Thin-Walled Beams: the Case of the Rectangular Cross-Section

In this paper we present an asymptotic analysis of the three-dimensional problem for a thin linearly elastic cantilever =×(0,l) with rectangular cross-section of sides and ², as goes to zero. Under suitable assumptions on the given loads, we show that the three-dimensional problem converges in a variational sense to the classical one-dimensional model for extension, flexure and torsion of thin-walled beams. Mathematics Subject Classifications (2000) 474K20, 74B10, 49J45.     

Lower semicontinuity of the attractor for a singularly perturbed hyperbolic equation

For a smooth, bounded domain R, n 3, and a real, positive parameter, we consider the hyperbolic equationu _tt +u _t –u=–f(u) –g in with Dirichlet boundary conditions. Under certain conditions onf, this equation has a global attractorA inH ₀ ¹ () ×L ²(). For=0, the parabolic equation also has a global attractor which can be naturally embedded into a compact setA ₀ inH ₀ ¹ () ×L ²(). If all of the equilibrium points of the parabolic equation are hyperbolic, it is shown that the setsA are lower semicontinuous at=0. Moreover, we give an estimate of the symmetric distance betweenA ₀ andA .     

Asymptotic analysis of equilibrium states for rotating turbulent flows

The equilibrium states of homogeneous turbulence simultaneously subjected to a mean velocity gradient and a rotation are examined by using asymptotic analysis. The present work is concerned with the asymptotic behavior of quantities such as the turbulent kinetic energy and its dissipation rate associated with the fixed point (/kS)=0, whereS is the shear rate. The classical form of the model transport equation for (Hanjalic and Launder, 1972) is used. The present analysis shows that, asymptotically, the turbulent kinetic energy (a) undergoes a power-law decay with time for (P/)<1, (b) is independent of time for (P/)=1, (c) undergoes a power-law growth with time for 1<(P/)<(C ₂–1), and (d) is represented by an exponential law versus time for (P/)=(C ₂–1)/(C ₁–1) and (/kS)>0 whereP is the production rate. For the commonly used second-order models the equilibrium solutions forP/,II, andIII (whereII andIII are respectively the second and third invariants of the anisotropy tensor) depend on the rotation number when (P/kS)=(/kS)=0. The variation of (P/kS) andII versusR given by the second-order model of Yakhot and Orzag are compared with results of Rapid Distortion Theory corrected for decay (Townsend, 1970).     

An absence of a certain class of periodic solutions in the Navier-Stokes equations

In the case of the 3D Navier-Stokes equations, it is proved that there exists a constant>0 with the following property: Every time-periodic solution with a period smaller than is necessarily a stationary solution. An explicit formula for is also provided.     

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.     

Heteroclinic orbits in singular systems: A unifying approach

We consider singularly perturbed systems , such that=f(, _o, 0). _o ^m , has a heteroclinic orbitu(t). We construct a bifurcation functionG(, ) such that the singular system has a heteroclinic orbit if and only ifG(, )=0 has a solution=(). We also apply this result to recover some theorems that have been proved using different approaches.     

Discrete time semigroup transformations with random perturbations

Let (X, ) and (Y,C) be two measurable spaces withX being a linear space. A system is determined by two functionsf(X): X X and:X×YX, a (small) positive parameter and a homogeneous Markov chain {y _n } in (Y,C) which describes random perturbations. States of the system, say {x _n X, n=0, 1,}, are determined by the iteration relations:x _n+1 =f(x _n )+(x _n ,Y_n+1) forn0, wherex ₀ =x ₀ is given. Here we study the asymptotic behavior of the solutionx _n as 0 andn under various assumptions on the data. General results are applied to some problems in epidemics, genetics and demographics.Supported in part by NSF Grant DMS92-06677.Supported in part by NSF Grant DMS93-12255.     

Numerical analysis of the transient conjugated heat transfer in a circular duct with a power-law fluid

We treat numerically in this paper, the transient analysis of a conjugated heat transfer process in the thermal entrance region of a circular tube with a fully developed laminar power-law fluid flow. We apply the quasi-steady approximation for the power-law fluid, identifying the suitable time scales of the process. Thus, the energy equation in the fluids is solved analytically using the well-known integral boundary layer technique. This solution is coupled to the transient energy equation for the solid where the transverse and longitudinal heat conduction effects are taken into account. The numerical results for the temporal evolution of the average temperature of the tube wall, _av, is plotted for different nondimensional parameters such as conduction parameter, , the aspect ratios of the tube, and ₀ and the index of power-law fluid, n.     

