Boundary layer effect on thermal NO decomposition behind incident shock waves

The thermal decomposition of nitric oxide (diluted in Argon) has been measured behind incident shock waves by means of IR diode laser absorption spectroscopy. In two independent runs the diode laser was tuned to the=0 =1² _3/2 R(18.5)-rotational vibrational transition and the=1 =2² _3/2 R(20.5)-rotational vibrational transition of nitric oxide, respectively. These two transitions originating from the vibrational ground state (=0) and the first excited vibrational state (=1) were selected in order to probe the homogeneity along the absorption path. The measured NO decomposition could satisfactorily be described by a chemical reaction mechanism after taking into account boundary layer corrections according to the theory of Mirels. The study forms a further proof of Mirels' theory including his prediction of the laminar-turbulent transition. It also shows, that the inhomogeneities from the boundary layer do not affect the IR linear absorption markedly.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.     

Exact Solutions of the Hydrodynamic Equations Derived from Partially Invariant Solutions

The paper proposes a heuristic approach to constructing exact solutions of the hydrodynamic equations based on the specificity of these equations. A number of systems of hydrodynamic equations possess the following structure: they contain a reduced system of n equations and an additional equation for an extra function w. In this case, the reduced system, in which w = 0, admits a Lie group G. Taking a certain partially invariant solution of the reduced system with respect to this group as a seed:rdquo; solution, we can find a solution of the entire system, in which the functional dependence of the invariant part of the seed solution on the invariants of the group G has the previous form. Implementation of the algorithm proposed is exemplified by constructing new exact solutions of the equations of rotationally symmetric motion of an ideal incompressible liquid and the equations of concentrational convection in a plane boundary layer and thermal convection in a rotating layer of a viscous liquid.     

Optical co-ordinates in general relativity 2 — the problem of distance

Summary Let denote the congruence of null geodesics associated with a given optical observer inV ₄. We prove that determines a unique collection of vector fieldsM₍₎ ( =1, 2, 3) and ₍₀₎ overV ₄, satisfying a weak version of Killing's conditions.This allows a natural interpretation of these fields as the infinitesimal generators of spatial rotations and temporal translation relative to the given observer. We prove also that the definition of the fieldsM₍₎ and ₍₀₎ is mathematically equivalent to the choice of a distinguished affine parameter f along the curves of, playing the role of a retarded distance from the observer.The relation between f and other possible definitions of distance is discussed.

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Sommario Sia la congruenza di geodetiche nulle associata ad un osservatore ottico assegnato nello spazio-tempoV ₄. Dimostriamo che determina un'unica collezione di campi vettorialiM₍₎ ( =1, 2, 3) e ₍₀₎ inV ₄ che soddisfano una versione in forma debole delle equazioni di Killing. Ciò suggerisce una naturale interpretazione di questi campi come generatori infinitesimi di rotazioni spaziali e traslazioni temporali relative all'osservatore assegnato. Dimostriamo anche che la definizione dei campiM₍₎, ₍₀₎ è matematicamente equivalente alla scelta di un parametro affine privilegiato f lungo le curve di, che gioca il ruolo di distanza ritardata dall'osservatore. Successivamente si esaminano i legami tra f ed altre possibili definizioni di distanza in grande.

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Work performed in the sphere of activity of: Gruppo Nazionale per la Fisica Matematica del CNR.     

Temperature measurement near the triple line during phase change using thermochromic liquid crystal thermography

The temperature generated by the evaporation of a volatile liquid in a confined space (tube =1,630 m) was mapped onto the tube surface with the use of unsealed thermochromic liquid crystals (TLCs). The strong evaporative cooling effect located near the meniscus triple line generates a temperature dip. Despite the thermal diffusion through the tubes thickness and its geometry, the TLC thickness and the inherent difficulties of working with unsealed TLCs, the present technique has revealed to be a suitable tool for accurate temperature measurement at the microscale size. The evaporation flux is deduced from the profile of temperature and comparison with the experimental measurement shows a very good agreement. The role of the nature and thickness of the tube wall material on the diffusion of the temperature profile from the inside to the outside is also investigated.     

