Methods of Small and Large δ in the Nonlinear Dynamics – A Comparative Analysis

New asymptotic approaches for dynamical systems containing a power nonlinear term x ⁿ are proposed and analyzed. Two natural limiting cases are studied: n 1 + , 1 and n . In the firstcase, the 'small method' (SM)is used and its applicability for dynamical problems with the nonlinearterm sin as well as the usefulness of the SMfor the problem with small denominators are outlined. For n , a new asymptotic approach is proposed(conditionally we call it the 'large method' –LM). Error estimations lead to the followingconclusion: the LM may be used, even for smalln, whereas the SM has a narrow application area. Both of the discussed approaches overlap all values ofthe parameter n.     

Supersonic flow past axisymmetric flared cones

Systematic data on the determination of the aerodynamic characteristics of axisymmetric bodies with a break in the generating line (Fig. 1a, b) in supersonic flow at zero angle of attack are presented in [1, 2, and others]. A characteristic feature of the flow past such bodies is the appearance of an extensive separation zone dec in the region of the break in the generator when the break angle exceeds some minimum value _min, which for a turbulent boundary layer depends basically on the Mach number M at the body surface ahead of the separation zone. In this case, compression waves which change into the oblique compression shocks dc and cc, emanate both from the beginning of the separation zone (point c) and from the end of it (point d). These shocks, intersecting at the point c, form the triple shock configuration acd and acc for which we introduce the notationac[c, d]. The maximum value (_max) of the generator break angle is limited by the possibility of the existence of an attached compression shock, dc. According to these data a change in the generator break angle for the range _minmax of the angle does not disrupt the nature of the flow in the separation zone, but only alters the size of this zone.We shall examine the flow past cones with values of the generator break angles (_max) for which the attached shock dc cannot exist.     

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 .     

Some characteristic features of diffusion of a magnetic field into a moving conductor

The study of the diffusion of a magnetic field into a moving conductor is of interest in connection with the production of ultra-high-strength magnetic fields by rapid compression of conducting shells [1,2]. In [3,4] it is shown that when a magnetic field in a plane slit is compressed at constant velocity, the entire flux enters the conductor. In the present paper we formulate a general result concerning the conservation of the sum current in the cavity and conductor for arbitrary motion of the latter. We also consider a special case of conductor motion when the flux in the cavity remains constant despite the finite conductivity of the material bounding the magnetic field.Notation ₁, * flux which has diffused into the conductor - ₂ flux in the cavity - ₀ sum flux - r radius - r* cavity boundary - thickness of the skin layer - (r) delta function of r - t time - q intensity of the fluid sink - v velocity - flux which has diffused to a depth larger than r - x self-similar variable - dimensionless fraction of the flux which has diffused to a depth larger than r - * fraction of the flux which has diffused into the conductor - a conductivity - c electrodynamic constant - R_m magnetic Reynolds number - dimensionless parameter     

Schwere symmetrische Kreisel kleiner Nutationen

Zusammenfassung Zur Integration der Eulerschen Bewegungsgleichungen schwerer symmetrischer Kreisel werden der Winkel (t) (Abb. 1) durch (t)=₀+(t) ersetzt und in sämtlichen Reihenentwicklungen von abhängiger Funktionen die Potenzen höheren als zweiten Grades vernachlässigt. Dadurch ist es möglich, die Eulerschen Winkel (t), (t) und (t) durch elementare Formeln zu beschreiben und somit sind die wesentlichsten Erscheinungen im Bewegungsablauf der schweren symmetrischen Kreisel einfach zu übersehen.     

