Similar solutions and the asymptotics of filtration equations

In this paper we consider the asymptotic behavior of solutions of the quasilinear equation of filtration as t. We prove that similar solutions of the equation u _t = (u )_xx asymptotically represent solutions of the Cauchy problem for the full equation u _t = [(u)]_xx if (u) is close to u for small u.     

Anwendung des Übertragungsverfahrens bei stark abklingenden Lösungsfunktionen

Übersicht Bei stark abklingenden Funktionen wird die Übertragungsmatrix U() aufgespalten in die Anteilc U ₁() e^– und U ₂() e. Der zweite Term spielt am Rand = 0 keinc Rolle. Die unbekannten Anfangswerte sind über die Matrix U ₁(0) an die bekannten gebunden und eindeutig bestimmbar.

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Summary For strongly decaying solution functions the transfer matrix U() is splitted into the parts U ₁() e^– and U ₂() e. The second term does not influence at the boundary = 0. The unknown initial values are related by the matrix U ₁(0) to the known values and they can be uniquely determined.

     

On the thermodynamics of elastic-plastic materials with temperature-dependent moduli and yield stresses

Summary The subject of this article is the thermodynamics of perfect elastic-plastic materials undergoing unidimensional, but not necessarily isothermal, deformations. The first and second laws of thermodynamics are employed in a form in which only the following quantities appear: the temperature , the elastic strain _e, the plastic strain _p, the elastic modulus (gq), the yield strain (gq), the heat capacity (_e, _p,), the latent elastic heat _e(_e, _p, ), and the latent plastic heat _p(_e, _p, ). Relations among the response functions , , , _e, and _p are derived, and it is shown that a set of these relations gives a necessary and sufficient condition for compliance with the laws of thermodynamics. Some observations are made about the existence and uniqueness of energy and entropy as functions of state.Dedicated to Clifford Truesdell on the occasion of his 60th birthdayThis research was supported by the U.S. National Science Foundation.     

Heat transfer from laminar flow in ducts with elliptic cross-section

Summary The cooling of a hot fluid in laminar Newtonian flow through cooled elliptic tubes has been calculated theoretically. Numerical data have been computed for the two values 1.25 and 4 of the axial ratio of the elliptic cross-section . For =1.25 the influence of non-zero thermal resistance between outmost fluid layer and isothermal surroundings has also been investigated. Special attention has been given to the distribution of heat flux around the perimeter; when increases the flux varies more with the position at the circumference. This positional dependence becomes less pronounced, however, as the (position-independent) thermal resistance of the wall increases.Flattening of the conduit, while maintaining its cross-sectional area constant, improves the cooling. Comparison with rectangular pipes shows that this improvement is not as marked with elliptic as with rectangular pipes.Nomenclature A _k =A _{m, n} coefficients of expansion (6) - a, b half-axes of ellipse, b<a - a _p =a _{r, s} coefficients of representation (V) - D hydraulic diameter, = 4S/P; S = cross-sectional area, P = perimeter - D _e equivalent diameter, according to (13) - n coordinate (outward) normal to the tube wall - T temperature of fluid - T _i temperature of fluid at the inlet - T _s temperature of surroundings - v ₀ mean velocity of fluid - v _z longitudinal velocity of fluid - x, y carthesian coordinates coinciding with axes of ellipse - z coordinate in flow direction - , dimensionless half-axes of ellipse, =a/D and =b/D - _t heat transfer coefficient from fluid at bulk temperature to surroundings; equation (11) - _w heat transfer coefficient at the wall; equation (3) - axial ratio of ellipse, = a/b = / - , , , dimensionless coordinates; =x/D, =y/D, =z/D, =n/D - dimensionless temperature, = (T–T _s)/(T _i–T _s) - ₀ cup-mixing mean value of ; equation (10) - thermal conductivity of fluid - _m,n = _k eigenvalue - c volumetric heat capacity of fluid - _{m, n} = _k = _k eigenfunction; equations (6) and (I) - Nu total Nusselt number, = _t D/ - Nusselt number at large distance from the inlet - Nu _w wall Nusselt number, = _w D/, based on _w - Pé Péclet number, = ₀ Dc/     

