Die Reduktion von nichteuklidischen geometrischen Objekten in eine euklidische Form und physikalische Deutung der Reduktion durch Eigenspannungszustände in Kristallen

Summary Self-stresses in crystals are closely related to tearing processes. These processes have a meaning only in connection with geometrical objects. The present paper deals with tearing processes connected with the geometrical objects metric and connexion. A tearing process is commonly associated with either one of these processes. It is therefore of interest to investigate the question whether a tearing process which is connected with both objects exists. The investigation of this problem leads to the subdivision of the self-stresses according to the metric properties of the connexion: metrical stresses caused by dislocations and disclinations and nonmetrical stresses caused by non-mechanical sources and by extra-matter.It should be mentioned that the mathematical part of this paper has an independent interest for the understanding of the structures associated with higher differential geometry.

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Vorgelegt von T. Meixner

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Herrn Professor Dr. A. Seeger danke ich für die verständnisvolle Förderung dieser Arbeit. Herrn Dr. C. Teodosiu bin ich für zahlreiche Diskussionen sehr dankbar.

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Thermal Growth of a Vapor Bubble Moving in a Superheated Liquid

The problem of thermal growth of a vapor bubble moving in a superheated liquid is solved for two models of the phase interface, namely: a rigid (no-slip condition) and a pliable (slip condition) spherical surface. The second self-similar solution of the problem of the motion of a vapor bubble with a pliable surface is found. On the basis of this solution, an approximate dependence of the nondimensional heat flux into the bubble on the Jacob and Péclet numbers is constructed. For two limiting cases, namely, for a bubble growing at rest and a moving bubble of constant radius, this dependence coincides with the known solutions. The calculation results are compared with the experimental data obtained for vapor bubbles rising in a superheated liquid.     

Semi-similar solutions to the three-dimensional laminar boundary layer

The theory of semi-similar solutions is developed for and applied to the problem of three-dimensional laminar boundary layer flow. A number of specific examples are calculated. Particular attention is given to certain flows in which separation is approached and the nature of three-dimensional laminar boundary layer separation is inferred from the behavior of these solutions close to separation. Two types of separation are observed: singular separation characterized by the vanishing of the total shear along the line of separation and ordinary separation characterized by limiting streamlines which become parallel to the line of separation.     

Chaos and integrability in a nonlinear wave equation

We consider the parametrized family of equations _tt ,u- _xx u-au+u ₂ ² u=O,x(0,L), with Dirichlet boundary conditions. This equation has finite-dimensional invariant manifolds of solutions. Studying the reduced equation to a four-dimensional manifold, we prove the existence of transversal homoclinic orbits to periodic solutions and of invariant sets with chaotic dynamics, provided that =2, 3, 4,.... For =1 we prove the existence of infinitely many first integrals pairwise in involution.     

Some aspects of dilute suspension theory

A theory proposed by the author as representative of the flow of a general suspension contains three interaction forces, f, S and N. For a quasi-concentrated suspension and for a dilute suspension, N and S, N are omitted, respectively. For the latter special case, we treat diffusion of a fluid through an elastic solid. For a quasi-concentrated suspension, we show that F and S depend on the gradient of the motion gradient. We demonstrate the existence of interesting phenomena: non-simple behavior, dissipative effects, generalized lift and drag forces.Presented at the second conference Recent Developments in Structured Continua, May 23 – 25, 1990, in Sherbrooke, Québec, Canada.     

Effect of Porous Coatings on Stability of Hypersonic Boundary Layers

The influence of ultrasoundabsorbing coatings on stability of hypersonic boundary layers is considered. Two types of coatings were used in experiments: feltmetal with a random porous microstructure and a sheet perforated by blind cylindrical microchannels. The experiments were performed in a wind tunnel at a Mach number M = 5.95 on sharp cones with a 7° apex halfangle. Evolution of natural disturbances and artificially induced wave packets in the boundary layer was studied with the help of hotwire anemometry. Spatial characteristics of artificial disturbances were obtained. It is demonstrated that such coatings exert a stabilizing effect on secondmode disturbances.     