Strömung und Wärmeübergang bei der Oberflächenverdampfung und Filmkondensation

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

---

Flow and heat transfer in surface evaporation and film condensation

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

---

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

An experimental investigation of separating flow on a convex surface

The mean and turbulent characteristics of an incompressible turbulent boundary layer developing on a convex surface under the influence of an adverse pressure gradient are presented in this paper.The turbulence quantities measured include all the components of Reynolds stresses, auto-correlation functions and power spectra of the three components of turbulence. The results indicate the comparative influence of the convex curvature and adverse pressure gradient which are simultaneously acting on the flow. The investigation provides extensive experimental information which is much needed for a better understanding of turbulent shear flows.Nomenclature a, b constants in equation for velocity defect profile (Fig. 6) - c _f skin-friction coefficient (= _w/F _1/2 U ₁ ² ) - E(k ₁) one-dimensional wave number spectra - f frequency in Hz - G Clauser's equilibrium parameter = (H–1)/H(c _f /2) - H shape parameter (= ₁/ ₂) - k ₁ wave number (=2f/U) - L _u, L _v, L _w length scales of u, v and w fluctuations - p _s static pressure on the measurement surface - p _w reference tunnel wall static pressure - q ² total turbulent kinetic energy - R radius of curvature of the convex surface - R() auto-correlation function - T _u, T _v, T _w time scales of u, v and w fluctuations - U local mean velocity - U ₁ local free stream velocity - U _* friction velocity - u, v, w velocity fluctuations in x, y and z directions respectively - X streamwise coordinate measured along the surface from A (Fig. 1b) - x streamwise coordinate measured along the surface reckoned from station 9 - y coordinate normal to the surface - z spanwise coordinate - ₁/ _w · dp/dx - - boundary layer thickness - ₁ displacement thickness - ₂ momentum thickness - ₃ energy thickness - kinematic viscosity - density - time delay - _w wall shear stress     

Eine analytische Näherungslösung für die eingefrorene laminare Grenzschichtströmung eines binären Gemisches

Zusammenfassung Für die eingefrorene laminare Grenzschichtströmung eines teilweise dissoziierten binären Gemisches entlang einer stark gekühlten ebenen Platte wird eine analytische Näherungslösung angegeben. Danach läßt sich die Wandkonzentration als universelle Funktion der Damköhler-Zahl der Oberflächenreaktion angeben. Für das analytisch darstellbare Konzentrationsprofil stellt die Damköhler-Zahl den Formparameter dar. Die Wärmestromdichte an der Wand bestehend aus einem Wärmeleitungs- und einem Diffusionsanteil wird angegeben und diskutiert. Das Verhältnis beider Anteile läßt sich bei gegebenen Randbedingungen als Funktion der Damköhler-Zahl ausdrücken.

---

An analytical approximation for the frozen laminar boundary layer flow of a binary mixture

An analytical approximation is derived for the frozen laminar boundary layer flow of a partially dissociated binary mixture along a strongly cooled flat plate. The concentration at the wall is shown to be a universal function of the Damkohler-number for the wall reaction. The Damkohlernumber also serves as a parameter of shape for the concentration profile which is presented in analytical form. The heat transfer at the wall depending on a conduction and a diffusion flux is derived and discussed. The ratio of these fluxes is expressed as a function of the Damkohler-number if the boundary conditions are known.

---

Formelzeichen A Atom - A₂ Molekül - C Konstante in Gl. (20) - c₁=1/(2C) Konstante in Gl. (35) - c_p spezifische Wärme bei konstantem Druck - D binärer Diffusionskoeffizient - Ec=u ² /(2h_f) Eckert-Zahl - h spezifische Enthalpie - h_t=h+u²/2 totale spezifische Enthalpie - h _A ⁰ spezifische Dissoziationsenthalpie - K_w Reaktionsgeschwindigkeitskonstante der heterogenen Wandreaktion - 1= /( ) Champman-Rubesin-Parameter - Le=Pr/Sc Lewis-Zahl - M Molmasse - p statischer Druck - Pr= c_pf/ Prandtl-Zahl - q_w Wärmestromdichte an der Wand - q_cw, q_dw Wärmeleitungsbzw. Diffusionsanteil der Wärmestromdichte an der Wand - universelle Gaskonstante - R=/(2M_a) individuelle Gaskonstante der molekularen Komponente - Re_x= u x/ Reynolds-Zahl - Sc=/( D) Schmidt-Zahl - T absolute Temperatur - T_d=h _A ⁰ /R charakteristische Dissoziationstemperatur - u, v x- und y-Komponenten der Geschwindigkeit - U=u/u normierte x-Komponente der Geschwindigkeit - x, y Koordinaten parallel und senkrecht zur Platte Griechische Symbole - =A/ Dissoziationsgrad - Grenzschichtdicke - ₂ Impulsverlustdicke - Damköhler-Zahl der Oberflächenreaktion - =T/T normierte Temperatur - =y/ normierter Wandabstand - Wärmeleitfähigkeit - dynamische Viskosität - , ^* Ähnlichkeitskoordinaten - Dichte - Schubspannung Indizes A auf ein Atom bezogen - M auf ein Molekül bezogen - f auf den eingefrorenen Zustand bezogen - w auf die Wand bezogen - auf den Außenrand der Grenzschicht bezogen     