The relaxation of polymers with linear flexible chains of uniform length

The analysis of dynamic mechanical data indicates that linear flexible polymer chains of uniform length follow a scaling relation during their relaxation, having a linear viscoelastic relaxation spectrum of the formH() = n ₁ G _N ⁰ × (/ _max) ⁿ¹ for _max. Data are well represented with a scaling exponent of about 0.22 for polystyrene and 0.42 for polybutadiene. The plateau modulusG _N ⁰ is a material-specific constant and the longest relaxation time depends on the molecular weight in the expected way. At high frequencies, the scaling behavior is masked by the transition to the glassy response. Surprisingly, this transition seems to follow a Chambon-Winter spectrumH() = C^–n2, which was previously adopted for describing other liquid/solid transitions. The analysis shows that the Rouse spectrum is most suitable for low molecular-weight polymersM M _c , and that the de Gennes-Doi-Edwards spectrum clearly predicts terminal relaxation, but deviates from the observed behavior in the plateau region.Dedicated to Prof. Richard S. Stein on the occasion of his 65th birthday.On sabbatical leave from the University of Linz, Austria.     

Loss of real characteristics for models of three-phase flow in a porous medium

In this paper we examine the generalized Buckley-Leverett equations governing threephase immiscible, incompressible flow in a porous medium, in the absence of gravitational and diffusive/dispersive effects. We consider the effect of the relative permeability models on the characteristic speeds in the flow. Using a simple idea from projective geometry, we show that under reasonable assumptions on the relative permeabilities there must be at least one point in the saturation triangle at which the characteristic speeds are equal. In general, there is a small region in the saturation triangle where the characteristic speeds are complex. This is demonstrated with the numerical results at the end of the paper.Symbols and Notation a, b, c, d entries of Jacobian matrix - A, B, C, D coefficients in Taylor expansion of _t, _v, _a - det J determinant of matrix J - dev J deviator of matrix J - J Jacobian matrix - L linear term in Taylor expansion for J near (s _v, s_a) = (0, 1) - m slope of r ₊ - p pressure - r± eigenvectors of Jacobian matrix - R real line - S intersection of saturation triangle with circle of radius centered at (1, 0) - S intersection of saturation triangle with circle of radius centered at (0, 1) - s _l, s_v, s_a saturations of phases (liquid, vapor, aqua) - tr J trace of matrix J - v _l , v _v , v _a phase flow rates (Darcy velocities) - v _T total flow rate - X, Y, Z entries of dev J - smooth closed curve inside saturation triangle - saturation triangle - _l, _v, _a phase density times gravitational acceleration times resevoir dip angle - K total permeability - _l, _v, _a three-phase relative permeabilities - _lv>, _la liquid phase relative permeabilities from two-phase data - _l, _v, _a mobilities of phases - _T total mobility - _l Corey mobility - _l, _v, _a phase viscosities - _± eigenvalues of Jacobian matrix - porosity Supported in part by National Science Foundation grant No. DMS-8701348, by Air Force Office of Scientific Research grant No. AFOSR-87-0283, and by Army Research Office grant No. DAAL03-88-K-0080.This work was performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48.     

Experimental and numerical analysis of oscillatory tube flow of viscoelastic fluids represented at the example of human blood

Summary The 4 constant parameters of an Oldroyd type constitutive equation for normal human blood of 45% haematocrit were determined by means of the steady flow curve and the material functions () and (), measured at 2 Hz in an oscillatory capillary viscometer. It was found, as other authors did before, that the phenomenological behaviour of blood flow can be reproduced qualitatively thereby. The quantitative behaviour, however, cannot be described by thus developed parameters. The parameters of the constitutive equationµ ₀, ₁ and ₂ were therefore generalized to become dependent of shear rate and frequency respectively.In itself this is nothing but a transformation of the material functions ( ),() and (), but these can be used as parameters in a constitutive equation though having lost the property of constancy.In this way the linear region of viscoelasticity and the steady flow curve can be reproduced quantitatively. A computer simulation of oscillatory flow for large amplitudes shows another tendency for the phase shift between pressure and flow than the experiment in the oscillatory capillary rheometer does.The applicability of a constitutive equation modified in this manner for other than oscillatory flow should be further examined especially for pulsatile flow.