Combined effect of external turbulence and strong acceleration on boundary layer transition in a nozzle

The effect of external turbulence on the boundary layer flow in a convergent-divergent nozzle with a high expansion ratio has been studied numerically. The external turbulence was simulated by the turbulent viscosity _e, for which we used the partial differential equation that serves to close the system of boundary layer equations [1–4]. It was found that there exists a critical value _cr such that for all _e< _cr the flow regime in the nozzle remains perfectly laminar, whereas for _ecr a laminar-turbulent transition takes place and the boundary layer in the supersonic part of the nozzle becomes turbulent. For postcritical values of _e the heat fluxes and friction losses are approximately an order greater than for precritical values. With increase in the Reynolds number, determined from the parameters in the nozzle throat, the value of _cr decreases; as the coordinate of the onset of boundary layer formation is displaced in the direction of flow the value of _cr increases.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 34–37, January–February, 1906.The authors are grateful to L. V. Gogish for participating in the discussion of the results.     

The bipore model in solid-liquid extraction: the batch process

This paper presents the exact analytical solution for the general case of transient mass transfer between a solid with a biporous structure (with a micro and a macroporosity) and the entouring finite fluid. The transport inside the solid is by molecular diffusion and outside of it the convective film resistance is included. A general expression is given which is valid for the infinite plate, for the infinite cylinder and for the sphere. The standard monopore case is obtained as a particular solution.

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Das Bipor-Modell in der fest-flüssig Extraktion: Das diskontinuierliche Verfahren

Zusammenfassung Es wird die exakte analytische Lösung für den allgemeinen Fall der instationären Stoffübertragung zwischen einem Festkörper mit biporöser Struktur (bestehend aus einer Mikro- und einer Makroporosität) und dem äußeren Fluid vorgestellt. Der Transport in dem Feststoff erfolgt mittels molekularer Diffusion. Außerhalb der Feststoffpartikel wird der konvektive Filmwiderstand berücksichtigt. Eine allgemeine Formel wird angegeben, die für die unendliche Platte, für den unendlichen Zylinder und für die Kugel anwendbar ist. Die Lösung für das übliche monopore Modell ergibt sich als Sonderfall.

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Nomenclature c _a concentration of liquid in micropores - c _b concentration of liquid in macropores - ¯c average concentration in the particle - c₁ initial value of ¯c - c _e concentration in liquid outside the particle - c _e1 initial value ofc _e - D _a ,D _b effective diffusivity in the micro resp. in the macro structure limit ofE for infinite time - f _n form-function defined in Eqs. (20), (21) and (22) - F _n function defined in Eq. (33) - f, g,h Laplace transforms ofc _a ^* ,c _b/* and ¯c^* resp. - I ₀ ,I ₁ modified Bessel functions of the first kind, order zero and first order resp. - J ₀ ,J ₁ Bessel functions of the first kind, order zero and first order resp. - k _c mass transfer coefficient - M _p mass of the solid particles - n numerical form constant, 1 for the plate, 2 for the cylinder and 3 for the sphere - N function defined in Eq. (19) - R _a radius of the microporus spheres - R _b size of the particle (for the plate2R _b is its thickness, for cylinder and sphere: the radius) - r radial coordinate inside the microporous sphere - r ^* =r/R _a adimensional forrt time - t ^* -t/ _b adimensional for time (Fourier Number) - V volume of fluid phase (exterior to solid) - x position coordinate inside the solid particle - x ^* =x/R _b adimensional forx - =(M_pp)/(V_p) volume of fluid inside the particles divided by volume of fluid outside - y=(R _b k _c )/D _b adimensional for the mass transfer coefficient - _a mircoporosity - _b microporosity - _p = _b + (1 _b ) _a total porosity of the particle - =_p/_b – 1=(1 -_b @#@) ( _a / _b ) adimensional parameter, characteristic for the biporous structure - _p density of particle - _a =R _a/2 / (D _a / _a) characteristic (micro) time - _b =R _b/2 / (D _b / _b) characteristic (macro) time - = _a / _b adimensional parameter, characteristic for the biporous particle     

