Homoclinic Saddle-Node Bifurcations in Singularly Perturbed Systems

In this paper we study the creation of homoclinic orbits by saddle-node bifurcations. Inspired on similar phenomena appearing in the analysis of so-called localized structures in modulation or amplitude equations, we consider a family of nearly integrable, singularly perturbed three dimensional vector fields with two bifurcation parameters a and b. The O() perturbation destroys a manifold consisting of a family of integrable homoclinic orbits: it breaks open into two manifolds, W ^s() and W ^u(), the stable and unstable manifolds of a slow manifold . Homoclinic orbits to correspond to intersections W ^s()W ^u(); W ^s()W ^u()= for a<a*, a pair of 1-pulse homoclinic orbits emerges as first intersection of W ^s() and W ^u() as a>a*. The bifurcation at a=a* is followed by a sequence of nearby, O( ²(log)²) close, homoclinic saddle-node bifurcations at which pairs of N-pulse homoclinic orbits are created (these orbits make N circuits through the fast field). The second parameter b distinguishes between two significantly different cases: in the cooperating (respectively counteracting) case the averaged effect of the fast field is in the same (respectively opposite) direction as the slow flow on . The structure of W ^s()W ^u() becomes highly complicated in the counteracting case: we show the existence of many new types of sometimes exponentially close homoclinic saddle-node bifurcations. The analysis in this paper is mainly of a geometrical nature.     

Degenerate Lagrange densities involving geometric objects

The Euler-Lagrange equations corresponding to a Lagrange density which is a function of a symmetric affine connection, _{i j} ^h , and its first derivatives together with a symmetric tensor g_{i j}, are investigated. In general, by variation of the _{i j} ^h , these equations will be of second order in _{i j} ^h . Necessary and sufficient conditions for these Euler-Lagrange equations to be of order one and zero in _{i j} ^h are obtained. It is shown that if the g_{i j} may be regarded as independent then the only permissible zero order Euler-Lagrange equations are those which ensure that the _{i j} ^h are precisely the Christoffel symbols of the second kind.     

Über den Einfluß von Querkrümmung und Dichte bei inhomogenen turbulenten Freistrahlen

Zusammenfassung Zur Berechnung turbulenter Strömungen wird das k--Modell im Ansatz für die turbulente Scheinzähigkeit erweitert, so daß es den Querkrümmungs- und Dichteeinfluß auf den turbulenten Transportaustausch erfaßt. Die dabei zu bestimmenden Konstanten werden derart festgelegt, daß die bestmögliche Übereinstimmung zwischen Berechnung und Messung erzielt wird. Die numerische Integration der Grenzschichtgleichungen erfolgt unter Verwendung einer Transformation mit dem Differenzenverfahren vom Hermiteschen Typ. Das erweiterte Modell wird auf rotationssymmetrische Freistrahlen veränderlicher Dichte angewendet und zeigt Übereinstimmung zwischen Rechnung und Experiment.

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On the influence of transvers-curvature and density in inhomogeneous turbulent free jets

The prediction of turbulent flows based on the k- model is extended to include the influence of transverse-curvature and density on the turbulent transport mechanisms. The empirical constants involved are adjusted such that the best agreement between predictions and experimental results is obtained. Using a transformation the boundary layer equations are solved numerically by means of a finite difference method of Hermitian type. The extended model is applied to predict the axisymmetric jet with variable density. The results of the calculations are in agreement with measurements.

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Bezeichnungen Wirbelabsorptionskoeffizient - c_i Massenkonzentration der Komponente i - c_D, c_L, c, c₁, c₂ Konstanten des Turbulenzmodells - d Düsendurchmesser - E bezogene Dissipationsrate - f bezogene Stromfunktion - f Korrekturfunktion für die turbulente Scheinzähigkeit - j turbulenter Diffusionsstrom - k Turbulenzenergie - k_i Schrittweite in -Richtung - K dimensionslose Turbulenzenergie - L turbulentes Längenmaß - M_i Molmasse der Komponente i - p Druck - allgemeine Gaskonstante - r Querkoordinate - r_0,5 Halbwertsbreite der Geschwindigkeit - r_0,5c Halbwertsbreite der Konzentration - T Temperatur - u Geschwindigkeitskomponente in x-Richtung - v Geschwindigkeitskomponente in r-Richtung - x Längskoordinate - y allgemeine Funktion - Y_i diskreter Wert der Funktion y - Relaxationsfaktor für Iteration - turbulente Dissipationsrate - transformierte r-Koordinate - kinematische Zähigkeit - Exponent - transformierte x-Koordinate - Dichte - _k, Konstanten des Turbulenzmodells - Schubspannung - allgemeine Variable - Stromfunktion - Turbulente Transportgröße Indizes 0 Strahlanfang - m auf der Achse - r mit Berücksichtigung der Krümmung - t turbulent - mit Berücksichtigung der Dichte - im Unendlichen - Schwankungswert oder Ableitung einer Funktion - – Mittelwert Herrn Professor Dr.-Ing. R. Günther zum 70. Geburtstag gewidmet     