Plastic deformation of metals subjected to intensive sources

The elastoplastic strain of metals being formed when they melt under the effect of a point heat source with a pulse duration greater than 10^–6 sec is considered in this paper. The time development of the plastic strain and pressure domains in the melt is investigated. It is shown that two plastic strain domains occur during the interaction under consideration: a relatively broad domain of mechanical influence and a narrow domain of thermal influence. The stress-strain distributions as well as the hydrostatic pressure in the fluid are determined by a quasistationary temperature distribution starting with times corresponding to half (of the quasistationary) the value of the melt radius X 0.5. It is shown that the dimensions of the weak and strong plastic strain domains formed by heat and acoustic waves grow continuously to the quasistationary values, while the hydrostatic pressure in the fluid reaches the maximum value for X 0.3...0.4. The ratio between the radii of the plastic strain zones and of the liquid bath for a quasistationary temperature distribution in the first domain lies within the range 10–50, and does not exceed 1.7 for Cu, Ni, and Fe in the second. The anomalous nature of the development of the strong plastic strain domain in Al, because of migration of the metal grain boundaries to result in collapse of the domain for the values X 0.5 accompanied by a jumplike diminution in the hydrostatic pressure in the fluid, is noted.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 129–140, May–June, 1976.     

Asymptotic theory for calculating heat fluxes near the corner of a body

A method is presented for calculating the distribution of the thermal fluxes, friction stresses, and pressure near the corner point of a body contour in whose vicinity the outer supersonic flow passes through an expansion wave. The method is based on a study of the asymptotic solutions of the Navier-Stokes equations as the Reynolds number R approaches infinity for the flow region in which the longitudinal gradients of the flow functions are large, invalidating conventional boundary layer theory. This problem was examined in part in [1], in which the distribution of the friction and pressure in a region with length on the order of a few thicknesses of the approaching boundary layer was obtained in the first approximation. The leading term of the expansion for the thermal flux to the surface of the body vanishes for a value of the Prandtl number equal to unity and for other values of the Prandtl number does not match directly with its value in the undisturbed boundary layer.The thermal-flux distribution is obtained for values of the Prandtl number approaching unity. For this purpose it was necessary to consider a more general double passage to the limit as 1 and 0 for a finite value of the parameter B=[(–1)/] [–ln ^1/4/]^1/4 characterizing the ratio of the effects of thermal conduction, viscous dissipation, and convection. The solution obtained previously [1] corresponds to the particular case B and therefore for actual values of R=10⁴–10⁶, ~ 0.7 overestimates considerably the effect of the dissipative term on heat transfer, although even in first approximation it describes the pressure distribution well and the friction distribution satisfactorily. For smooth matching of the solutions with the corresponding flow functions in the undisturbed boundary layer it was necessary to introduce a flow region with free interaction for the expansion flow. Equations and boundary conditions which describe the flow as a whole are presented. Examples are given of numerical calculations and comparison with experiment.     

The application of a strain gauge technique to the measurement of the power characteristics of five impellers

The work described in this paper establishes the accuracy with which the power number can be measured within a mixing vessel and presents measurements for five impellers (Rushton, bucket, 45°-pitched blade, 60°-pitched blade and hyperboloid) as a function of rotational speed. The technique uses strain gauges and the telemetric transmission of the strain and is, therefore, more accurate than methods which involve mechanical determination of torque and that allows power measurements to be carried out in relatively small and more cost-effective models of mixing vessels. In the cases where previous data exists the results of this study compare well. This work provides new data for the bucket, 45°-pitched blade and hyperboloid (low Reynolds numbers) impellers and represented a systematic study for all five impellers with one measurement system.The authors would like to acknowledge the financial support of the EPSRC during the course of this work.     

Explanation of the “anomalous” dispersal of the discrete particles in a two-phase turbulent jet with allowance for their inertia

The anomalous dispersal of the discrete particles in a two-phase turbulent jet (growth of their concentration in the region of the axis, rapid damping of the concentration in the initial sections of the jet, and wavelike axial distribution) is studied with allowance for the initial conditions and the Magnus effect.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 36–41, January–February, 1984.I thank G. N. Abramovich and A. N. Sekundov for discussing the results.     