Correlations between porous medium flow data and turbulent tube flow results for aqueous hydroxypropylguar solutions (Part 2)

Turbulent tube flow and the flow through a porous medium of aqueous hydroxypropylguar (HPG) solutions in concentrations from 100 wppm to 5000 wppm is investigated. Taking the rheological flow curves into account reveals that the effectiveness in turbulent tube flow and the efficiency for the flow through a porous medium both start at the same onset wall shear stress of 1.3 Pa. The similarity of the curves = ( _w ) and = ( _w ), respectively, leads to a simple linear relation / =k, where the constantk or proportionality depends uponc. This offers the possibility to deduce (for turbulent tube flow) from (for flow through a porous medium). In conjunction with rheological data, will reveal whether, and if yes to what extent, drag reduction will take place (even at high concentrations).The relation of our treatment to the model-based Deborah number concept is shown and a scale-up formula for the onset in turbulent tube flow is deduced as well.     

Filament stretching equation in Lagrangian coordinates

The general integral of the very simple equation ^21/n/() was found to describe the cross sectional area of filaments of isothermal power law fluids while in transient stretching where is time and is the initial location of fluid molecules at time = 0 given as the distance from a reference point fixed in space. Any such stretching transient given as a solution of the above equation is physically realizable subject to the restrictions > 0 and/ < 0.     

Transcendentally small transversality in the rapidly forced pendulum

The rapidly forced pendulum equation with forcing sin((t/), where =<₀^p,p = 5, for ₀, sufficiently small, is considered. We prove that stable and unstable manifolds split and that the splitting distanced(t) in the ( ,t) plane satisfiesd(t) = sin(t/) sech(/2) +O( ₀ exp(–/2)) (2.3a) and the angle of transversal intersection,, in thet = 0 section satisfies 2 tan/2 = 2S _s = (/2) sech(/2) +O(( ₀ /) exp(–/2)) (2.3b) It follows that the Melnikov term correctly predicts the exponentially small splitting and angle of transversality. Our method improves a previous result of Holmes, Marsden, and Scheuerle. Our proof is elementary and self-contained, includes a stable manifold theorem, and emphasizes the phase space geometry.     

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     

Berechnung des Temperaturfeldes um eine im Erdboden verlegte Punkt- und Linienquelle bei Randbedingung III. Art an der Erdoberfläche

Zusammenfassung Für den Fall, daß sich in einem halbunendlichen Körper in der Tiefe L eine Punkt- bzw. Linienquelle befindet und daß an der Oberfläche des Körpers ein örtlich und zeitlich konstanter Wärmeübergangskoeffizient herrscht, wird das stationäre Temperaturfeld analytisch berechnet. Beim Vergleich mit einer Näherungslösung (Hilfsschicht) zeigt sich, daß nicht so sehr die Biot-Zahl Bi= · L/ als vielmehr der größte Winkel zwischen Wandnormale und Wärmestromdichte in der Hilfsschicht ein Maß für die Genauigkeit der Näherungslösung ist.

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Calculation of the temperature field around a buried point- and linesource, respectively, when the boundary condition is Newton's law

The steady state temperature field in a semiinfinite body caused by a buried point- and linesource, respectively, has been analytically calculated. The comparison with a simple approach (additional-layer) shows that the greatest angle between the normal of the wall and the heat flux density in the additional-layer, describes the quality of the approach better than the Biot-number Bi=L/ does.