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Zusammenfassung Mit Hilfe der Fließkurve für die stationäre Strömung und Messungen der Materialfunktion () im Oszillations-Kapillarrheometer bei 2 Hz wurden die Konstanten der 4-Konstanten-Oldroyd-Stoffgleichung für gesundes Humanblut von 45% Hämatokrit bestimmt. Es zeigte sich, daß sich — wie auch schon von anderen Autoren mitgeteilt wurde — mit den so ermittelten Modellkonstanten die Phänomenologie des Fließverhaltens des Blutes qualitativ gut beschreiben läßt. Zur quantitativen Beschreibung reicht dieses Modell jedoch nicht aus, wie man an der Wiedergabe der stationären Fließkurve und der Materialfunktionen der linearen Viskoelastizität erkennt. Aus diesem Grund wurden die Parameterµ ₀, ₁ und ₂ in Abhängigkeit der Schergeschwindigkeit bzw. der Frequenz angesetzt.Dies bedeutet zunächst nichts anderes als eine Transformation der Materialfunktionen, jedoch in einer Art, daß letztere als Parameter in einer Stoffgleichung verwendet werden können, was allerdings mit dem Verlust der Konstanz der Parameter verbunden ist. Mit einer derart modifizierten Stoffgleichung lassen sich der Bereich der linearen Viskoelastizität und die stationäre Fließkurve quantitativ beschreiben. Eine Computersimulation der oszillierenden Rohrströmung zeigt für große Amplituden eine andere Tendenz für die Phasenverschiebung zwischen Druck und Volumenstrom, als sie sich bei Messungen am Oszillations-Kapillarrheometer ergibt.Die Anwendbarkeit der modifizierten Stoffgleichung für andere Strömungsformen, wie z.B. pulsierende Rohrströmungen, muß noch geprüft werden.

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Paper, presented at the Annual Conference of the Deutsche Rheologische Gesellschaft in Berlin, May 8–10, 1978.

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With 8 figures     

On an inversion theorem for conical regions in elasticity theory

Summary Two-dimensional stress singularities in wedges have already drawn attention since a long time. An inverse square-root stress singularity (in a 360° wedge) plays an important role in fracture mechanics.Recently some similar three-dimensional singularities in conical regions have been investigated, from which one may be also important in fracture mechanics.Spherical coordinates are r, , . The conical region occupied by the elastic homogeneous body (and possible anisotropic) has its vertex at r=0. The mantle of the cone is described by an arbitrary function f(, )=0. The displacement components be u. For special values of (eigenvalues) there exist states of displacements (eigenstates) % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqef0uAJj3BZ9Mz0bYu% H52CGmvzYLMzaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasa% acOqpw0xe9v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8Wq% Ffea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dme% GabaqaaiGacaGaamqadaabaeaafiaakabbaaa6daaahjxzL5gapeqa% aiaadwhadaWgaaWcbaGaeqOVdGhabeaakiabg2da9iaadkhadaahaa% WcbeqaaiabeU7aSbaakiaadAgadaWgaaWcbaGaeqOVdGhabeaakiaa% cIcacqaH7oaBcaGGSaGaeqiUdeNaaiilaiabfA6agjaacMcaaaa!582B!\[u_\xi = r^\lambda f_\xi (\lambda ,\theta ,\Phi )\],which may satisfy rather arbitrary homogeneous boundary conditions along the generators.The paper brings a theorem which expresses that if is an eigenvalue, then also-1- is an eigenvalue. Though the theorem is related to a known theorem in Potential Theory (Kelvin's theorem), the proof has to be given along quite another line.

