On the existence of the derivative of the volume average

A general theorem on the derivative of the volume average is formulated and proved. Conditions for the existence of the derivative are presented and discussed. This is done in order to give a better base to the theory of spatial averaging.Latin Letters E ₃ three-dimensional vector space over the field of real numbers - K, K(x) averaging domain - G, G _w, G_s open sets in E ₃; components of the two-phase system - C ¹(G) the set of functions 1-times continuously differentiable in G - W1/2(G) Sobolev space - V volume of the domain K - f function defined in G, G _w - K _infi ^sup* (x), K _infi ^sup– (x) special parts of K(x) Greek Letters boundary of G, G _w, G_s; w-s interface - _ij Kronecker delta - v unit outward normal of G, G _w - _j j-dimensional Lebesgue measure Other M closure of a set M in the metric space E ₃ - f phase average of f for the w-phase - (u, v) scalar product of u, v in E ₃ - one-sided derivatives     

The importance of the sigma phenomenon in the study of the flow of blood

Summary During the past three decades, work on blood and on nonbiological systems concerning anomalies in theR ⁴ law of capillary flow has been proceeding along parallel and usually unconnected lines.The present paper attempts to draw together these diverse lines of work and stresses the importance to haemo-pathology of a fuller understanding of the sigmaphenomenon in blood.

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Zusammenfassung Während der letzten 3 Jahrzehnte wurden Untersuchungen an Blut und an nichtbiologischen Systemen, die die Abweichung vomR ⁴-Gesetz beim kapillaren Fließen betreffen, auf parallelen und im allgemeinen getrennten Wegen vorangetrieben.Die vorgelegte Arbeit versucht, diese verschiedenen Forschungsrichtungen zusammenzubringen und ein volleres Verständnis des Sigma-Phänomens beim Blut in seiner Bedeutung für die Hämopathologie zu schaffen.

     

On the relativistic extension of the equations of the classical gravitational potential

Summary The meaning of the vector V which characterizes the difference between the two tensor potentials of the Riemann and of the Weyl tensor is here studied. In the context of general relativity, the equations satisfied by such a vector and by the scalar potential characterizing its irrotational part, are connected to the equation proposed by Cattaneo as a relativistic extension of the Gauss-Poisson classical one. The link between the vector potential V and the standard gravitational field is thus obtained, for static universes.

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Sommario Si studia il significato del vettore V che caratterizza il divario fra i tensori potenziali dei tensori di Riemann e di Weyl. Le equazioni cui soddisfano, in relatività generale, detto vettore ed il potenziale scalare che ne individua la parte irrotazionale, vengono collegate con l'equazione relativistica proposta da Cattaneo quale estensione di quella classica di Gauss-Poisson. Si perviene così, nel caso statico, a stabilire un legame fra il potenziale vettore V ed il campo gravitazionale standard.

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This work was done in the sphere of activity of the C.N.R. Groups for mathematical research.     

Experimental investigation of the influence of the flow structure on the aerodynamic coefficients of the IXV vehicle

In the framework of the ESA Future Launchers Preparatory Program (FLPP) an experimental study on the aerodynamic behavior during the re-entry phase of the Intermediate eXperimental Vehicle (IXV) configuration was conducted in the DLR hypersonic wind tunnel H2K in Cologne. Tests were carried out at Mach 6.0 and 8.7 with different flap deflection angles and the angle of attack varied continuously between 20° and 55° to investigate the flow topology as well as the aerodynamic forces and moments and the surface pressure distribution. The experimental data show that depending on the combination of the flap deflection angle (δ _L/R) and angle of attack (α) the complex flow structure in the vicinity of the flaps significantly influences the vehicle’s aerodynamic coefficients. An analysis of this shock/shock and shock/boundary layer interaction causing flow separation with reattachment is performed.     

