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.     

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.     

Über den Einfluß von Querkrümmung und Dichte bei inhomogenen turbulenten Freistrahlen

Zusammenfassung Zur Berechnung turbulenter Strömungen wird das k--Modell im Ansatz für die turbulente Scheinzähigkeit erweitert, so daß es den Querkrümmungs- und Dichteeinfluß auf den turbulenten Transportaustausch erfaßt. Die dabei zu bestimmenden Konstanten werden derart festgelegt, daß die bestmögliche Übereinstimmung zwischen Berechnung und Messung erzielt wird. Die numerische Integration der Grenzschichtgleichungen erfolgt unter Verwendung einer Transformation mit dem Differenzenverfahren vom Hermiteschen Typ. Das erweiterte Modell wird auf rotationssymmetrische Freistrahlen veränderlicher Dichte angewendet und zeigt Übereinstimmung zwischen Rechnung und Experiment.

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On the influence of transvers-curvature and density in inhomogeneous turbulent free jets

The prediction of turbulent flows based on the k- model is extended to include the influence of transverse-curvature and density on the turbulent transport mechanisms. The empirical constants involved are adjusted such that the best agreement between predictions and experimental results is obtained. Using a transformation the boundary layer equations are solved numerically by means of a finite difference method of Hermitian type. The extended model is applied to predict the axisymmetric jet with variable density. The results of the calculations are in agreement with measurements.

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Bezeichnungen Wirbelabsorptionskoeffizient - c_i Massenkonzentration der Komponente i - c_D, c_L, c, c₁, c₂ Konstanten des Turbulenzmodells - d Düsendurchmesser - E bezogene Dissipationsrate - f bezogene Stromfunktion - f Korrekturfunktion für die turbulente Scheinzähigkeit - j turbulenter Diffusionsstrom - k Turbulenzenergie - k_i Schrittweite in -Richtung - K dimensionslose Turbulenzenergie - L turbulentes Längenmaß - M_i Molmasse der Komponente i - p Druck - allgemeine Gaskonstante - r Querkoordinate - r_0,5 Halbwertsbreite der Geschwindigkeit - r_0,5c Halbwertsbreite der Konzentration - T Temperatur - u Geschwindigkeitskomponente in x-Richtung - v Geschwindigkeitskomponente in r-Richtung - x Längskoordinate - y allgemeine Funktion - Y_i diskreter Wert der Funktion y - Relaxationsfaktor für Iteration - turbulente Dissipationsrate - transformierte r-Koordinate - kinematische Zähigkeit - Exponent - transformierte x-Koordinate - Dichte - _k, Konstanten des Turbulenzmodells - Schubspannung - allgemeine Variable - Stromfunktion - Turbulente Transportgröße Indizes 0 Strahlanfang - m auf der Achse - r mit Berücksichtigung der Krümmung - t turbulent - mit Berücksichtigung der Dichte - im Unendlichen - Schwankungswert oder Ableitung einer Funktion - – Mittelwert Herrn Professor Dr.-Ing. R. Günther zum 70. Geburtstag gewidmet     

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).     

Effect of free-stream turbulence on surface friction in a turbulent boundary layer

The combined effect of the turbulence intensity , the turbulence scaleL, and the Reynolds number Re^** on the surface friction coefficientc _f in a turbulent boundary layer is studied. The dependence of the relative friction increment on the equivalent turbulence level _cq, which takes into account the simultaneous variation in ,L and Re^**, is determined. The threshold value _cq ^* below which the value ofc _f does not depend on _cq is found.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 65–75, March–April, 1995.     

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     

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.     

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.     

Application of Rayleigh-Ritz method to forced convection turbulent heat transfer

An analytical model to predict heat transfer rates to an incompressible fluid in turbulent flow, with fully developed velocity profile, between a heated plate and a parallel, insulated plate is developed. The model employs van Driest's mixing length expression near the wall, a constant eddy diffusivitiy in the core and a constant turbulent Prandtl number. An approximate solution obtained by employing Rayleigh-Ritz method is shown to compare well with the exact solution obtained by numerical integration of the differential equations. The results are compared with the available experimental data and analytical solutions.