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Zusammenfassung Zwei-dimensionale Spannungssingularitäten in keilförmigen Gebieten sind schon längere Zeit untersucht worden und neuerdings auch ähnliche drei-dimensionale Singularitäten in konischen Gebieten.Kugelkoordinaten sind r, , . Das konische Gebiet hat seine Spitze in r=0. Der Mantel des Kegels lässt sich beschreiben mittels einer willkürlichen Funktion f(, )=0. Die Verschiebungskomponenten seien u. Für spezielle Werte von (Eigenwerte) bestehen Verschiebunszustände % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqef0uAJj3BZ9Mz0bYu% H52CGmvzYLMzaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasa% acOqpw0xe9v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8Wq% Ffea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dme% GabaqaaiGacaGaamqadaabaeaafiaakabbaaa6daaahjxzL5gapeqa% aiaadwhadaWgaaWcbaGaeqOVdGhabeaakiabg2da9iaadkhadaahaa% WcbeqaaiabeU7aSbaakiaadAgadaWgaaWcbaGaeqOVdGhabeaakiaa% cIcacqaH7oaBcaGGSaGaeqiUdeNaaiilaiabfA6agjaacMcaaaa!582B!\[u_\xi = r^\lambda f_\xi (\lambda ,\theta ,\Phi )\],welche homogene Randwerte der Beschreibenden des Kegels entlang genügen.Das Bericht bringt ein Theorem, welches aussagt, das und =–1– beide Eigenwerte sind.

     

Deformation to breaking of deep water gravity waves

The results of laboratory observations of the deformation of deep water gravity waves leading to wave breaking are reported. The specially developed visualization technique which was used is described. A preliminary analysis of the results has led to similar conclusions than recently developed theories. As a main fact, the observed wave breaking appears as the result of, first, a modulational instability which causes the local wave steepness to approach a maximum and, second, a rapidly growing instability leading directly to the breaking.List of symbols L total wave length - H total wave height - crest elevation above still water level - trough depression below still water level - wave steepness =H/L - crest steepness =/L - trough steepness =/L - F ₁ forward horizontal length from zero-upcross point (A) to wave crest - F ₂ backward horizontal length from wave crest to zero-downcross point (B) - crest front steepness =/F ₁ - crest rear steepness =/F ₂ - vertical asymmetry factor=F ₂/F ₁ (describing the wave asymmetry with respect to a vertical axis through the wave crest) - µ horizontal asymmetry factor=/H (describing the wave asymmetry with respect to a horizontal axis: SWL) - T ₀ wavemaker period - L ₀ theoretical wave length of a small amplitude sinusoïdal wave generated at T _inf0 ^sup–1 frequency - ₀ average wave height     

Asymptotics of the homogenized moduli for the elastic chess-board composite

We find the asymptotic behavior of the homogenized coefficients of elasticity for the chess-board structure. In the chess board white and black cells are isotropic and have Lamé constants (, ,) and (, ) respectively. We assume that the black cells are soft, so 0. It turns out that the Poisson ratio for this composite tends to zero with .     

Remarks on the bifurcation of solutions of a nonlinear eigenvalue problem

Conclusions We have investigated solutions of equation (3) when ² is an eigenvalue of the linearized operator (13) and when it is not. In Section 4 we have shown that for 0 and ² = _i ² we have exactly two nontrivial solutions which bifurcate to the right of _i ² ; these solutions are shown to exist in an interval ( _i ² , _i ² + ₀). The method of Section 3 may then be used to extend these two solutions to the right of _i ² + ₀ providing that ²= _i ² + ₀ is not an eigenvalue of the linear operator (13) evaluated at = ± ₁. Either a solution can be uniquely extended, or there exists a value of ²where the bifurcation method must be applied again³.While the method used here gives the exact number of solutions bifurcating from _i ² , other problems remain open; for example, it is still not proven that the two bifurcating branches have i zeros, as is the case for Hammerstein operators with oscillation kernels [4]. The conjecture of Odeh and Tadjbakhsh that there are exactly 2(i+1) nontrivial solutions in the interval _i ² < _i ^+1/2 remains un-answered, although it would be proven if one could show that there is no secondary bifurcation as in the cases of Kolodner [7] and Coffman [8].     