Positively invariant regions for a problem in phase transitions

Positively invariant regions for the system v _t + p(W) _x = V _xx , W _t –V _x = W _xx are constructed where p < 0, w < , w > , p(w) = 0, w , > 0. Such a choice of p is motivated by the Maxwell construction for a van der Waals fluid. The method of an analysis is a modification of earlier ideas of Chueh, Conley, & Smoller [1]. The results given here provide independent L bounds on the solution (w, v).Dedicated to Professor James Serrin on the occasion of his sixtieth birthday     

Zur Variationsformulierung nichtlinearer Randwertprobleme

Übersicht Es werden verschiedene Bedingungen aufgestellt, die es erlauben, die durch die beiden (Systeme von) nichtlinearen DifferentialgleichungenA (u, ) = q, B (u, ) = und Randbedingungen zusammen mit den nichtlinearen algebraischen Relationenq = C(u, ), = D(u, ) beschriebene Aufgabe durch äquivalente Variationsprobleme zu ersetzen. Dabei zeigt sich ein enger Zusammenhang mit den in der Festkörpermechanik wohlbekannten Prinzipien der virtuellen Verschiebungen und der virtuellen Kräfte. Die auf systematischem Weg konstruierten Variationsfunktionale enthalten viele in der Physik bekannte Funktionale als Sonderfälle, insbesondere jene, die in der Elastomechanik nach Green, Castigliano, Hellinger, Reißner, Hu und Washizu benannt werden.

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Summary In this paper there are established various conditions which allow a variational formulation of the problem described by the two (systems of) nonlinear differential equationsA(u, ) = q, B(u, ) = and boundary conditions together with the nonlinear algebraic relationsq = C(u, ), = D(u, ). Besides a close relationship is revealed to the principles of virtual displacements and virtual forces which are wellknown in solid mechanics. The systematically constructed variational functional contain many functionals in physics as special cases, mainly those of Green, Castigliano, Hellinger, Reißner, Hu and Washizu in elastomechanics.

     

Application of dispersion theory to time domain reflectometry in soils

With time domain reflectometry (TDR) two dispersive parameters, the dielectric constant, , and the electrical conductivity, can be measured. Both parameters are nonlinear functions of the volume fractions in soil. Because the volume function of water ( _w) can change widely in the same soil, empirical equations have been derived to describe these relations. In this paper, a theoretical model is proposed based upon the theory of dispersive behaviour. This is compared with the empirical equations. The agreement between the empirical and theoretical aproaches was highly significant: the ( _w) relation of Topp et al. had a coefficient of determination r ² = 0.996 and the (_u) relation of Smith and Tice, for the unfrozen water content, _u, at temperatures below 0°C, had an r ² = 0.997. To obtain ( _w) relations, calibration measurements were performed on two soils: Caledon sand and Guelph silt loam. For both soils, an r ² = 0.983 was obtained between the theoretical model and the measured values. The correct relations are especially important at low water contents, where the interaction between water molecules and soil particles is strong.     

Emissivities and spectral absorptivities of combustion products

This paper is concerned with the determination of the spectral absorptivities K and emissivities of a thermodynamic-equilibrium mixture of combustion products consisting of CO₂, H₂O, CO, OH, HCl, HF, H₂, and NO molecules on the temperature interval 2000–5000° K. The proposed calculation method, based on the use of a graph of the molecular absorptivities reduced to atmospheric pressure, enables K and to be calculated for any gas mixture composed of the above-mentioned molecules.     

Scheme of Investigation of the Spectrum of a Family of Perturbed Operators and Its Application to Spectral Problems in Thick Junctions

We develop a scheme for the investigation of the asymptotic behavior of eigenvalues and eigenvectors of a family of self-adjoint compact operators {A: > 0} that act in different spaces and lose their compactness in the limit case 0. We prove the Hausdorff convergence of the spectrum of the operator A to the spectrum of the limit operator A₀, obtain asymptotic estimates for this convergence both to points of the discrete spectrum and to points of the essential spectrum of the operator A₀, and prove asymptotic estimates for eigenvectors of A. This scheme is applied to the investigation of the asymptotic behavior of eigenvalues and eigenfunctions of the Neumann problem in a thick singularly degenerate junction that consists of two domains connected by an -periodic system of thin rods of fixed length.     