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Formelzeichen A Fläche - Bi Biot-Zahl - C Eigenwertfunktion - E₁ Exponentialintegral - exp Exponentialfunktion - i komplexe Einheit - J₀ Besselfunktion nullter Ordnung und 1. Grades - L Verlegungstiefe der Punkt- bzw. Linienquelle - Q Quellstärke - r Radius - Re Realteil eines Ausdruckes - T Temperatur - t Integrationsvariable - x, y, z Ortskoordinaten - Wärmeübergangskoeffizienten an der Erdoberfläche - Laplace-Operator - Wärmeleitfähigkeit des Erdbodens - dimensionslose Temperatur - Integrationsvariable - dimensionsloser Radius - komplexe Ortskoordination Indizes 0 Erdoberfläche, senkrecht über der Quelle - 1 Lösung für das 1. Randwertproblem - 3 Lösung für das 3. Randwertproblem - 13 Zusatzfunktion - w Erdoberfläche - Umgebungstemperatur - Näherungslösung     

The rheology of polymeric liquid crystals

The molecular theory of Doi has been used as a framework to characterize the rheological behavior of polymeric liquid crystals at the low deformation rates for which it was derived, and an appropriate extension for high deformation rates is presented. The essential physics behind the Doi formulation has, however, been retained in its entirety. The resulting four-parameter equation enables prediction of the shearing behavior at low and high deformation rates, of the stress in extensional flows, of the isotropic-anisotropic phase transition and of the molecular orientation. Extensional data over nearly three decades of elongation rate (10^–2–10¹) and shearing data over six decades of shear rate (10^–2–10⁴) have been correlated using this analysis. Experimental data are presented for both homogeneous and inhomogeneous shearing stress fields. For the latter, a 20-fold range of capillary tube diameters has been employed and no effects of system geometry or the inhomogeneity of the flow-field are observed. Such an independence of the rheological properties from these effects does not occur for low molecular weight liquid crystals and this is, perhaps, the first time this has been reported for polymeric lyotropic liquid crystals; the physical basis for this major difference is discussed briefly. A Semi-empirical constant in eq. (18), N/m² - c rod concentration, rods/m³ - c ^* critical rod concentration at which the isotropic phase becomes unstable, rods/m³ - C interaction potential in the Doi theory defined in eq. (3) - d rod diameter, m - D semi-empirical constant in eq. (19), s^–1 - D _r lumped rotational diffusivity defined in eq. (4), s^–1 - rotational diffusivity of rods in a concentrated (liquid crystalline) system, s^–1 - D _ro rotational diffusivity of a dilute solution of rods, s^–1 - f distribution function defining rod orientation - F tensorial term in the Doi theory defined in eq. (7) (or eq. (19)), s^–1 - G tensorial term in the Doi theory defined in eq. (8) - K _B Boltzmann constant, 1.38 × 10^–23 J/K-molecule - L rod length, m - S scalar order parameter - S tensor order parameter defined in eq. (5) - t time, s - T absolute temperature, K - u unit vector describing the orientation of an individual rod - rate of change ofu due to macroscopic flow, s^–1 - v fluid velocity vector, m/s - v velocity gradient tensor defined in eq. (9), s^–1 - V mean field (aligning) potential defined in eq. (2) - x coordinate direction, m - Kronecker delta (= 0 if = 1 if = ) - _r ratio of viscosity of suspension to that of the solvent at the same shear stress - _s solvent viscosity, Pa · s - ^* viscosity at the critical concentrationc ^*, Pa · s - v ₁, v₂ numerical factors in eqs. (3) and (4), respectively - deviatoric stress tensor, N/m² - volume fraction of rods - ⁰ constant in eq. (16) - ^* volume fraction of rods at the critical concentrationc ^* - average over the distribution functionf(u, t) (= d ²u f(u, t)) - gradient operator - d ²u integral over the surface of the sphere (|u| = 1)     

The mirror relations and nonlinear viscoelasticity of polymer melts

An attempt is made to incorporate into a quasilinear viscoelastic constitutive equation of the Boltzmann superposition type the two mirror relations of Gleissle, as well as his relation between the steady-state first normal-stress difference and the shear viscosity curve. It is shown that the three relations can hold separately within this constitutive model, but not simultaneously, because they require a different nonlinear strain measure, namelyS ₁₂ () = – a ( – 1) (a = 0 for 1,a = 1 for 1) for the mirroring of the viscosities,S ₁₂ () = – a (–k ²/) (a = 0 for k, a = 1 for k) for the mirroring of the first normal-stress coefficients, and for the third relation. Here denotes the shear strain and erf the error function. Experimental data on melts of a low-density polyethylene, a high-density polyethylene and a polypropylene show that the mirror relations are passable approximations, but that the third relation meets reality surprisingly close if the right value ofk is used.     