On the origin of the hole pressure

Summary One wall of a slit die apparatus contains a slot filled with the test liquid. Photographs of marked streamlines near the slot mouth show a pronounced curvature. A 0.5% aqueous solution of Separan AP 30 and molten polystyrene and polyethylene were investigated at low values of Reynolds number (Re < 1.0). In each case, streamlines are convex towards the slot mouth. This supports the hypothesis ofTanner andPipkin which attributes hole pressures in flowing viscoelastic liquids to the combined effect of holegenerated streamline curvature and shear-generated normal stress differences.

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Zusammenfassung In einer der Wände einer Schlitzdüsenvorrichtung ist eine mit Testflüssigkeit gefüllte Quernut angebracht. Es wurde die Strömung einer 0,5%-igen wässerigen Separan AP 30-Lösung, sowie einer Polystyrol- und einer Polyäthylenschmelze bei kleinen Reynoldszahlen (Re < 1.0) untersucht. In allen Fällen waren die Stromlinien konvex, also zur Schlitzmündung hin gekrümmt. Dadurch wird die Hypothese vonTanner andPipkin unterstützt, wonach die Lochdrücke in strömenden viskoelastischen Flüssigkeiten als Wirkung der durch die Scherung erzeugten Normalspannungsdifferenzen auf die gekrümmten Stromlinien erklärt werden.

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Department of Engineering Mechanics.

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Material Science Program.

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With 4 figures     

On the derivation of the Forchheimer equation by means of the averaging theorem

The averaging theorem is applied to the microscopic momentum equation to obtain the macroscopic flow equation. By examining some very simple tube models of flow in porous media, it is demonstrated that the averaged microscopic inertial terms cannot lead to a meaningful representation of non-Darcian (Forchheimer) effects. These effects are shown to be due to microscopic inertial effects distorting the velocity and pressure fields, hence leading to changes in the area integrals that result from the averaging process. It is recommended that the non-Darcian flow regime be described by a Forchheimer number, not a Reynolds number, and that the Forchheimer coefficient be more closely examined as it may contain information on tortuosity.English a _i gravitational acceleration (m/s²) - A _fs interfacial area between the fluid and solid phases (m²) - Fo Forchheimer number - k permeability (m²) - k ₀ permeability at zero velocity (m²) - p thermodynamic pressure (Pa) - r _i coordinate on the microscopic scale (m) - Re Reynolds number - t time (s) - u _i ,u bulk velocity (m/s) - V volume (m³) - V _f fluid volume (m³) - w _i ,w microscopic velocity (m/s) - x _i ,x coordinate on the macroscopic scale (m) Greek the Forchheimer coefficient (1/m) - _ij extra (viscous) stress tensor (Pa) - _ij stress tensor (Pa) - Viscosity (Pa. s) - density (kg/m³) - porosity - a general variable Symbols < > phase average - < > ^f intrinsic phase average - the fluctuating part of a variable     

A logrithmic bound on the location of the poles of the scattering matrix

It is shown that the complex poles z of the scattering matrix satisfy the inequality: Im z≧a+b log ¦z¦, b>0, in three instances of classical scattering in three space dimensions described by the wave equation u_{t t}?c²Δu+qu=0.

1. c and q smooth with c=1 and q=0 for ¦x¦>p, all rays going to infinity, and the energy form positive definite.
2. c=1 and q=0 outside of a convex body on which u=0.
3. c=1, q bounded and measurable, q=0 for ¦x¦>p, and the energy form not necessarily positive definite.

     

On the solution of the sector problem

A rigorous study of the sector problem is presented by using the Mellin transform technique. The stress function is obtained as an asymptotic expansion of the complex inversion integral. The number of terms of this expansion, as well as the differentiability of the stress function, depend on the differential properties of boundary conditions on the radial edges. If these boundary conditions belong toC , this asymptotic expansion is transformed to a uniformly convergent infinite series. The coefficients of the series, which depend only on the boundary conditions along the circumferential edges, are calculated by applying a bi-orthogonality condition, or, by a technique based on the Betti formula.