Hydrodynamic instability of the contact zone between accelerated gases

An experimental apparatus for investigating Rayleigh-Taylor instability in the transition layer between two gases at accelerations g 10⁵g₀ (g₀ is the acceleration of gravity) is described. The constantly acting acceleration is communicated to the contact zone by the compression wave formed ahead of a flame front. The linear stage of development is investigated together with the effect of the thickness of the contact zone. It is shown that on the interval 0.3 < <- ( is the wavelength of the disturbance at the edge of the contact zone) the rate of growth of the perturbation amplitude 0.5₀, where ₀ is the amplitude growth rate for media separated by an interface with a discontinuous change of density.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 15–21, November–December, 1991.     

Reservoir transport equations by compositional approach

The objective of this paper is to present an overview of the fundamental equations governing transport phenomena in compressible reservoirs. A general mathematical model is presented for important thermo-mechanical processes operative in a reservoir. Such a formulation includes equations governing multiphase fluid (gas-water-hydrocarbon) flow, energy transport, and reservoir skeleton deformation. The model allows phase changes due to gas solubility. Furthermore, Terzaghi's concept of effective stress and stress-strain relations are incorporated into the general model. The functional relations among various model parameters which cause the nonlinearity of the system of equations are explained within the context of reservoir engineering principles. Simplified equations and appropriate boundary conditions have also been presented for various cases. It has been demonstrated that various well-known equations such as Jacob, Terzaghi, Buckley-Leverett, Richards, solute transport, black-oil, and Biot equations are simplifications of the compositional model.Notation List B reservoir thickness - B formation volume factor of phase - Cⁱ mass of component i dissolved per total volume of solution - C ⁱ mass fraction of component i in phase - C heat capacity of phase at constant volume - C_p heat capacity of phase at constant pressure - D ⁱ hydrodynamic dispersion coefficient of component i in phase - D_MTf thermal liquid diffusivity for fluid f - F = F(x, y, z, t) defines the boundary surface - f_p fractional flow of phase - g gravitational acceleration - H_p enthalpy per unit mass of phase - J_p volumetric flux of phase - k_rf relative permeability to fluid f - k₀ absolute permeability of the medium - M_p ⁱ mass of component i in phase - n porosity - N rate of accretion - P_f pressure in fluid f - p_ca capillary pressure between phases and =p-p - Rⁱ rate of mass transfer of component i from phase to phase - Rⁱ _source source rate of component i within phase - S saturation of phase - s gas solubility - T temperature - t time - U displacement vector - u velocity in the x-direction - v velocity in the y-direction - V volume of phase - V_s velocity of soil solids - W_i body force in coordinate direction i - x horizontal coordinate - z vertical coordinate Greek Letters _p volumetric coefficient of compressibility - _T volumetric coefficient of thermal expansion - _ij Kronecker delta - volumetric strain - _m thermal conductivity of the whole matrix - internal energy per unit mass of phase - gf suction head - density of phase - _ij tensor of total stresses - _ij tensor of effective stresses - volumetric content of phase - f viscosity of fluid f     

The boundary-layer flow past a suddenly heated vertical surface in a saturated porous medium

The boundary-layer flow generated on an impermeable vertical surface in a saturated porous medium is considered in the case when wall heating at a rate proportional tox is switched on at timet=0, (x measures distance along the wall and is a constant). The similarity equations which hold in the limit of larget are discussed and are shown to have a solution only for >–1. The behaviour of the solution as –1 and as is obtained. Numerical solutions of the initial value problem are then obtained for a range of values of . A direct numerical integration is possible for 1, while an iterative procedure is required for <1, with the numerical scheme becoming unstable for =–0.5.

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Grenzschichtströmung an einer plötzlich aufgeheizten vertikalen Fläche, in einem gesättigten porösen Medium

Zusammenfassung Es wird die an einer undurchlässigen, vertikalen Fläche hervorgerufene Grenzschichtströmung im Falle eines Einschalten der Heizung beit=0 betrachtet. Die Stärke der Wandheizung is proportional zux , wobeix die Koordinate längs der Wand ist und eine Konstante. Die Ähnlichkeitsgleichungen werden für den Bereich von großen Zeitent besprochen und es wird gezeigt, daß eine Lösung nur für >–1 vorliegt. Es wird das Verhalten der Lösungen für –1 und erhalten. Numerische Lösungen für die Anfangsbedingungen des Problems werden für eine Reihe von -Werten errechnet. Eine direkte numerische Integration ist für 1 möglich, während für <1 eine Iteration erforderlich ist, wobei das numerische Verhalten für =–0.5 instabil wird.