Flow visualization and analysis of aerodynamically detuned supersonic rotors

The effect of aerodynamic detuning on the supersonic steady and unsteady blade passage flow field is experimentally investigated on a free surface water table by means of color Schlieren and shadowgraph flow visualization techniques. Two aerodynamic detuning mechanisms are considered: (1) alternate circumferential spacing of adjacent airfoils; (2) the replacement of alternate airfoils with splitters. The steady flow visualization demonstrates the significant effect of aerodynamic detuning on the passage flow field and, in particular, the shock wave-airfoil surface intersection locations. The unsteady flow visualization studies show the importance of the interblade phase angle. A mathematical model is also described and utilized to demonstrate the enhanced aeroelastic stability associated with the altered cascade passage shock wave structure due to these aerodynamic detuning mechanisms.List of symbols a dimensionless perturbation sonic velocity - C airfoil chord - I _x mass moment of inertia - k reduced frequency, k = C/2 - K spring constant - M _R dimensionless unsteady aerodynamic moment - M Mach number - P _split split splitter circumferential spacing - P _start splitter leading edge location - R reference full chord airfoil - R _s reference splitter - Sp _chord splitter chord length - u dimensionless perturbation chordwise velocity - v dimensionless perturbation normal velocity - amplitude of oscillation - interblade phase angle - level of aerodynamic detuning - undamped natural torsional frequency - ₀ reference frequency - flutter frequency     

Unsteady effects in a thin viscous shock layer near a three-dimensional stagnation point of a body moving with given acceleration and deceleration

An unsteady viscous shock layer near a stagnation point is studied. The Navier-Stokes equations are analyzed in the limit 1, Re₀ , df/dt = ^n-mF(t/^m). The Reynolds number Re₀ is defined in the paper by Eq. (1.3) (df/dt is the velocity of the body with respect to an inertial frame of reference moving with the original steady velocity –V'_t8, ₂ = ( – 1)/( + 1)). Various flow regimes in the case 1, l, n max(2m, m + 1), m 0, where ² = 1/Re₀ are analyzed. Equations are derived that generalize the asymptotic analysis to the case of a viscous unsteady flow of gas in a thin three-dimensional shock layer. The problem of a thin unsteady viscous shock layer near the stagnation point of a body with two curvatures is formulated. Examples of numerical solution are given for different ratios of the principal curvatures of the body, the wall temperature, the parameters of the original steady flow, and the acceleration and deceleration regimes.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 100–111, March–April, 1981.I thank Yu. D. Shevelev for a fruitful discussion of the work.     

Investigations of mean and turbulent flow characteristics of a two dimensional plane diffuser

This paper reports the investigation of mean and turbulent flow characteristics of a two-dimensional plane diffuser. Both experimental and theoretical details are considered. The experimental investigation consists of the measurement of mean velocity profiles, wall static pressure and turbulence stresses. Theoretical study involves the prediction of downstream velocity profiles and the distribution of turbulence kinetic energy using a well tested finite difference procedure. Two models, viz., Prandtl's mixing length hypothesis and k- model of turbulence, have been used and compared. The nondimensional static pressure distribution, the longitudinal pressure gradient, the pressure recovery coefficient, percentage recovery of static pressure, the variation of U _max/U _bar along the length of the diffuser and the blockage factor have been valuated from the predicted results and compared with the experimental data. Further, the predicted and the measured value of kinetic energy of turbulence have also been compared. It is seen that for the prediction of mean flow characteristics and to evaluate the performance of the diffuser, a simple turbulence model like Prandtl's mixing length hypothesis is quite adequate.List of symbols C ₁ , C ₂ ,C turbulence model constants - F _x body force - k kinetic energy of turbulence - l _m mixing length - L length of the diffuser - u, v, w rms value of the fluctuating velocity - u, v, w turbulent component of the velocity - mean velocity in the x direction - _A average velocity at inlet - U _bar average velocity in any cross section - U _max maximum velocity in any cross section - V mean velocity in the y direction - W local width of the diffuser at any cross section - x, y coordinates - dissipation rate of turbulence - _m eddy diffusivity - Von Karman constant - mixing length constant - _l laminar viscosity - _eff effective viscosity - v kinematic viscosity - density - _k effective Schmidt number for k - effective Schmidt number for - stream function - non dimensional stream function     