The extension of Beltrami's ellipsoid to anisotropic bodies

Summary The spectral decomposition of the compliance, stiffness, and failure tensors for transversely isotropic materials was studied and their characteristic values were calculated using the components of these fourth-rank tensors in a Cartesian frame defining the principal material directions. The spectrally decomposed compliance and stiffness or failure tensors for a transversely isotropic body (fiber-reinforced composite), and the eigenvalues derived from them define in a simple and efficient way the respective elastic eigenstates of the loading of the material. It has been shown that, for the general orthotropic or transversely isotropic body, these eigenstates consist of two double components, ₁ and ₂ which are shears (₂ being a simple shear and ₁, a superposition of simple and pure shears), and that they are associated with distortional components of energy. The remaining two eigenstates, with stress components ₃, and ₄, are the orthogonal supplements to the shear subspace of ₁ and ₂ and consist of an equilateral stress in the plane of isotropy, on which is superimposed a prescribed tension or compression along the symmetry axis of the material. The relationship between these superimposed loading modes is governed by another eigenquantity, the eigenangle .The spectral type of decomposition of the elastic stiffness or compliance tensors in elementary fourth-rank tensors thus serves as a means for the energy-orthogonal decomposition of the energy function. The advantage of this type of decomposition is that the elementary idempotent tensors to which the fourth-rank tensors are decomposed have the interesting property of defining energy-orthogonal stress states. That is, the stress-idempotent tensors are mutually orthogonal and at the same time collinear with their respective strain tensors, and therefore correspond to energy-orthogonal stress states, which are therefore independent of each other. Since the failure tensor is the limiting case for the respective _x, which are eigenstates of the compliance tensor S, this tensor also possesses the same remarkable property.An interesting geometric interpretation arises for the energy-orthogonal stress states if we consider the projections of _x in the principal₃D stress space. Then, the characteristic state ₂ vanishes, whereas stress states ₁, ₃ and ₄ are represented by three mutually orthogonal vectors, oriented as follows: The ₃ and ₄ lie on the principal diagonal plane (₃₁₂) with subtending angles equaling (–/2) and (-), respectively. On the positive principal ₃-axis, is the eigenangle of the orthotropic material, whereas the ₁-vector is normal to the (₃₁₂)-plane and lies on the deviatoric -plane. Vector ₂ is equal to zero.It was additionally conclusively proved that the four eigenvalues of the compliance, stiffness, and failure tensors for a transversely isotropic body, together with value of the eigenangle , constitute the five necessary and simplest parameters with which invariantly to describe either the elastic or the failure behavior of the body. The expressions for the _x-vector thus established represent an ellipsoid centered at the origin of the Cartesian frame, whose principal axes are the directions of the ₁-, ₃- and ₄-vectors. This ellipsoid is a generalization of the Beltrami ellipsoid for isotropic materials.Furthermore, in combination with extensive experimental evidence, this theory indicates that the eigenangle alone monoparametrically characterizes the degree of anisotropy for each transversely isotropic material. Thus, while the angle for isotropic materials is always equal to _i = 125.26° and constitutes a minimum, the angle || progressively increases within the interval 90–180° as the anisotropy of the material is increased. The anisotropy of the various materials, exemplified by their ratiosE _L/2G_L of the longitudinal elastic modulus to the double of the longitudinal shear modulus, increases rapidly tending asymptotically to very high values as the angle approaches its limits of 90 or 180°.     