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Anwendung der Rayleigh-Ritz-Methode auf die Wärmeübertragung bei erzwungener turbulenter Strömung

Zusammenfassung Es wird ein analytisches Modell zur Berechnung der Wärmeübertragung an ein inkompressibles Fluid in turbulenter Strömung mit voll ausgebildetem Geschwindigkeitsprofil zwischen einer beheizten Platte und einer dazu parallelen isolierten Platte angegeben. Das Modell verwendet van Driest's Ausdruck für die wandnahe Mischungslänge, eine konstante Wirbeldiffusivität im Kern und eine konstante turbulente PrandtlZahl. Eine Näherungslösung nach der Rayleigh-Ritz-Methode läßt sich gut mit der exakten Lösung vergleichen, die durch numerische Integration der Differentialgleichungen erhalten wurde. Die Ergebnisse werden mit verfügbaren Versuchswerten und analytischen Lösungen verglichen.

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Nomenclature A⁺ dimensionless constant in van Driest formula - a⁺ dimensionless distance from the wall after which the eddy diffusivity of momentum is constant - b half-gap of passage - b⁺ dimensionless half-gap=bu^*/ - C_f skin friction coefficient - C_p constant pressure specific heat - d hydraulic mean diameter defined as 4xarea/perimeter=4b - h convective heat transfer coefficient - K⁺ dimensionless constant in van Driest formula - k fluid thermal conductivity - m mass flow rate of fluid - Nu Nusselt number hd/k - P pressure - Pr Prandtl number=/ - Pr_t turbulent Prandtl number=_m/ - q_w heat flux at wall - Re Reynolds number=v_md/ - T Temperature - u⁺ dimensionless velocity=V_x/u^* - u^* friction velocity= - V_x axial velocity - x axial distance from the entrance - x⁺ dimensionless distance=x/d - y distance from the heated wall - y⁺ dimensionless distance=yu^*/ Greek Symbols thermal molecular diffusivity - function equal to (_H+)/ - boundary layer thickness - _H eddy diffusivity of heat - _m eddy diffusivity of momentum - _m0 uniform eddy diffusivity of momentum in the core - dimensionless temperature - T-T_i/q_wd/k uniform heat flux - T-T_w/T_i-T_w uniform temperature - fluid kinematic viscosity - fluid density - fluid shearing stress - bulk mean temperature—fully developed region - fully developed transverse temperature profile Suffixes 1 fully developed - 2 in the entrance region - i at the inlet - m bulk mean value - w at the heated wall     

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     

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.

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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.

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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     

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.     

Generation of vortices in a partially filled,rapidly rotating cylinder

We describe a system in which vortices are shed from a cylindrical free surface approximately centered in a rotating flow. Shedding is controlled by the parameter =2 g/ ² d, where g, , d denote gravity, rotation rate and the diameter of the free surface. We find vortex shedding for >0.162 and no vortex shedding for < 0.0847. The range depends on the aspect ratio L/d, where L is the column length, in a nonmonotonic fashion. These results are independent of viscosity and surface tension for small values of these parameters.Now at Martin Marietta, Orlando Aerospace, PO Box 5837, Mail Point 150, Orlando, FL 32855, USA     

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.     

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.     

On the shape of a slender axisymmetric nonstationary cavity

Very few studies have been made of three-dimensional nonstationary cavitation flows. In [1, 2], differential equations were obtained for the shape of a nonstationary cavity by means of a method of sources and sinks distributed along the axis of thin axisymmetric body and the cavity. In the integro-differential equation obtained in the present paper, allowance is made for a number of additional terms, and this makes it possible to dispense with the requirement ¦ In ¦ 1 adopted in [1, 2]. The obtained equation is valid under the weaker restriction 1. In [3], the problem of determining the cavity shape is reduced to a system of integral equations. Examples of calculation of the cavity shape in accordance with the non-stationary equations of [1–3] are unknown. In [4], an equation is obtained for the shape of a thin axisymmetric nonstationary cavity on the basis of a semiempirical approach. In the present paper, an integro-differential equation for the shape of a thin axisymmetric nonstationary cavity is obtained to order ² ( is a small constant parameter which has the order of the transverse-to-longitudinal dimension ratio of the system consisting of the cavity-forming body, the cavity, and the closing body). A boundary-value problem is formulated and an analytic solution to the corresponding differential equation is obtained in the first approximation (to terms of order ² In ), A number of concrete examples is considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 38–47, July–August, 1980.I thank V. P. Karlikov and Yu. L. Yakimov for interesting discussions of the work.     

