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.     

Singular Limit and Homogenization for Flame Propagation in Periodic Excitable Media

This paper is concerned with a class of singular equations modelling the combustion of premixed gases in periodic media. The model involves two parameters: the period of the medium |L| and a singular parameter related to the activation energy. The existence of pulsating travelling fronts for fixed and |L| was proved by Berestycki & Hamel in [BH]. In the present paper, we investigate the behaviour of such solutions when More precisely, we establish that pulsating travelling fronts behave like travelling waves, when the period |L| is small and . We also study the convergence of the solution, as goes to zero (and |L| is fixed), toward a solution of a free boundary problem.     

Fractal porous media III: Transversal Stokes flow through random and Sierpinski carpets

The transversal Stokes flow of a Newtonian fluid through random and Sierpinski carpets is numerically calculated and the transversal permeability derived. In random carpets derived from site percolation, the average macroscopic permeability varies as (- _c)^3/2, close to the critical porosity _c. This exponent is found to be slightly different from the conductivity exponent. Results for Sierpinski carpets are presented up to the fourth generation. The Carman equation is not verified in these two model porous media.     

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.     

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     

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.     

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).     

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.     

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.

     

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.     

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     

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     

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     

Scalar Minimizers with Fractal Singular Sets

Lack of regularity of local minimizers for convex functionals with non-standard growth conditions is considered. It is shown that for every >0 there exists a function aC() such that the functional admits a local minimizer uW^1,p() whose set of non-Lebesgue points is a closed set with dim()>N–p–, and where 1<p<N<N+<q<+.     

Turbulent heat and mass transfer for Pr ≫ 1 and law of turbulent diffusion damping at a solid wall

The question of determining the law of damping for the turbulent diffusion coefficient at a smooth wall according to data on mass and heat transfer for Pr 1 is discussed. It is proved that the hypothesis that this law is determined by the first member of the Taylor series expansion of , namely, / = yⁿ ₊ is valid in the Pr range from 10³ to 10⁵ only under the assumption that the subsequent terms in the expansion have smaller coefficients. A statistical analysis of electrochemical and other experiments devoted to this problem shows that apparently n = 3, but singularities in the experimental results do not permit making a final conclusion. Requirements on a conclusive experiment are formulated on the basis of the analysis made.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 172–175, March–April, 1977.     

The problems of nonlinear bending for orthotropic rectangular plate with four clamped edges

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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     

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.