Flow stability of a Grad-model liquid layer on an inclined plane

A study is presented of the flow of stability of a Grad-model liquid layer [1, 2] flowing over an inclined plane under the influence of the gravity force.It is assumed that at every point of the considered material continuum, along with the conventional velocity vector v, there is defined an angular velocity vector , the internal moment stresses are negligibly small, and in the general case the force stress tensor _kj is asymmetric. The model is characterized by the usual Newtonian viscosity , the Newtonian rolling viscosity _r, and the relaxation time = J/4 _r, where J is a scalar constant of the medium with dimensions of moment of inertia per unit mass, is the density. It is assumed that the medium is incompressible, the coefficients , _r, J are constant [2].The exact solution of the equations of motion, corresponding to flow of a layer with a plane surface, coincides with the solution of the Navier-Stokes equations in the case of flow of a layer of Newtonian fluid. The equations for three-dimensional periodic disturbances differ considerably from the corresponding equations for the problem of the flow stability of a layer of a Newtonian medium. It is shown that the Squire theorem is valid for parallel flows of a Grad liquid.The flow stability of the layer with respect to long-wave disturbances is studied using the method of sequential approximations suggested in [3, 4].     

On natural convection from a vertical plate with a prescribed surface heat flux in porous media

This paper presents a theoretical and numerical investigation of the natural convection boundary-layer along a vertical surface, which is embedded in a porous medium, when the surface heat flux varies as (1 +x ²)), where is a constant andx is the distance along the surface. It is shown that for > -1/2 the solution develops from a similarity solution which is valid for small values ofx to one which is valid for large values ofx. However, when -1/2 no similarity solutions exist for large values ofx and it is found that there are two cases to consider, namely < -1/2 and = -1/2. The wall temperature and the velocity at large distances along the plate are determined for a range of values of .Notation g Gravitational acceleration - k Thermal conductivity of the saturated porous medium - K Permeability of the porous medium - l Typical streamwise length - q _w Uniform heat flux on the wall - Ra Rayleigh number, =gK(q _w /k)l/(v) - T Temperature - Too Temperature far from the plate - u, v Components of seepage velocity in the x and y directions - x, y Cartesian coordinates - Thermal diffusivity of the fluid saturated porous medium - The coefficient of thermal expansion - An undetermined constant - Porosity of the porous medium - Similarity variable, =y(1+x ) ^/3/x ^1/3 - A preassigned constant - Kinematic viscosity - Nondimensional temperature, =(T – T )Ra^1/3 k/qw - Similarity variable, = =y(log_e x)^1/3/x ^2/3 - Similarity variable, =y/x ^2/3 - Stream function     

Nonstationary filtration in partially saturated porous media

From the mathematical formulation of a one-dimensional flow through a partially saturated porous medium, we arrive at a nonlinear free boundary problem, the boundary being between the saturated and the unsaturated regions in the medium. In particular we obtain an equation which is parabolic in the unsaturated part of the domain and elliptic in the saturated part.Existence, uniqueness, a maximum principle and regularity properties are proved for weak solutions of a Cauchy-Dirichlet problem in the cylinder {(x,t): 0x1, t0} and the nature, in particular the regularity, of the free boundary is discussed.Finally, it is shown that solutions of a large class of Cauchy-Dirichlet problems converge towards a stationary solution as t and estimates are given for the rate of convergence.     

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°.     