A priori Estimates and Existence for Elliptic Systems via Bootstrap in Weighted Lebesgue Spaces

We present a new general method to obtain regularity and a priori estimates for solutions of semilinear elliptic systems in bounded domains. This method is based on a bootstrap procedure, used alternatively on each component, in the scale of weighted Lebesgue spaces L^p()=L^p((x)dx), where (x) is the distance to the boundary. Using this method, we significantly improve the known existence results for various classes of elliptic systems.     

Prediction of stagnation point heat transfer for a single round jet impinging on a concave hemispherical surface

This paper deals with a systematic procedure for assessment of fluid flow and heat transfer parameters for a single round jet impinging on a concave hemispherical surface. Based on Scholkemeier's modifications of the Karman-Pohlhausen integral method, expressions are derived for evaluation of the momentum thickness, boundary layer thickness and the displacement thickness at the stagnation point. This is followed by the estimation of thermal boundary layer thickness and local heat transfer coefficients. A correlation is presented for the Nusselt number at the stagnation point as a function of the Reynolds number for different non-dimensional distances from the exit plane of the jet to the impingement surface.

---

Bestimmung des Staupunktes bei der Wärmeübertragung für einen einzelnen Strahl, der auf eine konkave halbkugelige Oberfläche trifft

Zusammenfassung Diese Arbeit beschäftigt sich mit dem systematischen Verfahren der Bewertung von Fluidströmungen und Wärmeübertragungsparametern für einen einzelnen runden Strahl, der auf eine konkave halbkugelförmige Oberfläche trifft. Das Verfahren beruht auf Scholkemeiers Modifikation des Karman-Pohlhausen Integrationsverfahrens. Ausdrücke sind für die Berechnung der Impuls-Dicke, der Grenzschichtdicke und der Verschiebungsdicke am Staupunkt hergeleitet worden. Dies ist aus der Berechnung der thermischen Grenzschichtdicke und des lokalen Wärmeübertragungskoeffizienten abgeleitet worden. Es wird eine Gleichung für die Nusselt-Zahl am Staupunkt als Funktion der Reynolds-Zahl für verschiedene dimensionslose Abstände vom Strahlaustrittspunkt bis zum Auftreffpunkt auf die Oberfläche vorgestellt.

---

Nomenclature c _p specific heat at constant pressure - d diameter of single round nozzle - h ₀ heat transfer coefficient at the stagnation point - H distance from the exit plane of the jet to the impingement surface - k thermal conductivity - Nu _0.5 Nusselt number based on impinging jet quantities=h _0.50/k - Nu _{0.5, 0} stagnation point Nusselt number=h _{0 0,50}/k - p pressure - p _a ambient pressure - p ₀ maximum pressure or stagnation pressure - p(x) static pressure at a distancex from the stagnation point - R radius of curvature of the hemisphere - Re _J jet Reynolds number=U _Jd/ - Re _0.5 Reynolds number based on impinging jet quantities=u _{m0 0.50}/ - T temperature - T _a room temperature - T _J jet temperature - T _W wall temperature - u velocity component inx andx directions (Fig. 1) - u _m jet centerline (or maximum) free jet velocity: external (or maximum) boundary layer velocity aty= _m - u _m0 arrival velocity defined as the maximum velocity the free jet would have at the plane of impingement if the plane were not there - U _J jet exit velocity - x* non-dimensional coordinate starting at the stagnation point=x/2 _0.50 - x, y rectangular Cartesian coordinates - y coordinate normal to the wall starting at the wall - ratio of thermal to velocity boundary layer thickness= _T/_m - ₀ ratio of thermal to velocity boundary layer thickness at the stagnation point - * inner layer displacement thickness - _0.50 jet half width at the plane of impingement if the plate were not there - _m inner boundary layer thickness atu=u _m - Pohlhausen's form parameter - dynamic viscosity - kinematic viscosity=/ - fluid density - momentum thickness - ₀ momentum thickness at the stagnation point     

A similarity law for acute circular cones at large angles of attack in a supersonic flow