The relaxation of a double-well energy

This paper studies coherent, energy-minimizing mixtures of two linearly elastic phases with identical elastic moduli. We derive a formula for the relaxed or macroscopic energy of the system, by identifying microstructures that minimize the total energy when the volume fractions and the average strain are fixed. If the stress-free strains of the two phases are incompatible then the relaxed energy is nonconvex, with double-well structure. An optimal microstructure always exists within the class of layered mixtures. The optimal microstructure is generally not unique, however; we show how to construct a large family of optimal, sequentially laminated microstructures in many circumstances. Our analysis provides a link between the work of Khachaturyan and Roitburd in the metallurgical literature and that of Ball, James, Pipkin, Lurie, and Cherkaev in the recent mathematical literature. We close by explaining why the corresponding problem for three or more phases is fundamentally more difficult.Supported in part by ARO contract DAAL03-89-K-0039, DARPA contract F49620-87-C-0065, ONR grant N00014-88-K-0279, NSF grant DMS-8701895 and AFOSR grant 90-0090     

Zur Rheometrie viskoelastischer Fluide mit der Kegel-Platte-Anordnung

Zusammenfassung Es wird das Schub- und Normalspannungsverhalten verschiedener konzentrierter Polymerlösungen bei stationärer Scherung mit einer offenen Kegel-Platte-Anordnung in einem Mechanischen Spektrometer der Firma Rheometrics Inc. untersucht. Dabei wird festgestellt, daß an der freien Oberfläche des Meßfluids im Randbereich Einschnürungen auftreten, wenn das Geschwindigkeitsgefälle einen charakteristischen Wert überschreitet. Durch Bestimmung der Temperatur an der freien Oberfläche mit einem Strahlungsthermometer wird eine Beanspruchungsgrenze ermittelt, ab welcher mit merklichen Temperaturerhöhungen gerechnet werden muß.Es wird gefunden, daß beim Anlaufvorgang bei Anwendung hinreichend hoher Geschwindigkeitsgefälle nach dem Überschwingen (overshoot) der Spannungen noch ein Unterschwingen (undershoot) erfolgt, das auf das Wirksamwerden von Verfestigungsmechanismen in dem strömenden Fluid zurückzuführen ist. Diese Strömungsverfestigung wird in den untersuchten Polymerlösungen immer dann beobachtet, wenn das Verhältnis von erster Normalspannungsdifferenz zu Schubspannung einen Wert von ca. 10 überschreitet. Dieser Effekt läßt sich besonders eindrucksvoll an Lösungen von hochmolekularem Polyisobutylen (Oppanol B 200) nachweisen. Unter Heranziehung von Ergebnissen anderer Autoren wird erläutert, wieso bei Dehn- und Scherströmungen zwar ein sehr unterschiedliches Fließverhalten vorliegt, trotzdem aber ganz ähnliche Verfestigungsphänomene — Aggregation, Kristallisation, Entmischung — auftreten können. Weiterhin wird der Einfluß des gewählten Lösungsmittels sowie der Temperatur auf den Spannungsverlauf untersucht.In einem Anhang wird aus der Schubspannung und der ersten Normalspannungsdifferenz eine signifikante Vergleichsspannung sowie der Orientierungswinkel berechnet. Bei Zugrundelegung dieser Größen wird die bekannte Tatsache, daß die erste Normalspannungsdifferenz erst dann dominierenden Einfluß auf das Fließverhalten gewinnt, wenn sie die Schubspannung wesentlich übersteigt, ohne weiteres verständlich.

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Summary The shear and normal stress behavior of several concentrated polymer solutions in steady shear flow is investigated by means of an open cone and plate arrangement of the Mechanical Spectrometer (Rheometrics Inc.) type. Deformations of the free gap surface are found if the shear rate is raised above a critical value. Sensitive temperature measurements of the liquid surface, carried out by means of a radiative pyrometer, yield a shear rate above which dissipation effects become appreciable.Start-up experiments in regions of sufficiently high shear rate show an additional undershoot after the initial overshoot. This undershot has to be attributed to mechanisms leading to a stiffening of the flowing liquid. Such a flow hardening can be observed in the investigated polymer solutions if the ratio of normal and shear stresses is of the order of 10 or more. It may be observed particularly impressively with concentrated solutions of a high molecular mass polyisobutylene (Oppanol B 200). From the results of other authors it is shown that elongational and shear flows despite their different kinematics, may produce similar hardening phenomena — aggregation, cristallisation, demixing. Furthermore, the influence of different solvents and temperature on the stress curves is discussed.In an appendix two quantities, a significant stress and an orientation angle are calculated from the shear and normal stresses. These quantities give an insight into the well known fact that the first normal stress difference has a dominating influence on the flow only when the normal stress difference is much greater than the shear stress.