Approximate solution of the boundary problem for the equations of a stationary electrostatic charged-particle beam

The solution is given of the equations of a three-dimensional stationary electrostatic beam of charged particles of like sign filling the region between two nearby curvilinear surfaces. We assume that the flow is nonrotational and nonrelativistic and that the velocity vector is a single-valued function. The solution is constructed in the form of an asymptotic series in powers of the small parameter , which is the ratio of the characteristic transverse (a) and longitudinal (l) dimensions of the problem. The first dimension is taken to be the distance between the electrodes, andl defines the scale at which the geometric and physical parameters (emitter curvature, electric field E on the emitter, and the emission current density J) change noticeably. The emission regimes limited by the space charge (-regime), temperature (T-regime), and the case of nonzero initial velocity (U-regime) are studied. The asymptotic behavior is given by the formulas for the corresponding one-dimensional flow between parallel surface.The solution of the boundary problem for emission in the-regime reduces to determination of the emission current density J for fixed electrode geometry and given accelerating voltage. The corresponding formulas are presented, retaining terms of order ³.Two approximations with respect to are performed for the T- and U-regimes. Here the unknown quantity for given properties of the emitting surface (J) will be the electric field E.The results provided by the constructed expansions are compared with the exact solution for flow from a planar emitter along circular trajectories [1]. As an example we examine the two-dimensional problem of flow between two nearby circular cylindrical electrodes with disruption of the coaxiality.The conventional tensor notations are used.     

Electrical parameters of a channel with finite electrodes with allowance for the electrode potential drop

Spatial problems involving the electric field in an MHD channel were formulated in [1] with allowance for the electrode potential drop. It was assumed that the electrode layer had a small thickness, so that relationships on the boundary of the layer could be applied to the surface of the electrode. It was assumed that the electrode potential drop ° could be represented as a function of the current density jn at the electrode in the form of a known function ° =f (j_n) determined experimentally or deduced from the appropriate electrode-layer theory. An approximate method was then put forward for solving such problems by reducing them to the determination of the electric field from a known distribution of the magnetic field and the gas-dynamic parameters. It was shown that when =°/ E is small (E is the characteristic induced or applied potential difference), the solution can be sought in the form of series in powers of . In the zero-order approximation, the electric field is determined without taking into account the electrode processes. The first approximation gives a correction of the order of . The quantity °, which is present in the boundary conditions on the electrode in the first-order approximation, is determined from the current density calculated in the zero-order approximation.One of the problems discussed in [1] was concerned with the electric current in a channel with one pair of symmetric electrodes. Its solution was found in the first approximation in the form of the integral Keldysh-Sedov formula. In this paper we report an analysis of the solution for ° taken in the form of a step function.     

The virtual origin of riblets in an open channel flow

In this paper, a method using the mean velocity profiles for the buffer layer was developed for the estimation of the virtual origin over a riblets surface in an open channel flow. First, the standardized profiles of the mixing length were estimated from the velocity measurement in the inner layer, and the location of the edge of the viscous layer was obtained. Then, the virtual origins were estimated by the best match between the measured velocity profile and the equations of the velocity profile derived from the mixing length profiles. It was made clear that the virtual origin and the thickness of the viscous layer are the function of the roughness Reynolds number. The drag variation coincided well with other results.Nomenclature f _r skin friction coefficient - f _ro skin friction coefficient in smooth channel at the same flow quantity and the same energy slope - g gravity acceleration - H water depth from virtual origin to water surface - H ⁺ u*H/ - H false water depth from top of riblets to water surface - H ⁺ u*H/ - I _e streamwise energy slope - I _b bed slope - k riblet height - k ⁺ u*k/ - l mixing length - l _s standardized mixing length - Q flow quantity - Re Reynolds number volume flow/unit width/v - s riblet spacing - u mean velocity - u* friction velocity = - u* false friction velocity = - y distance from virtual origin - y distance from top of riblet - y ₀ distance from top of riblet to virtual origin - y _v distance from top of riblet to edge of viscous layer - y ⁺ u*y/ - y ⁺ u*y/ - y ₀ ⁺ u*y ₀/ - u ⁺ u*y/ - shifting coefficient for standardization - thickness of viscous layer=y ₀+y - ⁺ u*/ - ⁺ u*/ - eddy viscosity - ridge angle - v kinematic viscosity - density - shear stress     

On a generalized nonlinearK-ɛ model for turbulence that models relaxation effects

In this paper, based on a similarity that exists between the constitutive relations for turbulent mean flow of a Newtonian fluid and that for the laminar flow of a non-Newtonian fluid, and making use of extended thermodynamics, we develop a generalized nonlinearK- model, whose approximate form includes the standardK- model and the nonlinearK- model of Speziale (1987) as special cases. Our nonlinearK- model, which is frame indifferent, can predict relaxation of the Reynolds stress, unlike most standardK- models. Also, our model is in keeping with that of Yakhotet al. (1992). Most interestingly, the linearized form of our model bears a striking resemblance to the model due to Yoshizawa and Nisizima (1993); however, it has been obtained from a totally different perspective.