For thin bodies placed in a hypersonic flow at a small angle of attack the similarity law is known. From this law it follows that for various numbers M, angles of attack , and relative thicknesses the similarity conditions will be observed if in the flows under consideration the parameters M and / are the same. This similarity law is obtained with the assumption M 1, 1. But even for M=3 and 1/3 the results of solving the complete system of gasdynamic equations for affino-similar bodies is in a good agreement with the similarity law [1], In [2] it is shown that this similarity law is generalized for the case of a flow around a thin pointed body at large angles of attack. According to the similarity law, at large angles of attack the flows near bodies with an identical distribution of cross-sectional shapes will be similar if the parameters K₁= cotan and K₂=m sin for all cases have one and the same value. As the angle of attack decreases, the requirements of constancy of K₁ and K₂ become analogous to the conditions M=const, /=const.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 78–83, May–June, 1976.The authors thank V. V. Lunev for the useful discussions and valuable observations.     

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.     

A turbulent immersed air wall jet on a burning graphite surface

The turbulent boundary layer in an immersed air jet traveling along a burning graphite wall is analyzed. In order to study the heat and mass transfer and the friction in the boundary layer, a method of calculation based on the solution of the integrated energy and momentum relations is employed, allowing for the conservative properties of the turbulent boundary layer at the wall [1].Notation x longitudinal coordinate - y transverse coordinate - s height of slot - thickness of boundary layer - * displacement thickness - ** momentum-loss thickness - velocity - j transverse mass flow (flux) - density - T temperature - i total enthalpy - temperature factor - ₁ enthalpy factor - k reduced concentration of the i-th component - tangential stress - b permeability parameter - C₁ form (shape) parameter - dynamic viscosity coefficient - c _f friction coefficient - relative friction coefficient Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 60–67, January–February, 1971.     

Prediction of stagnation point heat transfer for a slot jet impinging on a concave semicylindrical surface

A systematic procedure has been laid out for assessment of fluid flow and heat transfer parameters for a slot jet impinging on a concave semicylindrical surface. Based on Walz's modifications of the Karman-Pohlhausen integral method, expressions have been derived for evaluation of the momentum thickness, boundary layer thickness and the displacement thickness at the stagnation point. The work then has been extended for the estimation of thermal boundary layer thickness and local heat transfer coefficients. A correlation has been presented for the Nusselt number at the stagnation point as a function of the Reynolds number for different non-dimensional distances from the exit plane of the jet to the impingement surface.

---

Berechnung des Wärmeübergangs im Staupunkt eines Strahles, der aus einer rechteckigen öffnung auf eine konkave halbzylindrische Fläche auftrifft

Zusammenfassung Es wurde eine systematische Prozedur für die Abschätzung von Strömungs- und Wärmeübergangsparametern für einen Strahl, der auf eine konkave halbzylindrische Fläche auftrifft, aufgestellt. Basierend auf Walz's Modifikationen der Karman-Pohlhausen Integral-Methode, wurden Ausdrücke für die Berechnung der Impulsdicke, der Grenzschichtdicke und die Versetzungsdicke am Staupunkt abgeleitet. Die Arbeit wurde dann auf die Abschätzung der thermischen Grenzschichtdicke und der lokalen Wärmeübertragungskoeffizienten ausgedehnt. Es wird eine Beziehung für die Nusselt-Zahl am Staupunkt als eine Funktion der Reynolds-Zahl für verschiedene dimensionslose Abstände von der Austrittsfläche des Schlitzes bis zur Aufprallfläche aufgestellt.

