimensionslose Temperatur, Tabelle 1 - ₀ (_S) dimensionslose Temperatur bei konstanten Stoffwerten - _1i imensionslose Temperaturen, Gl. (26) i=1,2, 3, 4 - Wärmeleitfähigkeit - kinematische Viskosität - Dichte _W Wandschubspannung - Stromfunktion - _i Exponenten, Gl. (69) i=,,,c_p Indizes c.p. konstante Stoffwerte - L an der StelleL ^* - m mittlerer Wert - W Wand - mgebungszustand     

ITERATIVE PROCESS TO -HEMICONTRACTIVE OPERATOR AND-STRONGLY ACCRETIVE OPERATOR EQUATIONS

ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔＥｂｅａｎａｒｂｉｔｒａｒｙＢａｎａｃｈｓｐａｃｅ ,Ｅ ｂｅｉｔｓｄｕａｌｓｐａｃｅａｎｄ〈ｘ ,ｆ 〉ｂｅｔｈｅｇｅｎｅｒａｌｉｚｅｄｄｕａｌｉｔｙｐａｉｒｉｎｇｂｅｔｗｅｅｎｘ∈Ｅａｎｄｆ ∈Ｅ .ＴｈｅｍａｐｐｉｎｇＪ :Ｅ→ 2 Ｅ ｄｅｆｉｎｅｄｂｙＪ(ｘ) =ｆ ∈Ｅ :〈ｘ ,ｆ 〉 =‖ｆ ‖‖ｘ‖ ,‖ｆ ‖ =‖ｘ‖ｉｓｃａｌｌｅｄｔｈｅｎｏｒｍａｌｉｚｅｄｄｕａｌｉｔｙｍａｐｐｉｎｇ .ＩｆＥ ｉｓｓｔｒｉｃｔｌｙｃｏｎｖｅｘ ,ｔｈｅｎＪｉｓｓｉｎｇｌｅ＿ｖａｌｕｅｄ .Ｗｅｓｈ…