Response of an elastic Bingham fluid to oscillatory shear

The response of an elastic Bingham fluid to oscillatory strain has been modeled and compared with experiments on an oil-in-water emulsion. The newly developed model includes elastic solid deformation below the yield stress (or strain), and Newtonian flow above the yield stress. In sinusoidal oscillatory deformations at low strain amplitudes the stress response is sinusoidal and in phase with the strain. At large strain amplitudes, above the yield stress, the stress response is non-linear and is out of phase with strain because of the storage and release of elastic recoverable strain. In oscillatory deformation between parallel disks the non-uniform strain in the radial direction causes the location of the yield surface to move in-and-out during each oscillation. The radial location of the yield surface is calculated and the resulting torque on the stationary disk is determined. Torque waveforms are calculated for various strains and frequencies and compared to experiments on a model oil-in-water emulsion. Model parameters are evaluated independently: the elastic modulus of the emulsion is determined from data at low strains, the yield strain is determined from the phase shift between torque and strain, and the Bingham viscosity is determined from the frequency dependence of the torque at high strains. Using these parameters the torque waveforms are predicted quantitatively for all strains and frequencies. In accord with the model predictions the phase shift is found to depend on strain but to be independent of frequency.Notation A plate strain amplitude (parallel plates) - A _R plate strain amplitude at disk edge (parallel disks) - G elastic modulus - m torque (parallel disks) - M normalized torque (parallel disks) = 2m/R ³₀ - N ratio of viscous to elastic stresses (parallel plates) =µ A/ ₀ ratio of viscous to elastic stresses (parallel disks) =µ A _R/₀ - r normalized radial position (parallel disks) =r/R - r radial position (parallel disks) - R disk radius (parallel disks) - t normalized time = t — /2 - t time - _E elastic strain - _P plate strain (displacement of top plate or disk divided by distance between plates or disks) - _PR plate strain at disk edge (parallel disks) - ₀ yield strain - _E normalized elastic strain = _E/₀ - _P normalized plate strain = _P/₀ - _PR normalized plate strain at disk edge (parallel disks) = _PR/₀ - ₀ normalized plate strain amplitude (parallel plates) =A/ ₀ — normalized plate strain amplitude at disk edge (parallel disks) =A _R/₀ - phase shift between _P andT (parallel plates) — phase shift between _PR andM (parallel disks) - µ Bingham viscosity - stress - ₀ yield stress - T normalized stress =/ ₀ - frequency     

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     

Determination of permeability tensors for two-phase flow in homogeneous porous media: Theory

In this paper we continue previous studies of the closure problem for two-phase flow in homogeneous porous media, and we show how the closure problem can be transformed to a pair of Stokes-like boundary-value problems in terms of pressures that have units of length and velocities that have units of length squared. These are essentially geometrical boundary value problems that are used to calculate the four permeability tensors that appear in the volume averaged Stokes' equations. To determine the geometry associated with the closure problem, one needs to solve the physical problem; however, the closure problem can be solved using the same algorithm used to solve the physical problem, thus the entire procedure can be accomplished with a single numerical code.Nomenclature a a vector that maps V onto , m^-1. - A a tensor that maps V onto . - A area of the - interface contained within the macroscopic region, m². - A area of the -phase entrances and exits contained within the macroscopic region, m². - A area of the - interface contained within the averaging volume, m². - A area of the -phase entrances and exits contained within the averaging volume, m². - Bo Bond number (= (=(–)g²/). - Ca capillary number (= v/). - g gravitational acceleration, m/s². - H mean curvature, m^-1. - I unit tensor. - permeability tensor for the -phase, m². - viscous drag tensor that maps V onto V. - ^* dominant permeability tensor that maps onto v , m². - ^* coupling permeability tensor that maps onto v , m². - characteristic length scale for the -phase, m. - l characteristic length scale representing both and , m. - L characteristic length scale for volume averaged quantities, m. - n unit normal vector directed from the -phase toward the -phase. - n unit normal vector representing both n and n . - n unit normal vector representing both n and n . - P pressure in the -phase, N/m². - p superficial average pressure in the -phase, N/m². - p intrinsic average pressure in the -phase, N/m². - p –p , spatial deviation pressure for the -phase, N/m². - r ₀ radius of the averaging volume, m. - r position vector, m. - t time, s. - v fluid velocity in the -phase, m/s. - v superficial average velocity in the -phase, m/s. - v intrinsic average velocity in the -phase, m/s. - v –v , spatial deviation velocity in the -phase, m/s. - V volume of the -phase contained within the averaging volmue, m³. - averaging volume, m³. Greek Symbols V /, volume fraction of the -phase. - viscosity of the -phase, Ns/m². - density of the -phase, kg/m³. - surface tension, N/m. - (v +v ^T ), viscous stress tensor for the -phase, N/m².     