Ordinary differential equations (ODEs) on inertial manifolds for reaction-diffusion systems in a singularly perturbed domain with several thin channels

Dynamics of solutions to a reaction-diffusion system in a domain of specific shape is investigated under the homogeneous Neumann boundary conditions. It is assumed that the domain hasN large regionsD _i ,i=1,...,N, and thin channelsQ _i,j () connectingD _i andD _j , which approach a line segment as 0 in some sense. In such a domain the firstN eigenvalues of – with the Neumann boundary conditions tend to zero as 0, while the (N + 1)-th eigenvalue is bounded away from zero. By virtue of this gap of the eigenvalues, an inertial manifold which is invariant and attracts every solution exponentially can be constructed under a certain condition. Moreover, the ODE describing the dynamics on the inertial manifold can be given in quite an explicit form through the analysis of the limit of the manifold as 0.     

Messung des binären Diffusionskoeffizienten in einem Entmischungssystem mit Hilfe der Photonen-Korrelationsspektroskopie

Zusammenfassung Die dynamische Lichtstreuung in Form der Photonen-Korrelationsspektroskopie wird in einem Entmischungssystem beispielhaft zur Messung des binären Diffusionskoeffizienten eingesetzt. Mit einem Versuchsaufbau wird nahe dem kritischen Entmischungspunkt mit der Homodyn-Methode und weit davon entfernt mit der Heterodyn-Methode gearbeitet. Ein Verfahren ermöglicht die Korrektur der Störeinflüsse des Heterodyn-Anteils bei der Homodyn-Messung. Fürn-Hexan/Nitrobenzol wird der Diffusionskoeffizient für vier unterschiedliche Konzentrationswerte als Funktion der Temperatur ausgemessen. Bei der kritischen Konzentration zeigt er bei gleicher Temperatur den kleinsten Wert und läßt sich nahe dem kritischen Entmischungspunkt durch einen einfachen Potenzansatz mit der reduzierten Temperaturdifferenz =T (– T_c)/T_c ausdrücken. Die statistische Genauigkeit ist besser als 1 %. Die Übereinstimmung mit Literaturwerten ist gut.

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Measuring of the binary diffusion coefficient in a separation system with photon-correlation-spectroscopy

The dynamic light scattering in form of photon-correlation spectroscopy is examplary used in a separation system for measuring the binary diffusion coefficient. In a test setup the homodyntechnique is used near the critical separation point and in distance the heterodyn-technique is used. A special method allows the correction of the disturbing influences of the heterodyn-part using the homodyn-measuring. Forn-hexane/nitrobenzene the diffusion coefficient is measured for four different concentration values as a function of temperature. At the critical concentration with constant temperature the coefficient shows the minimum value and it is expressed near the critical separation point with an elementary exponential equation with the reduced temperature difference =T (– T_c)/T_c. The statistical precision is better than 1%. The conformity with the literature is well.

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Abbreviations

Formelzeichen A Konstante - B Konstante - b Konstante - b ₁ Konstante - b₂ Konstante - C Konstante - c Konzentration - c _c kritische Konzentration - D Konstante - D ₁₂ binärer Diffusionskoeffizient - E ₀ elektrisches Feld des Laserlichts - E _s elektrisches Feld des Streulichts - G () Korrelationsfunktion - I ₀ Intensität des Referenzlichts - _s mittlere Intensität des Streulichts - k ₀ Wellenvektor des Laserlichts - k _s Wellenvektor des Streulichts - n Brechungsindex der zu untersuchenden Flüssigkeit - p Druck - q Streuvektor - R Ortsvektor - r Ortsvektor - T Temperatur, Zeit - t Zeit - x Molenbruch Griechische Buchstaben reduzierte Temperaturdifferenz; Dielektrizität - Frequenz des Laserlichts - statistische Schwankungen der Dielektrizitätskonstante - ₀ Wellenlänge des Laserlichts - Streuwinkel - Zeit - _c Zeitkonstante - kritischer Exponent