---

Nomenclature c _p specific heat at constant pressure - h ₀ heat transfer coefficient at the stagnation point - H distance from the exit plane of the jet to the impingement surface - k thermal conductivity - Nu _.5 Nusselt number based on impinging jet quantities =h _0.50/k - Nu _.5,0 stagnation point Nusselt number =h _{0 0.50}/k - p pressure - p _a ambient pressure - p ₀ maximum pressure or stagnation pressure - p(x) static pressure at a distancex from the stagnation point - p(x*) static pressure at nondimensional distancex* from the stagnation point - Re _J jet Reynolds number =U _J W/ - Re _0.5 Reynolds number based on impinging jet quantities =u _{m0 0.50}/ - T temperature - T* nondimensional temperature =(T–T _W)/(T _J–T _W) - T _a room temperature - T _J jet temperature - T _W wall temperature - u velocity component inx andx directions - u _m jet centerline (or maximum) free jet velocity: external (or maximum) boundary layer velocity aty = _m - u _m0 arrival velocity defined as the maximum velocity the free jet would have at the plane of impingement if the plane were not there - U _J jet exit velocity - W jet nozzle width - x* nondimensional coordinate starting at the stagnation point =x/2 _0.50 - x, y rectangular cartesian coordinates - y coordinate normal to the wall and starting at the wall - ratio of thermal to velocity boundary layer thickness = _T/ _m - ₀ ratio of thermal to velocity boundary layer thickness at the stagnation point - * inner layer displacement thickness - _.50 jet half width at the plane of impingement if the plate were not there - d_.5 free jet (half width) thickness whereu=u _m/2 - _m inner boundary layer thickness atu =u _m - _T thermal boundary layer thickness - nondimensional coordinate normal to wall =y/ _m - _T nondimensional coordinate normal to wall =y/ _T - Pohlhausen's form parameter - dynamic viscosity - kinematic viscosity = / - fluid density - momentum thickness - ₀ momentum thickness at the stagnation point     

Peristaltic pumping of a second-order fluid in a planar channel

The peristaltic motion of a non-Newtonian fluid represented by the constitutive equation for a second-order fluid was studied for the case of a planar channel with harmonically undulating extensible walls. A perturbation series for the parameter ( half-width of channel/wave length) obtained explicit terms of 0(²), 0(²Re²) and 0(₁Re²) respectively representing curvature, inertia and the non-Newtonian character of the fluid. Numerical computations were performed and compared to the perturbation analysis in order to determine the range of validity of the terms.Presented at the second conference Recent Developments in Structured Continua, May 23–25, 1990, in Sherbrooke, Québec, Canada     

ON THE UNIFICATION OF THE HAMILTON PRINCIPLES IN NONHOLONOMIC SYSTEM AND IN HOLONOMIC SYSTEM

ＯＮＴＨＥＵＮＩＦＩＣＡＴＩＯＮＯＦＴＨＥＨＡＭＩＬＴＯＮＰＲＩＮＣＩＰＬＥＳＩＮＮＯＮＨＯＬＯＮＯＭＩＣＳＹＳＴＥＭＡＮＤＩＮＨＯＬＯＮＯＭＩＣＳＹＳＴＥＭ（梁立孚）（韦扬）ＯＮＴＨＥＵＮＩＦＩＣＡＴＩＯＮＯＦＴＨＥＨＡＭＩＬＴＯＮＰＲＩＮＣＩＰＬ...     

Zum Impuls-, Wärme- und Stoffaustausch in turbulenten Strömungen reagierender Binärgemische

Zusammenfassung Die in Teil I vorgestellten Reynolds 'schen Gleichungen und Transportgleichungen werden für Strömungen mit Grenzschichtcharakter angegeben. Weiter werden Integralbedingungen mitgeteilt. Nach einer Diskussion über die Schließung des Gleichungssystems werden Lösungsverfahren besprochen. Dabei wird speziell auf Integralverfahren eingegangen.

---

About the transfer of momentum, heat and mass in turbulent flows of binary mixturesPart II: Thin shear flow layers

The Reynolds equations and transport equations given in part I are presented for thin shear flow layers. Integral relations are given. After a discussion of the closure problem methods of solution are described. Specially integral methods are discussed.