Turbulent wall pressure fluctuations over wavy surfaces

Measurements of the spectral characteristics of the wall pressure fluctuations produced by a turbulent boundary layer flow over solid sinusoidal surfaces of moderate wave amplitude to wave-length ratios have been obtained. The wave amplitudes were sufficiently small so that the flow remained attatched. The results show that the root mean square pressure level reaches a maximum on the adverse pressure gradient side of the wave at a position somewhat before the trough. Spectral analysis of the pressure fluctuations in narrow frequency bands reveals considerable differences in low and high frequency behavior. At low frequencies, the peak fluctuation amplitude was found at the trough whereas at high frequencies, the peak occurs just after the crest and a minimum is found at the trough. Pressure fluctuations having streamwise correlation lengths on the order of or larger than the wavelength of the surface do not return to their equilibrium (crest) amplitudes as they travel the length of a wave. Pressure fluctuations having streamwise correlation lengths about one order of magnitude less than a wavelength return exactly to their equilibrium amplitudes. Two-point correlation measurements show a decrease in longitudinal coherence on the adverse pressure gradient side of the wave at low frequencies and a considerable increase over a broad frequency range on the positive pressure gradient side. No change is found in the lateral coherence.List of symbols C _f skin friction coefficient - C _p pressure coefficient - C _n Fourier amplitudes of the pressure coefficient - C _dp pressure drag coefficient - d pinhole diameter - f frequency - h half the crest to trough distance - h ₊ nondimensional wave amplitude = - k _n wavenumber = - k fundamental wavenumber = - l _p pressure correlation length - p _s mean surface pressure - P ambient pressure - p fluctuating pressure - p ² mean square pressure - q dynamic head = 1/2 U ² - R space-time correlation - P Reynolds number based on wavelength = - R Reynolds number based on momentum thickness = - t time - R free stream velocity - U mean streamwise velocity - U _e streamwise velocity at the edge of the boundary layer - u ^* friction velocity = - x streamwise coordinate - y wall-normal coordinate - z spanwise coordinate - ⁺ non-dimensional wavelength = ^*) - phase of the cross-spectral density - ^* boundary layer displacement thickness - _long longitudinal coherency - _lat lateral coherency - wavelength of wavy surface - v kinematic viscosity - radian frequency = 2 f - spectral or cross-spectral density - _n phase of the Fourier series - density - time delay - _w wall shear stress - boundary layer momentum thickness     

Ein Rechenmodell für reagierende turbulente Scherströmungen im chemischen Nichtgleichgewicht

Zusammenfassung Zur Berechnung turbulenter Strömungen mit chemischen Reaktionen wird ein Schlie\ungsmodell 2. Ordnung vorgeschlagen, das auch die Berücksichtigung von chemischem Nichtgleichgewicht erlaubt. Es besteht aus dem k- Modell zur Schlie\ung der gemittelten Impulsgleichungen, einem thermodynamischen Modell zur Schlie\ung der Zustandsgieichungen und der Energiegleichung und einem Mischungsmodell, das den Grad der Vermischung der Komponenten beschreibt und damit die Schlie\ung der gemittelten Stofferhaltungsgleichungen erlaubt.Für die Behandlung der gemittelten reaktionskinetischen Quellterme der Stofferhaltungsgleichungen wird eine Modifikation des Reihenansatzes von Borghi [7] vorgeschlagen, der die AnnÄherung an den Gleichgewichtszustand besser beschreibt. Das Modell wird auf die von Batt [11, 12] vermessene ebene Scherströmung angewendet und zeigt gute übereinstimmung zwischen Rechnung und Experiment.

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A method for predicting reacting turbulent shear flows with chemical non-equilibriums

A prediction model based on second order closure for the calculation of reacting turbulent flows including chemical non-equilibrium is put forward. It consists of the k- model for the closure of the mean momentum equations, the thermodynamic model for the closure of the mean equations of state and the mean energy-equation and the mixing model that describes the degree of mixedness of the components and consequently leads to the closure of the mean mass transport equations. A modification of the series truncation method of Borghi [7] is suggested that improves the representation of the mean chemical source terms as equilibrium is approached. The results of the calculations are compared with the measurements of Batt [11, 12] in a turbulent plane shear layer with and without reaction and show good agreement.

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Bezeichnungen C_i,j Turbulenzkonstante - D Divergenz der Geschwindigkeit (m/s) - D_b, D_f Vorexponentielle Faktoren im Arrheniusansatz (m³/kmol) - E_b, E_f Exponenten im Arrheniusansatz - F_b, F_f Aktivierungsenergie (K) - H_i Bezugsenthalpie der i-ten Komponente (kJ/kg) - K_i,j Turbulenzkonstante - KON Anzahl der im System vorkommenden Komponenten - M_i Molmasse (kg/kmol) - NR Anzahl der im System vorkommenden Elementargleichungen - Q_ij Konzentrationskorrelation zwischen der i-ten und der j-ten Komponente (kg²/m⁶) - R Restglied der Taylorentwicklung für die Geschwindigkeitskoeffizienten - T Temperatur (K) - a_b, a_f,b_b, b_f Koeffizienten der Taylorreihenentwicklung der Geschwindigkeitskoeffizienten - c_i Massenkonzentration der i-ten Komponente - c₁ Turbulenzkonstante der -Gleichung - c₂ Turbulenzkonstante der -Gleichung - c₃ Turbulenzkonstante der -Gleichung - c_pi spezifische WÄrmekapazitÄt der i-ten Komponente (kJ/kg/K) - h_i spezifische Enthalpie des Gesamtsystems (kJ/kg/K) - k Turbulenzenergie (m²/s²) - k_fj, k_bj Geschwindigkeit der j-ten Elementarreaktion (m³/kmol) - p Druck (N/m²) - v, (u,v,w) Geschwindigkeit (m/s) - x, (x,y,z) Raumkoordinate (m) - Molekularer Diffusionskoeffizient (m²/s) - Allgemeine Gaskonstante (kJ/kmol K) - w_i Quellterm der Konzentrationserhaltungsgleichungen (kg/m³/s) - _ijij stöchiometrische Koeffizienten deriten Komponente in der j-ten Elementarreaktion (VorwÄrtsreaktion ,RückwÄrtsreaktion) - _ij Kronecker-Symbol - Turbulente Dissipation (m²/s³) - Molekulare WÄrmeleitfÄhigkeit (KJ/m/s/K) - Dynamische ZÄhigkeit (kg/m/s) - _t Turbulente dynamische ZÄhigkeit(kg/ m/s) - kinematische ZÄhigkeit (m²/s) - _t Turbulente kinematische ZÄhigkeit (m²/s) - Turbulente Prandtlzahl Mittelwert und Schwankungsgrö\en Unbewichteter Mittelwert - · Unbewichtete Schwankungsgrö\e - Bewichteter (Favre-) Mittelwert - · Bewichtete (Favre-)Schwankungsgrö\e Indizes i,j Komponentenindex - , Summations-oder Vektorindex (,=1,2,3) - b RückwÄrtsreaktion - f VorwÄrtsreaktion     