Temperaturverteilung in tiefen und flachen Seen

Zusammenfassung Die Temperaturverteilung in tiefen und flachen Seen lÄ\t sich unter gewissen Vereinfachungen als instationares, eindimensionales WÄrmeleitproblem behandeln. Konvektion in bestimmten Bereichen des Sees kann durch Mittelung der den instabilen Temperaturprofilen entsprechenden inneren Energie berücksichtigt werden. Ma\gebliche Parameter für den örtlichen und zeitlichen Temperaturverlauf sind der vertikale Absorptionskoeffizient und das VerhÄltnis von WÄrmeaustausch an der OberflÄche einerseits zur sichtbaren Sonnen- und Himmelsstrahlung andererseits. Zeitlich verÄnderlicher Energieaustausch lÄ\t sich durch eine schrittweise Berechnung berücksichtigen. Messungen anderer Autoren an Seen sowie eigene Modellexperimente bestÄtigen die im theoretischen Modell enthaltenen Annahmen.

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Temperature distribution in deep and shallow lakes

For certain simplifications the temperature distribution in deep as well as shallow lakes can be treated as a non-steady, one-dimensional heat conduction problem. Convection in certain regions of the lake can be taken into account by averaging the internal energy corresponding to the calculated unstable profiles. The leading parameters for the spatial and temporal temperature distribution are the vertical absorption coefficient on one hand and the ratio of heat exchange at the surface to visible solar and sky radiation on the other. Transient energy exchange can be considered by stepwise calculation. Field measurements of other authors as well as own model experiments confirm the assumptions made in the theoretical model.

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Formelzeichen

1 Lateinische Buchstaben a Temperaturleitzahl - c spezifische WÄrmekapazitÄt - I IntensitÄt - L Seetiefe - n Brechungsindex - q Energiestromdichte - q_g Gesamtstrahlungsstromdichte an der OberflÄche - q_o sichtbare Strahlungsstromdichte an der OberflÄche - q_v WÄrmestromdichte an der OberflÄche (durch Konvektion, Verdunstung und/oder langwellige Zu- und Abstrahlung) - r Reflexionsver mögen der SeeoberflÄche - s Strahlweg - t Zeit - w volumenbezogene WÄrmequelle - z vertikale Ortskoordinate 2 Griechische Buchstaben Zenitwinkel - Zenitwinkel unter Wasser - Absorptionskoeffizient - dimensionslose vertikale Ortskoordinate - dimensionslose Variable - Temperatur - dimensionslose Temperatur - WellenlÄnge - WÄrmeleitfÄhigkeit (Index eff für turbulente WÄrmeleitfÄhigkeit) - dimensionslose Tiefe - Dichte - EnergiestromdichteverhÄltnis - dimensionslose Zeit - _k dimensionslose Variable 3 Indices e Epilimnion - h Hypolimnion - o SeeoberflÄche - spektrale Grö\e Auszug aus der vom Fachbereich Maschinenwesen der Technischen UniversitÄt München zur Erlangung des akademischen Grades eines Doktor-Ingenieurs genehmigten Dissertation über Temperaturverteilung in Seen des Diplom-Ingenieurs Siegfried Blo\. Berichterstatter Prof. Dr.-Ing. U. Grigull und Prof. Dr. rer. nat. D. Vortmeyer.- Die Dissertation wurde am 27. Mai 1977 bei der Technischen UniversitÄt München eingereicht und durch den Fachbereich Maschinenwesen am 19. Juli 1977 angenommen. Tag der Promotion 26. Juli 1977     

Transport in ordered and disordered porous media IV: Computer generated porous media for three-dimensional systems

In the method of volume averaging, the difference between ordered and disordered porous media appears at two distinct points in the analysis, i.e. in the process of spatial smoothing and in the closure problem. In theclosure problem, the use of spatially periodic boundary conditions isconsistent with ordered porous media and the fields under consideration when the length-scale constraint,r ₀L is satisfied. For disordered porous media, spatially periodic boundary conditions are an approximation in need of further study.In theprocess of spatial smoothing, average quantities must be removed from area and volume integrals in order to extractlocal transport equations fromnonlocal equations. This leads to a series of geometrical integrals that need to be evaluated. In Part II we indicated that these integrals were constants for ordered porous media provided that the weighting function used in the averaging process contained thecellular average. We also indicated that these integrals were constrained by certain order of magnitude estimates for disordered porous media. In this paper we verify these characteristics of the geometrical integrals, and we examine their values for pseudo-periodic and uniformly random systems through the use of computer generated porous media.