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Erster Teil einer von der Abteilung Chemietechnik der Universität Dortmund genehmigten Dissertation; auszugsweise vorgetragen auf der Jahrestagung der Deutschen Rheologischen Gesellschaft in Dortmund vom 9.–11. März 1977.

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Mit 12 Abbildungen und 1 Tabelle     

Asymptotic solution to the problem of hypersonic flow over bodies in the neighborhood of the separation point of a thin shock layer

A study is made of the problem of hypersonic flow of an inviscid perfect gas over a convex body with continuously varying curvature. The solution is sought in the framework of the asymptotic theory of a strongly compressed gas [1–4] in the limit M when the specific heat ratio tends to 1. Under these assumptions, the disturbed flow is situated in a thin shock layer between the body and the shock wave. At the point where the pressure found by the Newton-Buseman formula vanishes there is separation of the flow and formation of a free layer next to the shock wave [1–4]. The singularity of the asymptotic expansions with respect to the parameter ₁ = ( –1)/( + 1) associated with separation of the strongly compressed layer has been investigated previously by various methods [3–9]. Local solutions to the problem valid in the neighborhood of the singularity have been obtained for some simple bodies [3–7]. Other solutions [7, 9] eliminate the singularity but do not give the transition solution entirely. In the present paper, an asymptotic solution describing the transition from the attached to the free layer is constructed for a fairly large class of flows.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 99–105, January–February, 1982.     

On the sensitive dynamical system and the transition from the apparently deterministic process to the completely random process

The detailed analysis of the dynamical process of coin tossing is made. Through calculations, it is illustrated how and why the result is extremely sensitive to the initial conditions. It is also shown that, as the initial height of the mass center of the coin increases, the final configuration, i.e. head or tail, becomes more and more sensitive to the initial parameters (the initial velocity angular velocity, and the initial orientation), the coefficient of the air drag, and the energy absorption factor of the surface on which the coin bounces. If we keep the head upward initially but allow a small range for the change of some other initial parameters, the frequency that the final configuration is head, would be 1 if the initial height h of the mass center is sufficiently small, and would be clo to 1/2 if h is sufficiently large. An interesting question is how this frequency changes continuously from 1 to 1/2 as h increases. Detailed calculations show that such a transition is very similar to the transition from laminar to turbulent flows. A basic difference between the transition stage and the completely random stage is indicated: In the completely random stage, the deterministic process of the individual case is extremely sensitive to the initial conditions and the dynamical parameters, out the statistical properties of the ensemble are insensitive to the small changes of the initial conditions and the dynamical parameters. On the contrary, in the transition stage, both the deterministic process of the individual case and the statistical properties of the ensemble are sensitive to the initial conditions and the dynamical parameters. The mechanism for this feature of the transition stage is the existence of the long-train structure in the parameter space. The illuminations of this analysis on some other random phenomena are discussed.     

Flow through a helically coiled annulus

In this paper the flow is studied of an incompressible viscous fluid through a helically coiled annulus, the torsion of its centre line taken into account. It has been shown that the torsion affects the secondary flow and contributes to the azimuthal component of velocity around the centre line. The symmetry of the secondary flow streamlines in the absence of torsion, is destroyed in its presence. Some stream lines penetrate from the upper half to the lower half, and if is further increased, a complete circulation around the centre line is obtained at low values of for all Reynolds numbers for which the analysis of this paper is valid, being the ratio of the torsion of the centre line to its curvature.Nomenclature A =constant - a outer radius of the annulus - b unit binormal vector to C - C helical centre line of the pipe - D rL - g 1000 - K Dean number=Re² - L 1+r sin - M (L ²+ ² r ²)^1/2 - n unit normal vector to C - P, P pressure and nondimensional pressure - p ₀, p pressures of O(1) and O() - Re Reynolds number=aW ₀/ - (r, , s), (r, , s) coordinates and nondimensional coordinates - nonorthogonal unit vectors along the coordinate directions - r ₀ radius of the projection of C - t unit tangent vector to C - V _r, V , V _s velocity components along the nonorthogonal directions - V_r, V, V _s nondimensional velocity components along - W ₀ average velocity in a straight annulus Greek symbols , curvature and nondimensional curvature of C - U, V, W lowest order terms for small in the velocity components along the orthogonal directions t - _r, , _s first approximations to V _r , V, V _s for small - =/=/ - kinematic viscosity - density of the fluid - , torsion and nondimensional torsion of C - , stream function and nondimensional stream function - nondimensional streamfunction for U, V - a inner radius of the annulus After this paper was accepted for publication, a paper entitled On the low-Reynolds number flow in a helical pipe, by C.Y. Wang, has appeared in J. Fluid. Mech., Vol 108, 1981, pp. 185–194. The results in Wangs paper are particular cases of this paper for =0, and are also contained in [9].     