A light transmittance probe for interfacial area measurements of dispersed two phase flows with small particles

An optical probe measuring interfacial area () by light attenuation has been designed with a special emphasis on flows with sub-millimetric particles. It permits measurements in liquid-liquid or gas-liquid dispersions without need of introducing empirical correcting factors for the standard exponential decay law of light intensity while keeping an extended application range. This probe was successfully tested with an air-glass particle flow, the parameters of which were carefully determined basically by hold-up methods. The volume fraction of the dispersed phase was varied between 0.05% and 5%, and the particle size between 10 m and 300 m.List of symbols D diameter of spherical particle - D _S Sauter diameter - E ₀ irradiance on a surface perpendicular to light propagation 226E;=(1/l) averaged density function along y axis - f density function of a dispersion - f ₁, f ₂ focal length of the lenses L ₁, L ₂ - g granulometry function of a powder (probability density) - h granulometry function of a powder (unnormalized) - I ₀, I light beam intensity respectively before and after passing through the dispersion - j volumetric powder flow - K ₁, K ₂, K ₃ dimensionless constants - l optical path length of the beam in the dispersion - L experimental pipe width along x axis - m mass of a sample - n optical index of the continuous phase - p _a, p ₀ respectively slope of _a and ₀ straight line - r distance between particles - S _d scattering cross-section - V volume of dispersion - averaged particles velocity - x, y, z spatial coordinates - interfacial area - _a absolute interfacial area (by unit volume of dispersion) - ₀ interfacial area measured by light attenuation method - _d angle (around the initial direction of light propagation) within which a particle diffracts - _dr detector aperture angle - light wavelength - _d scattering cross section by unit volume of dispersion - light beam diameter - ₁, ₂ L₁, and L₂ lenses diameters - local volumetric fraction of dispersed phase - averaged fraction of dispersed phase along x axis - ₂ averaged fraction of dispersed phase along x and y axis - volumetric mass of particles     

Time discretization schemes for the Stefan problem in a concentrated capacity

We consider two different time discretization algorithms for a nonlinear parabolic PDE arising in heat conduction phenomena with phase changes in two adjoining bodies and , where can be considered as the boundary of . Stability, convergence and error estimate results are given for both algorithms.

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Sommario Si studiano due algoritmi di discretizzazione nel tempo di un sistema di equazioni a derivate parziali non lineari paraboliche che governa la conduzione del calore, in presenza di cambiamento di fase, in due corpi congiunti e , di cui possa essere considerato come la frontiera di , Vengono dati risultati di stabilità, convergenza e maggiorazione dell'errore per entrambi gli algoritmi.

     

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).     

Time-resolved, 3D, laser-induced fluorescence measurements of fine-structure passive scalar mixing in a tubular reactor