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Résumé En utilisant la technique de la transformation de Mellin on présente une étude rigoureuse du problème d'un secteur. La fonction des contraintes est obtenue par un développement asymptotique de l'intégral complexe d'inversion. Le nombre de termes de ce développement, ainsi que la differentiabilité de la fonction des contraintes, dépendent de proprietés différentielles des conditions aux limites sur les bords radiaux. Si ces conditions aux limites sont dansC , ce développement asymptotique est transformé en série infinie uniformement convergente. Les coefficients de cette série, qui dépendent uniquement de conditions aux limites sur le bord circulaire, sont calculés à l'aide d'une condition de bi-orthogonalité que nous démontrons, ou, à l'aide d'une technique basée sur la formule de Betti.

     

Contributions to the theory of the method of steepest descent

Applying the method of steepest descent to F(x ₁,..., x _n ) one obtains a sequence of points _v . To obtain conditions for convergence of _v , the derived set H of the _v in the case of divergence is studied. In this case H is a continuum on which not only grad F vanishes everywhere, but also the rank of the Hessian of F is everywhere less than n-1.     

Morawetz's method for the decay of the solution of the exterior initial-boundary value problem for the linearized equation of dynamic elasticity

In this paper, the behavior of the solution of the time-dependent linearized equation of dynamic elasticity is examined.For the homogeneous problem, it is proved that in the exterior of a star-shaped body on the surface of which the displacement field is zero, the solution decays at the rate t ^-1 as the time t tends to infinity.For the non-homogeneous problem with a harmonic forcing term, it is proved that for large times, the elastic material in the exterior of the body, tends to a harmonic motion, with the period of the external force.The convergence to the steady harmonic state solution is at the rate t ^-1/2 as t tends to infinity, and is uniform on bounded sets.     

On the use of the rolling-ball viscometer for the measurement of rheological parameters of power law fluids

Summary An attempt has been made to determine the flow curve of fluids obeying the power law rheological model on a rolling ball viscometer. The theoretical analysis is based on the hydrodynamical model developed byLewis in 1956 for the purpose of rolling ball viscometer calibration and extended in 1964 byTurian andBird for the two-parametric power law model. The shear rate variation has been accomplished on varying the tube angle of inclination. Experiments performed on an adapted, commercialHoeppler viscometer with aqueous solutions of carboxymethylcellulose and polyacrylamide reproduced the flow curves obtained simultaneously on rotational and capillary instruments with reasonable accuracy.

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Zusammenfassung Die Möglichkeit der Bestimmung von FließkurvenOstwaldscher Flüssigkeiten in einem Fallkugelviskosimeter wurde überprüft. Die analytische Behandlung des Problems beruht auf dem hydrodynamischen Modell, das 1956Lewis zur Eichung von Fallkugel-viskosimetern entwickelte und das später vonBird undTurian verallgemeinert wurde. Die Änderung der Schergeschwindigkeit wurde durch Neigung des Fallrohrs erzielt. Versuche wurden an einem handelsüblichenHöppler-Viskometer mit wässerigen Lösungen von Karboxymethylzellulose und Polyakrylamid durchgeführt. Vergleiche der gewonnenen Ergebnisse mit Fließkurven, die gleichzeitig an Rotations- und Kapillar-Viskosimetern gemessen wurden, ergaben im Gültigkeitsbereich der Theorie befriedigende Übereinstimmung.

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Nomenclature d ball diameter - D tube diameter - f function ofz defined in eq. [16] - F function ofn defined in eq. [30] - g gravity acceleration - H clearance between the ball and tube - I definite integral defined in eq. [17] - J _L = 3/4J(1) definite integral defined in [4] - J (n) definite integral defined in eq. [21] - K instrument constant defined forNewtonian fluids in eq. [1] - L distance marked on the viscometer tube - m consistency parameter - n flow behaviour index - p pressure - r radial variable - S area - t time - T temperature - u _z velocity component - U velocity of the ball - V flow rate - y, z spatial rate - tube angle of inclination - shear rate - gamma function - azimuthal variable - µ dynamic viscosity - fluid density - _s ball density - shear stress - dimensionless variable defined in eq. [9] - — denotes mean value - w denotes conditions at the solid surface With 9 figures and 2 tables     