Nomenclature

Roman Letters A interfacial area of the- interface associated with the local closure problem, m² - A _e area of entrances and exits for the-phase contained within the averaging system, m² - a _i i=1, 2, 3 gaussian probability distribution used to locate the position of particles - I unit tensor - L general characteristic length for volume averaged quantities, m - L characteristic length for , m - L characteristic length for , m - characteristic length for the -phase particles, m - ⁰ reference characteristic length for the-phase particles, m - characteristic length for the-phase, m - _i i=1, 2, 3 lattice vectors, m - m convolution product weighting function - m _v special convolution product weighting function associated with the traditional volume average - n _i i=1, 2, 3 integers used to locate the position of particles - n unit normal vector pointing from the-phase toward the-phase - n _e outwardly directed unit normal vector at the entrances and exits of the-phase - r _p position vector locating the centroid of a particle, m - r gaussian probability distribution used to determine the size of a particle, m - r ₀ characteristic length of an averaging region, m - r position vector, m - r _m support of the weighting functionm, m - averaging volume, m³ - V volume of the-phase contained in the averaging volume,, m³ - x positional vector locating the centroid of an averaging volume, m - x ₀ reference position vector associated with the centroid of an averaging volume, m - y position vector locating points relative to the centroid, m - y position vector locating points in the-phase relative to the centroid, m Greek Letters indicator function for the-phase - Dirac distribution associated with the- interface - V /V, volume average porosity - /L, small parameter in the method of spatial homogenization - standard deviation ofa _i - _r standard deviation ofr - _r intrinsic phase average of      

Die Temperaturabhängigkeit der Oberflächenspannung von reinen Kältemitteln vom Tripelpunkt bis zum kritischen Punkt

Zusammenfassung Die Oberflächenspannung von sechs reinen Substanzen — SF₆, CCl₃F, CCl₂F₂, CClF₃, CBrF₃ und CHClF₂ — wurde mit Hilfe einer modifizierten Kapillarmethode gemessen. Die zur Berechnung der Oberflächenspannung erforderlichen Sättigungsdichten und wurden teils aus vorhandenen Zustandsgleichungen, teils aus ebenfalls gemessenen Brechungsindizes bestimmt. Die Temperaturabhängigkeit der Oberflächenspannung läßt sich durch einen erweiterten Ansatz nach van der Waals =_O (T_c-T)(1+...) darstellen, wobei bei einfachen Stoffen ein eingliedriger, bei polaren und assoziierenden Stoffen ein zweigliedriger Ansatz notwendig und ausreichend ist. Für den kritischen Exponenten der Oberflächenspannung wurde ein von der molekularen Substanz weitgehend unabhängiger Wert von =1.284±0.005 gefunden.

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Temperature dependence of surface tension of pure refrigerants from triple point up to the critical point

The surface tension of six fluids (SF₆, CCl₃F, CCl₂F₂, CClF₃, CBrF₃, CHClF₂) have been measured by means of a modified capillary rise method. The liquid vapor densities, which are needed to calculate the surface tension, have partly been determined by means of refractive indices simultaneously measured in the same apparatus. The temperature dependence of the surface tension is described by an extended van der Waals power law =_O(T_c-T)(1+...). For simple fluids one term and for polar and associating fluids two terms are necessary and sufficient. The critical exponent is found to be 1.284 ± 0.005 and nearly independent of the molecular structure.