A three-dimensional, time-resolved, laser-induced fluorescence (3D-LIF) technique was developed to measure the turbulent (liquid-liquid) mixing of a conserved passive scalar in the wake of an injector inserted perpendicularly into a tubular reactor with Re=4,000. In this technique, a horizontal laser sheet was traversed in its normal direction through the measurement section. Three-dimensional scalar fields were reconstructed from the 2D images captured at consecutive, closely spaced levels by means of a high-speed CCD camera. The ultimate goal of the measurements was to assess the downstream development of the 3D scalar fields (in terms of the full scalar gradient vector field and its associated scalar energy dissipation rate) in an industrial flow with significant advection velocity. As a result of this advection velocity, the measured 3D scalar field is artificially skewed during a scan period. A method to correct for this skewing was developed, tested and applied. Analysis of the results show consistent physical behaviour.List of symbols  A  Deformation tensor - D_t, D_f  Reactor and injector diameter - L_x, L_y, L_z  Dimensions of the 3D-LIF measurement volume - N_x, N_y, N_z  Number of data samples per measurement volume - Re_m  Reynolds number based on mean velocity - Sc  Schmidt number - f  Focal length - f_c,lens, f_c,array  Cut-off frequency for camera lens and sensor array - f, f  Marginal probability density function for and - f  Joint probability density function of and -  Temporal separation of the 2D data planes -  Temporal resolution of the measurement volume -  Spatial resolution of the measurement volume - ,  Deformation angle and deformation, where =tan -  Fluid energy dissipation rate - ,  Strain limited vorticity and scalar diffusion layers -  Scalar concentration - , _B Kolmogorov and Batchelor length scale - ,  Spherical angles of the scalar gradient vector, -  Kinematic viscosity - _e^–2 Half-thickness (1/e²) of the laser sheet - , _a Kolmogorov and Kolmogorov advection time scales -  Scalar energy dissipation rate -  Scalar diffusivity - 2D, 3D Two- and three-dimensional - DNS Direct numerical simulation - LIF Laser-induced fluorescence - SED Scalar energy dissipation rate - TR Tubular reactor

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Flow in porous media III: Deformable media

Stokes flow in a deformable medium is considered in terms of an isotropic, linearly elastic solid matrix. The analysis is restricted to steady forms of the momentum equations and small deformation of the solid phase. Darcy's law can be used to determine the motion of the fluid phase; however, the determination of the Darcy's law permeability tensor represents part of the closure problem in which the position of the fluid-solid interface must be determined.Roman Letters A interfacial area of the- interface contained within the macroscopic system, m² - A interfacial area of the- interface contained within the averaging volume, m² - A _e area of entrances and exits for the-phase contained within the macroscopic system, m² - A ^* interfacial area of the- interface contained within a unit cell, m² - A _e ^* area of entrances and exits for the-phase contained within a unit cell, m² - E Young's modulus for the-phase, N/m² - e i unit base vectors (i = 1, 2, 3) - g gravity vector, m²/s - H height of elastic, porous bed, m - k unit base vector (=e ₃) - characteristic length scale for the-phase, m - L characteristic length scale for volume-averaged quantities, m - n unit normal vector pointing from the-phase toward the-phase (n = -n ) - p pressure in the-phase, N/m² - P p – g·r, N/m² - r ₀ radius of the averaging volume, m - r position vector, m - t time, s - T total stress tensor in the-phase, N/m² - T ⁰ hydrostatic stress tensor for the-phase, N/m² - u displacement vector for the-phase, m - V averaging volume, m₃ - V volume of the-phase contained within the averaging volume, m³ - v velocity vector for the-phase, m/s Greek Letters V /V, volume fraction of the-phase - mass density of the-phase, kg/m³ - shear coefficient of viscosity for the-phase, Nt/m² - first Lamé coefficient for the-phase, N/m² - second Lamé coefficient for the-phase, N/m² - bulk coefficient of viscosity for the-phase, Nt/m² - T –T ⁰ , a deviatoric stress tensor for the-phase, N/m²     

Strömung und Wärmeübergang bei der Oberflächenverdampfung und Filmkondensation

Zusammenfassung Mit Hilfe der Mischungswegtheorie wurden Gleichungen zur Berechnung der Geschwindigkeitsprofile und des Druckabfalles bei der turbulenten, abwärtsterichteten Gas/Film-Strömung aufgestellt. Zur Berechnung des Wärmeübergangs wurde die turbulente Temperaturleitfähigkeit aus einem halbempirischen Ansatz bestimmt. Es konnte eine befriedigende Übereinstimmung zwischen den berechneten und gemessenen Nußelt-Zahlen bei der Oberflächenverdampfung erzielt werden. Zur Auslegung von Fallstromverdampfern wurde ein Computerprogramm erstellt. Damit lassen sich Einflußgrößen wie Wandtemperatur, Filmdicke, Verdampfungsrate usw. in Abhängigkeit von der Lauflänge bestimmen.

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Flow and heat transfer in surface evaporation and film condensation

Using the mixing length model, equations were established to calculate the velocity profiles and pressure drop in turbulent downward directed gas/film flow. The thermal diffusivity needed for the calculation of heat transfer was determined from a semiempirical model. The calculated Nußelt-numbers agreed very well with experiments. For the design of falling-film evaporators, a computer program was developed, which enables to evaluate wall temperature, film thickness, evaporation rate etc. as a function of flow-path length.