     

The Gibbs-Thompson relation within the gradient theory of phase transitions

This paper discusses the asymptotic behavior as 0+ of the chemical potentials associated with solutions of variational problems within the Van der Waals-Cahn-Hilliard theory of phase transitions in a fluid with free energy, per unit volume, given by ²¦¦²+ W(), where is the density. The main result is that is asymptotically equal to E/d+o(), with E the interfacial energy, per unit surface area, of the interface between phases, the (constant) sum of principal curvatures of the interface, and d the density jump across the interface. This result is in agreement with a formula conjectured by M. Gurtin and corresponds to the Gibbs-Thompson relation for surface tension, proved by G. Caginalp within the context of the phase field model of free boundaries arising from phase transitions.     

Theoretical derivation of tangential velocity profiles in a flat vortex chamber-influence of turbulence and wall friction

Summary As part of a study on the hydrodynamics of a cyclone separator, a theoretical investigation of the flow pattern in a flat box cyclone (vortex chamber) has been carried out. Expressions have been derived for the tangential velocity profile as influenced by internal friction (eddy viscosity) and wall friction. The most important parameter controlling the tangential velocity profile is = –u ₀ R/(v+ ), where u ₀ is the radial velocity at the outer radius R of the cyclone, the kinematic liquid viscosity and is the kinematic eddy viscosity. For values of greater than about 10 the tangential velocity profile is nearly hyperbolic, for smaller than 1 the tangential velocity even decreases towards the centre. It is shown how and also the wall friction coefficient may be obtained from experimental velocity profiles with the aid of suitable graphs. Because of the close relation between eddy viscosity and eddy diffusion, measurements of velocity profiles in flat box cyclones will also provide information on the eddy motion of particles in a cyclone, a motion reducing its separation efficiency.List of symbols A cross-sectional area of cyclone inlet - h height of cyclone - p static pressure in cyclone - p static pressure difference in cyclone between two points on different radius - r radius in cyclone - r ₁ radius of cyclone outlet - R radius of cyclone circumference - u radial velocity in cyclone - u ₀ radial velocity at circumference of flat box cyclone - v tangential velocity - v ₀ tangential velocity at circumference of flat box cyclone - w axial velocity - z axial co-ordinate in cyclone - friction coefficient in flat box cyclone (for definition see § 5) - ₁ value of friction coefficient for ₁<< ₂ - ₂ value of friction coefficient for ₂<<1 - = - ₁ value of for ₁<< ₂ - ₂ value of for ₂<<1 - thickness of laminar boundary layer - =/h - turbulent kinematic viscosity - ratio of z to h - k ratio of height of cyclone to radius R of cyclone - parameter describing velocity profile in cyclone =–u ₀ R/(+) - kinematic viscosity of fluid - density of fluid - ratio of r to R - ₁ value of at outlet of cyclone - ₂ value of at inner radius of cyclone inlet - _w shear stress at cyclone wall - angular momentum in cyclone/angular momentum in cyclone inlet - ₁ value of at = ₁ - ₂ value of at = ₂     

Gasdynamics of a plasma in a multiple-mirror magnetic trap with “point” mirrors

Equations are derived for the gasdynamics of a dense plasma confined by a multiple-mirror magnetic field. The limiting cases of large and small mean free paths have been analyzed earlier: ₀ and k, where is the length of an individual mirror machine, ₀ is the size of the mirror, and k is the mirror ratio. The present work is devoted to a study of the intermediate range of mean free paths ₀ k. It is shown that in this region of the parameters the process of expansion of the plasma has a diffusional nature, and the coefficients of transfer of the plasma along the magnetic field are calculated.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 14–19, November–December, 1974.The authors thank D. D. Ryutov for the statement of the problem and interest in the work.     