---

Formelzeichen c Massenkonzentration der Komponente - c_t charakteristische Konzentrationsschwankung - c_o Bezugskonzentration - c spezifische Wärme bei konstantem Druck - c_f Reibungsbeiwert - c_D Dissipationsintegral - c_E Entrainment-Funktion - c Schubspannungsintegral - D binsrer Diffusionskoeffizient - H Formparameter - H₁₂ Formparameter - H₃₂ Formparameter - j Kassendiffusionsstrom - L Bezugslänge - p Druck - p_t charakteristische Druckschwankung - p_o Bezugsdruck - Pr Prandtl-Zahl - q Wärmestrom - q²/2 kinetische Energie der Schwankungsbewegung - Re_L mit L gebildete Reynolds-Zahl - Re mit gebildete Reynolds-Zahl - Re₂ mit ₂ gebildete Reynolds-Zahl - Sc Schmidt-Zahl - T absolute Temperatur - T_t charakteristische TemperaturSchwankung - T_o Bezugstemperatur - u,v,w Geschwindigkeitskomponenten - u_t charakteristische Geschwindigkeitsschwankung - u_o Bezugsgeschwindigkeit - U=/ü dimensionslose. x-Komponente der Geschwindigkeit - x,y,z Komponenten des Ortsvektors Griechische Symbole Grenzschichtdicke - ₁ Verdrängungsdicke - ₂ Impulsverlustdicke - ₃ Energieverlustdicke - _T Enthalpieverlustdicke - _c Konzentrationsverlustdicke - =d/dx Parameter für die Grenzschichtabsch:atzung - turbulente Impulsaustauschgröße - _D turbulente Stoffaustauschgröße - _q turbulente Energieaustauschgröße - Dissipationsfunktion - Wärmeleitfähigkeit - dynamische Viskosität - v=/ kinematische Viskosität - Dichte - Produktionsdichte - Schubspannung Indizes mol molekularer Anteil - tur turbulenter Anteil - res resultierender Anteil - Außenrand der Grenzschicht - w Wand     

Method of mass-average quantities for the boundary layer in an outer flow with transverse nonuniformity

An engineering method is proposed for calculating the friction and heat transfer through a boundary layer in which a nonuniform distribution of the velocity, total enthalpy, and static enthalpy is specified across the streamlines at the initial section x₀. Such problems arise in the vortical interaction of the boundary layer with the high-entropy layer on slender blunt bodies, with sudden change of the boundary conditions for an already developed boundary layer (temperature jump, surface discontinuity), and in wake flow past a body, etc.Notation x, y longitudinal and transverse coordinates - u,, H, h gas velocity, stream function, total and static enthalpy - p,,, pressure, density, viscosity, Prandtl number - , q friction and thermal flux at the body surface - r(x), (x) body surface shape and boundary layer thickness - V, M freestream velocity and Mach number - u⁽⁰⁾(x₀,), H⁽⁰⁾(x₀,), h⁽⁰⁾(x₀,) parameter distributions at initial section - u⁽⁰⁾(x,), h⁽⁰⁾(x,), h⁽⁰⁾(x,) profiles of quantities in outer flow in absence of friction and heat transfer at the surface of the body The indices v=0, 1 relate to plane and axisymmetric flows - , w, b, relate to quantities at the outer edge of the inner boundary layer, at the body surface in viscid and nonviscous flows, and in the freestream, respectively. The author wishes to thank O. I. Gubanov, V. A. Kaprov, I. N. Murzinov, and A. N, Rumynskii for discussions and assistance in this study.     

Saint-Venant's principle in nonlinear plane elasticity with sufficiently small strains

A homogeneous, isotropic cylinder in an equilibrium state of plane strain, whose cross-section is a rectangle R : [0 < y ₁ < 2L; 0 < y ₂ < h] with h/L 1, is considered. There are no body forces and the long sides are stress free. At y ₁ = 0 and y ₁ = 2L, there are arbitrary loadings, each statically equivalent to a uniformly distributed tensile or compressive stress c. Within the theory of nonlinear elasticity and with the strains and strain gradients assumed to be sufficiently small (but with no such assumptions on the displacement gradients), it is proved that if (,=1,2) represents the Cauchy stress tensor and the Kronecker delta, then |–c₁₁| decays exponentially to zero in R with distance from the nearer end, and the decay constant depends only upon the material but is independent of L.     

The motion of a detaching elastic body

Conclusions The qualitative behavior of the displacement (t) and the radius R(t) during the different phases of the motion is illustrated in the diagram of Fig. 6.1.After the first impact at t = 0 the displacement (t) varies according to (5.2). If the first maximum of (t) is higher than ₁ then at time t ₁ the graph of (t) intersects the straight line = _cand detachment first occurs. In the second phase the dependance of on t is expressed by (5.6). The detachment will end at the instant t ₂ when vanishes.The radius R remains equal to R ₀ until (t) reaches the critical value ₁ = _c that is at t = t ₁. After t ₁, R(t) will decrease according to (4.4) up to its final value ₂.A rather unexpected property of the solution is that the greatest elongation is finite for every non-vanishing value of the ratio .To Jerry Ericksen for his 60^th birthday