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Formelzeichen a Temperaturleitfähigkeit - c spez. Wärmekapazität - d Durchmesser - f_m bezogene mittlere turbulente Temperaturleitfähigkeit - Fi /(3²/g)^1/3) Filmkennzahl - Fr Froude-Zahl - g Fallbeschleunigung - Ka ³/g⁴ Kapitza-Zahl - L Rohrlänge - l Mischungsweg - m Massenstrom - Nu (²/g)^1/3/ Nußelt-Zahl - Nu / Nußelt-Zahl des Filmes - p Druck - Pr /a Prandtl-Zahl - q Wärmestromdichte - R Radius - Re Reynolds-Zahl - Re_ü Übergangs-Reynolds-Zahl - Re_w Schubspannungs-Reynolds-Zahl der Flüssigkeit - r radiale Koordinate - T Temperatur - u Geschwindigkeit - u_w Schubspannungsgeschwindigkeit der Flüssigkeit - u Grenzflächengeschwindigkeit - u_T Schubspannungsgeschwindigkeit des Gases - y Wandabstand - y^* y/ dimensionsloser Wandabstand - z axiale Koordinate Griechische Zeichen Wärmeübergangskoeffizient - Filmdicke - dyn. Viskosität - dimensionslose Temperatur - Wärmeleitfähigkeit - kin. Viskosität - Dichte - Oberflächenspannung - Schubspannung Zusatzzeichen und Indizes G Gas - K Kondensation - s Sättigung - t turbulent - w Wand - wi Welleninstabilität - Phasengrenze - - mittlere Größe     

Heat conduction in a cylinder crossed by an electric current with skin effect

The steady periodic temperature distribution in an infinitely long solid cylinder crossed by an alternating current is evaluated. First, the time dependent and non-uniform power generated per unit volume by Joule effect within the cylinder is determined. Then, the dimensionless temperature distribution is obtained by analytical methods in steady periodic regime. Dimensionless tables which yield the amplitude and the phase of temperature oscillations both on the axis and on the surface of copper or nichrome cylindrical electric resistors are presented.

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Wärmeleitung in einem stromdurchflossenen Zylinder unter Berücksichtigung des Skin-Effektes

Zusammenfassung Es wird die periodische Temperaturverteilung für den eingeschwungenen Zustand in einem unendlich langen, von Wechselstrom durchflossenen Vollzylinder ermittelt. Zuerst erfolgt die Bestimmung der zeitabhängigen, nichgleichförmigen Energiefreisetzung pro Volumeneinheit des Zylinders infolge Joulescher Wärmeentwicklung und anschließend die Ermittlung der quasistationären Temperaturverteilung auf analytischem Wege. Amplitude und Phasenverzögerung der Temperaturschwingungen werden für die Achse und die Oberfläche eines Kupfer- oder Nickelchromzylinders tabellarisch in dimensionsloser Form mitgeteilt.

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Nomenclature A integration constant introduced in Eq. (2) - ber, bei Thomson functions of order zero - Bi Biot numberhr ₀/ - c speed of light in empty space - c ₁,c ₂ integration constants introduced in Eq. (46) - c _p specific heat at constant pressure - E electric field - E _z component ofE alongz - E time independent part ofE, defined in Eq. (1) - f function ofs and defined in Eq. (11) - g function ofs and defined in Eq. (37) - h convection heat transfer coefficient - H magnetic field - i imaginary uniti=(–1)^1/2 - I electric current - I _eff effective electric currentI _eff=I/2^1/2 - Im imaginary part of a complex number - J _n Bessel function of first kind and ordern - J electric current density - q _g power generated per unit volume - time average of the power generated per unit volume - time averaged power per unit length - r radial coordinate - R electric resistance per unit length - r ₀ radius of the cylinder - Re real part of a complex number - s dimensionless radial coordinates=r/r ₀ - s, s integration variables - t time - T temperature - time averaged temperature - T _f fluid temperature outside the boundary layer - time average of the surface temperature of the cylinder - u, functions ofs, and defined in Eqs. (47) and (48) - W Wronskian - x position vector - x real variable - Y _n Bessel function of second kind and ordern - z unit vector parallel to the axis of the cylinder - z axial coordinate - · modulus of a complex number - equal by definition Greek symbols amplitude of the dimensionless temperature oscillations - electric permittivity - dimensionless temperature defined in Eq. (16) - ₀, ₁, ₂ functions ofs defined in Eq. (22) - thermal conductivity - dimensionless parameter=(2)^1/2 - magnetic permeability - ₀ magnetic permeability of free space - function of defined in Eq. (59) - dimensionless parameter=c _p/() - mass density - electric conductivity - dimensionless time=t - phase of the dimensionless temperature oscillations - function ofs:= ₁+i ₂ - angular frequency - dimensionless parameter=()^1/2 r ₀     

Approximate solutions for drag coefficient of bubbles moving in shear-thinning elastic fluids