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Formelzeichen a² Laplace-Koeffizient - a Parameter - B_O, B_on Koeffizient der Koexistenzkurve - g Erdbeschleunigung - H Höhe, kapillare Steighöhe - LL Lorentz-Lorenz-Funktion oder Refraktionskonstante - M molare Masse - M Zahl der Meßwerte - N Zahl der unbekannten Parameter - n Brechungsindex - p Druck - R,r Radius - s Entropie - SD Standardabweichung - T, t Temperatur - u innere Energie Griechische Formelzeichen Exponent des Laplace-Koeffizienten - Exponent der Koexistenzkurve - 2. Exponent der Oberflächenspannung - Wellenlänge des Lichts - Exponent der Oberflächenspannung - _D Dipolmoment - , Dichte der Flüssigkeit bzw. des Dampfes - Oberflächenspannung - reduzierte Temperatur (1-T/T_c) - ² gewichtete Varianz Indizes c kritischer Zustand - D Differenz - m Mittelwert - T Isotherme - t Zustand am Tripelpunkt - S Zustand am Schmelzpunkt - bezogen auf Oberfläche     

Effect of measuring volume length on two-component laser velocimeter measurements in a turbulent boundary layer

The effects of finite measuring volume length on laser velocimetry measurements of turbulent boundary layers were studied. Four different effective measuring volume lengths, ranging in spanwise extent from 7 to 44 viscous units, were used in a low Reynolds number (Re=1440) turbulent boundary layer with high data density. Reynolds shear stress profiles in the near-wall region show that u v strongly depends on the measuring volume length; at a given y-position, u v decreases with increasing measuring volume length. This dependence was attributed to simultaneous validations on the U and V channels of Doppler bursts coming from different particles within the measuring volume. Moments of the streamwise velocity showed a slight dependence on measuring volume length, indicating that spatial averaging effects well known for hot-films and hot-wires can occur in laser velocimetry measurements when the data density is high.List of symbols time-averaged quantity - u wall friction velocity, ( _w /)^1/2 - v kinematic viscosity - d _p pinhole diameter - l _eff spanwise extent of LDV measuring volume viewed by photomultiplier - l ⁺ non-dimensional length of measuring volume, l _eff u /v - y ⁺ non-dimensional coordinate in spanwise direction, y u /v - z ⁺ non-dimensional coordinate in spanwise direction, z u /v - U ⁺ non-dimensional mean velocity, /u - u instantaneous streamwise velocity fluctuation, U &#x2329;U - v instantaneous normal velocity fluctuation, V–V - u RMS streamwise velocity fluctuation, u ^21/2 - v RMS normal velocity fluctuation, v ^21/2 - Re Reynolds number based on momentum thickness, U 0/v - R _uv cross-correlation coefficient, u v/u v - R₁₂(0, 0, z) two point correlation between u and v with z-separation, <u(0, 0, 0) v (0, 0, z)>/<u(0, 0, 0) v (0, 0, 0)> - N rate at which bursts are validated by counter processor - T Taylor time microscale, u (dv/dt²)^–1/2     

Construction of exact numerical solutions of the stationary traveling wave type for viscous thin films

An effective numerical procedure, based on the Galerkin method, for finding solutions of the stationary traveling wave type in the complete formulation is proposed for the case of viscous liquid films. Examples of a viscous film flowing freely down a vertical surface have been calculated. The calculations have been made for various values of the dimensionless surface tension , including =0. The method makes it possible to predict a number of bifurcations that occur as decreases. The existence of numerous families of stationary traveling waves when 1 was demonstrated in [6]. The present study shows that as 1 all but one of these families of wave solutions disappear. The shape of the periodic and solitary waves and the pressure distribution in the film are found for various . When =0 and the wave number is fairly small, the periodic solution has a singularity, as predicted in [14]: at the crest of the wave a corner point appears; the angle between the tangents at this point =140–150. The method proposed can be used to calculate other wavy film flows.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 94–100, May–June, 1990.