Boundary layer diagnostics by means of an infrared scanning radiometer

A computerized infrared (IR) scanning radiometer is employed to characterize the boundary layer development over a model wing, having a Göttingen 797 cross-section, by measuring the temperature distribution over its heated surface. The Reynolds analogy is used to relate heat transfer measurements to skin friction. The results show that IR thermography is capable of rapidly detecting location and extent of transition and separation regions of the boundary layer over the whole surface of the tested model wing. Thus, the IR technique appears to be a suitable and effective diagnostic tool for aerodynamic research in wind tunnels.List of symbols c airfoil chord - c _f local skin friction coefficient = 2/( V ²) - c _p specific heat coefficient at constant pressure - h local convective heat transfer coefficient - Nu Nusselt number = h x/ - Nu _c Nusselt number based on airfoil chord = h c/ - Pr Prandtl number c _p / - Q _j wall heat flux due to Joule heating - Q _l heat flux loss - Re Reynolds number V x/ - Re _c Reynolds number based on airfoil chord = V c/ - St Stanton number = h/c _p V - T _w wall temperature - T _aw adiabatic wall temperature - V velocity of the free stream - x chordwise spatial coordinate - angle of attack - thermal conductivity coefficient - dynamic viscosity coefficient - mass density - wall shear stress     

Das erweiterte Korrespondenzgesetz für die dynamische Idealviskosität und die Idealwärmeleitfähigkeit reiner Stoffe

Zusammenfassung Zur Berechnung der dynamischen Idealviskosität Ideal (T) und der Idealwärmeleitfähigkeit ideal (T) benötigt man die kritische TemperaturT _kr, das kritische spezifische Volum _kr, die MolmasseM, den kritischen Parameter _kr und die molare isochore WärmekapazitätC _v(T). Sowohl das theoretisch, als auch das empirisch abgeleitete erweiterte Korrespondenzgesetz ergeben eine für praktische Zwecke ausreichende Genauigkeit für die Meßwertwiedergabe, die bei den assoziierenden Stoffen und den Quantenstoffen jedoch geringer ist als bei den Normalstoffen.

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The extended correspondence law for the ideal dynamic viscosity and the ideal thermal conductivity of pure substances

For the calculation of the ideal dynamic viscosity Ideal (T) and the ideal thermal conductivity ideal (T) the critical temperatureT _kr, the critical specific volumev _kr, the molecular massM, the critical parameter _kr, and the molar isochoric heat capacityC _v(T) is needed. Not only the theoretically determined but also the empirically determined extended correspondence law gives for practical use a good representation of the measured data, which for the associating substances and the quantum substances is not so good as for the normal substances.

     

Stochastic response of structures with small geometric imperfections

Summary A probabilistic model of the geometric imperfections of a real structure is proposed, in order to provide a general theory of the stochastic response of structures in presence of small random deviations from the perfect scheme. The main statistical measures of the stochastic response are derived and an application to the study of a particular conservative elastic system is developed.

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Sommario Si propone una teoria generale della risposta probabilistica di strutture, in presenza di piccole deviazioni aleatorie dei dati iniziali rispetto allo schema geometrico perfetto. Si deducono le principali proprietà statistiche della risposta della struttura a sollecitazioni esterne deterministiche, e si sviluppa una applicazione riguardante il comportamento aleatorio di un particolare sistema elastico conservativo.

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List of symbols element of the sample space of events - _kn random variables modelling the structural imperfections - P(o) probability density of random variables - random imperfection of the unloaded structure - u additional displacement of the loaded structure - u_o deterministic fundamental solution for the perfect structure - difference between the additional displacement of the loaded structure and the deterministic fundamental solution for the perfect structure - V₁=u₁ buckling mode of the perfect structure - _i intrinsic coordinates of the structure - suitable measure of the magnitude of the random imperfections - scalar geometric variable representing the internal product - random imperfection divided by - single scalar variable denoting the magnitude of the prescribed loads - potential energy of the structure - potential energy of the perfect structure - difference between and - _c lowest critical load - _s real local maximum for the magnitude of the prescribed loads - _c divided by _S - E{} expected value of a random variable - ² variance of a random variable - , random variables defined by Eq. (21)