The drag coefficient for bubbles with mobile or immobile interface rising in shear-thinning elastic fluids described by an Ellis or a Carreau model is discussed. Approximate solutions based on linearization of the equations of motion are presented for the highly elastic region of flow. These solutions are in reasonably good agreement with the theoretical predictions based on variational principles and with published experimental data. C _D Drag coefficient - E ^* Differential operator [E ^{* 2} = ²/² + (sin/ ²)/(1/sin /)] - El Ellis number - F _D Drag force - K Consistency index in the power-law model for non-Newtonian fluid - n Flow behaviour index in the Carreau and power-law models - P Dimensionless pressure [=(p – p ₀)/₀ (U /R)] - p Pressure - R Bubble radius - Re ₀ Reynolds number [= 2R U /₀] - Re Reynolds number defined for the power-law fluid [= (2R) ⁿ U ^2–n /K] - r Spherical coordinate - t Time - U Terminal velocity of a bubble - u Velocity - Wi Weissenberg number - Ellis model parameter - Rate of deformation - Apparent viscosity - ₀ Zero shear rate viscosity - Infinite shear rate viscosity - Spherical coordinate - Parameter in the Carreau model - ^* Dimensionless time [=/(U /R)] - Dimensionless length [=r/R] - Second invariant of rate of deformation tensors - ^* Dimensionless second invariant of rate of deformation tensors [=/(U /R)²] - Second invariant of stress tensors - ^* Dimensionless second invariant of second invariant of stress tensor [= / ₀ ² (U /R)²] - Fluid density - Shear stress - ^* Dimensionless shear stress [=/ ₀ (U /R)] - _1/2 Ellis model parameter - ₁ ^2/* Dimensionless Ellis model parameter [= _1/2/ ₀(U /R)] - Stream function - ^* Dimensionless stream function [=/U R ²]     

Material functions generated by the complex viscosity

Based on the complex viscosity model various steady-state and transient material functions have been completed. The model is investigated in terms of a corotational frame reference. Also, BKZ-type integral constitutive equations have been studied. Some relations between material functions have been derived. C ^–1 Finger tensor - F[], (F ^–1[]) Fourier (inverse) transform - rate of deformation tensor in corotating frame - h(I, II) Wagner's damping function - J (x) Bessel function - m parameter inh (I, II) - m(s) memory function - m _k, n_k integers (powers in complex viscosity model) - P principal value of the integral - parameter in the complex viscosity model - rate of deformation tensor - shear rates - [], [] incomplete gamma function - (a) gamma function - steady-shear viscosity - ^* complex viscosity - , real and imaginary parts of ^* - ₀ zero shear viscosity - ⁺, ₁ ⁺ stress growth functions - ^–, ₁ ^- stress relaxation functions - (s) relaxation modulus - ₁(s) primary normal-stress coefficient - ø(a, b; z) degenerate hypergeometric function - ₁, ₂ time constants (parameters of ^*) - frequency - extra stress tensor     

The virtual origin of riblets in an open channel flow

In this paper, a method using the mean velocity profiles for the buffer layer was developed for the estimation of the virtual origin over a riblets surface in an open channel flow. First, the standardized profiles of the mixing length were estimated from the velocity measurement in the inner layer, and the location of the edge of the viscous layer was obtained. Then, the virtual origins were estimated by the best match between the measured velocity profile and the equations of the velocity profile derived from the mixing length profiles. It was made clear that the virtual origin and the thickness of the viscous layer are the function of the roughness Reynolds number. The drag variation coincided well with other results.Nomenclature f _r skin friction coefficient - f _ro skin friction coefficient in smooth channel at the same flow quantity and the same energy slope - g gravity acceleration - H water depth from virtual origin to water surface - H ⁺ u*H/ - H false water depth from top of riblets to water surface - H ⁺ u*H/ - I _e streamwise energy slope - I _b bed slope - k riblet height - k ⁺ u*k/ - l mixing length - l _s standardized mixing length - Q flow quantity - Re Reynolds number volume flow/unit width/v - s riblet spacing - u mean velocity - u* friction velocity = - u* false friction velocity = - y distance from virtual origin - y distance from top of riblet - y ₀ distance from top of riblet to virtual origin - y _v distance from top of riblet to edge of viscous layer - y ⁺ u*y/ - y ⁺ u*y/ - y ₀ ⁺ u*y ₀/ - u ⁺ u*y/ - shifting coefficient for standardization - thickness of viscous layer=y ₀+y - ⁺ u*/ - ⁺ u*/ - eddy viscosity - ridge angle - v kinematic viscosity - density - shear stress     

Qualitative behavior of dissipative wave equations on bounded domains

The qualitative behavior of solutions of the mixed problem u_tt = u-a(x)u_t in IR x , u=0 on IR x , is studied in the case when a>0 and IRⁿ is bounded. Roughly speaking, if aa_min>0, then solutions decay at least as fast as exp t( –1/2a_min), with the possible exception of a finite dimensional set of smooth solutions whose existence is associated with a phenomenon of overdamping. If a_max is sufficiently small, depending on , then no overdamping occurs.Partially supported by NSF grant NSF GP 34260.This work was partially supported by the National Science Foundation under Grant No. GP 34260