Opportunities of the elastometry method for investigating the elastic properties of the eyeball shell

On the basis of a maximally simple mechanical model, the possibility to estimate the stiffness of the eyeball shell using the intraocular pressures measured in the presence of different weights applied to the cornea (elastometry method) is investigated. On the basis of general considerations of dimensional theory, it is shown that the pressure difference used in ophthalmology for estimating the rigidity of the eye ball and determined in the elastometry procedure depends on the ratio E/p ₀, where E is a quantity characterizing the shell stiffness and p ₀ is the intraocular pressure in the unloaded eye. An experimental procedure, which can be treated as a development of the elastometry method and makes it possible to estimate a certain parameter dependent only on the eyeball shell stiffness, is proposed. The practical realization of the method proposed needs to be checked experimentally.     

On the singularity index and the unfolding of the diffraction catastrophes

The caustics of high-frequency wave propagation may be classified using catastrophe theory. The wavefield in the vicinity of any caustic is described by the corresponding diffraction catastrophe. The singularity index, β, is a measure of the rate at which such a wavefield diverges as ω→∞ at the point where all control parameters and moduli are set equal to zero. It is shown that away from this point β also describes a balance between two different measures of the unfolding of the wavefield in each control direction, β = σ_nϱ_n. The indices σ_n and ϱ_n describe, respectively, the rate at which individual ray arrivals separate in time and decay as a function of control parameter.     

Influence of the atomic weight of the target on the magnitude of the energy accommodation coefficient of ions of a partially ionized gas flow

One of the most essential parameters in the investigation of singularities in molecule interaction with the surface of a solid is the energy accommodation coefficient. Data on the positive ion accommodation coefficients in the energy range 1–100 eV are scarce [1], hence the value of _i is often taken close to one, although this condition is not satisfied in practice for the majority of working gases. In this paper we discuss the resultant measurements of the value of _i in a high-speed flow of rarified plasma.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 83–87, May–June, 1979.     

Determination of the Closure of the Set of Elasticity Functionals

We determine the closure for Mosco-convergence in L ² (,³) of the set of elasticity functionals. We prove that this closure coincides with the set of all non-negative lower-semicontinuous quadratic functionals which are objective, i.e., which vanish for rigid motions. The result is still valid if we consider only the set of isotropic elasticity functionals which have a prescribed Poisson coefficient. This shows that a very large family of materials can be reached when homogenizing a composite material with highly contrasted rigidity coefficients.     

Dependence of the stability criterion of nucleate boiling on the compressibility of the vapor

The article gives the results of a study of the motion of bubbles and their deformation near the heating surface at different pressures. It was observed that, during the time of their growth, the gaseous medium in the bubbles is in a compressed state.Nomenclature R) radius of bubble - R_h) maximul radius of a deformed bubble in the horizontal plane - R_v) maximal radius of a deformed bubble in the vertical plane - ) specific weight - B) universal gas constant - ) surface-tension coefficient - p) pressure - ) edge wetting angle - g) acceleration due to gravity - V) volume - ) molecular weight - C_T) isothermal velocity of sound Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 77–81, July–August, 1971.     

On the solution of the model of two co-affected species

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On the theory of the dynamic viscometer

Summary It is shown that by using a dynamic viscometer, complex dynamic viscosities can be determined rather easily over a wide frequency range. Expansions of the exact solution which takes inertia effects into account are in the case of small gap widths almost identical for low and high frequencies. For intermediate frequencies the deviation from this limiting behavior is only a few percent.

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Zusammenfassung Die Anwendung von dynamischen Viskometern ermöglicht eine einfache Bestimmung von komplexen dynamischen Viskositäten über einen weiten Frequenzbereich. Reihenentwicklungen der exakten Lösung, welche Trägheitseffekte berücksichtigt, sind für den Fall enger Zylinder-Spalte für niedrige und hohe Frequenzen fast identisch. Aber auch bei Frequenzen im Zwischenbereich weichen die Ergebnisse nur um einige Prozent ab.

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With 1 figure