On universal solutions in initially stressed bodies

A saddle point (or maximum-minimum) principle is set up for the quasi-static boundary-value problem in linear viscoelasticity. The appropriate class of convolution-type functionals for it is taken in terms of bilinear forms with a weight function involving the Fourier transform. The minimax property is shown to hold as a direct consequence of thermodynamic restrictions on the relaxation function. This approach can be extended to further linear evolution problems where initial data are not prescribed.     

Angle Plates

An angle plate is a multistructure consisting of two plate-like substructures meeting at a right angle. We regard it as obtained from a single plate by a thought process of folding such as to affect the material behavior only in the elbow region. We model the three-dimensional plate-like material region corresponding to the plate to fold as being transversely isotropic with respect to an axis orthogonal to its base surface, with admissible displacements in the Reissner-Mindlin's form. After folding, we require that in the elbow region the material be doubly transversely isotropic and the displacement be the common restriction of the admissible displacements in the two arm plates. Under these assumptions, from a three-dimensional virtual-work formulation of equilibrium we deduce by mere thickness integration the field, boundary, and transmission equations of our two-dimensional model of a linearly elastic angle plate. Various generalizations of this model are possible, some of which we occasionally indicate.     

Some self-similar solutions of the Leibenzon equation

Self-similar one-dimensional solutions of the Leibenzon equation c²_t= _zz ^k (z 0, k 2) are considered. Approximate solutions are constructed for the two cases in which the initial value = ₁ = const > 0 and on the boundary either a constant value = ₂ < ₁ is maintained or the flow (directed outwards) is given. In the first problem the dependence of the boundary flow on the governing parameters is determined. A characteristic property of the types of motion in question is the existence near the boundary of a region, expanding with time, in which the flow is almost independent of the coordinate.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 145–150, September–October, 1991.     

Whose Thoughts Are They, Anyway? Dimensionally Exploding Bion's “Double-Headed Arrow” into Coadapting, Transitional Space

The physics and biology that found psychoanalysis account for discontinuous experience only in the presence of nonmeasurable, metaphysical operators; these include the ego and its subsystems as well as biological experience inherited through Lamarckian principles. Complex, self-organizing systems, however, can link biology to experience without metaphysics. They can also account for psychoanalytically relevant behaviors without appealing to stable internal representations. These behaviors include what W. R. Bion called transformation in O and its corollary, the appearance of the selected fact. By dimensionally exploding the double-headed arrow that he used to link the states Ps and D in his model for thinking (Ps D), we can generate a space that is, at once, psychoanalytically imaginal and dynamically coadapting. Isomorphic to D. W. Winnicott's transitional space, it is self-organizing. It is describable according to dynamics formulated by W J. Freeman, S. Kauffman and C. Langton and it can generate instantaneous conscious contents by way of a selective process analogous to spatio-temporal binding. As a whole, this model supports a clinical stance advanced by D. W. Winnicott as play, within transitional space.     

Indentation of the semi-infinite elastic solid by a concave rigid punch

A solution is obtained for the relationship between load, displacement and inner contact radius for an axisymmetric, spherically concave, rigid punch, indenting an elastic half-space. Analytic approximations are developed for the limiting cases in which the ratio of the inner and outer radii of the annular contact region is respectively small and close to unity. These approximations overlap well at intermediate values. The same method is applied to the conically concave punch and to a punch with a central hole. , , . , . . .