The onset of Taylor-like vortices in the flow induced by an impulsively started rotating cylinder

The onset of instability in the flow by an impulsively started rotating cylinder is analyzed under linear theory. It is well-known that at the critical Taylor number T_c=1695 the secondary flow in form of Taylor vortices sets in under the narrow-gap approximation. Here the dimensionless critical time _c to mark the onset of instability for TT_c is presented as a function of the Taylor number T. Available experimental data of water indicate that deviation of the velocity profiles from the primary flow occurs starting from a certain time 4_c. It seems evident that during _c4_c the secondary flow is very weak and the primary state of time-dependent annular Couette flow is maintained.     

Viscometric properties of viscous theta solutions of monodisperse polystyrene

Theta solvents for polystyrene are prepared from high-viscosity blends of styrene and low-molecular-weight polystyrene, and then used to make dilute solutions with monodisperse polystyrene solutes of high-M = 2.3, 6.0, 9.0, 18.0 · 10⁵. A Weissenberg rheogoniometer is used to measure the non-Newtonian viscosity as a function of shear stress, for low values, and also the complex viscosity components and as functions of frequency. A capillary viscometer is used for high- measurements of(). Viscometric properties, at room temperature, are analyzed as functions of high-molecular-weight solute concentrationc with parameters of constant or to obtain [()], [ ()], and [ ()]. Such a collection of data has apparently not previously been available for polymers in theta solvents (in which Gaussian chain statistics prevail). Also unique is the achievement of high stress ( = 2 10⁴ Pa) at low shear rate, by virtue of high solvent viscosity which is not characteristic of other known theta solvents.     

Modal theory of elastic-viscoelastic composite structures

The present paper investigates the vibration modal theory of composite structures constructed with elastic and viscoelastic materials. The equation of motion that comes in the form of integrodifferential equations is transformed into the first order differential equation in state space. Then modal analysis is carried out. The concepts of vibrating modal set and creeping modal set are proposed. And impulsive response matrix and transfer function matrix are defined and discussed in detail. Finally sample problems are given to support the theory developed in this paper.     

Existence and Uniqueness of Time-Periodic Physically Reasonable Navier-Stokes Flow Past a Body

Let be a three-dimensional exterior domain of class C^2,, 0<<1. Assume that a Navier-Stokes liquid is moving in under the action of a body force F that is time-periodic of period T, and that the velocity of the liquid is zero at spatial infinity. In this paper we show that, if F satisfies suitable conditions, and its norm, in appropriate function spaces, is sufficiently small, there is at least one time-periodic strong solution. Furthermore, the velocity field v of such a solution decays to zero for large |x| as |x|^–1 and its spatial gradient decays as |x|^–2, both uniformly in time. In addition, the pressure p decays like |x|^–2 and its gradient like |x|^–3, for almost all t[0,T]. In the special case where F is time-independent, these solutions are also time-independent and coincide with Finns physically reasonable solutions [4]. Moreover, we show that our time-periodic solutions are unique in a very large class, namely, the class of time-periodic weak solutions satisfying the energy inequality and with corresponding pressure fields verifying mild summability conditions in ×[0,T].     

On surface waves in generalized thermoelasticity

Knowles' representation theorem for harmonically time-dependent free surface waves on a homogeneous, isotropic elastic half-space is extended to include harmonically time-dependent free processes for thermoelastic surface waves in generalized thermoelasticity of Lord and Shulman and of Green and Lindsay.r , , r , , .This work was done when author was unemployed.     

Einige Beziehungen zwischen Transportkoeffizienten und thermodynamischen Zustandsgrö\en

Zusammenfassung Eine früher mitgeteilte Beziehung [1] zwischen der ViskositÄt und dem isenthalpen Joule-Thomson-Koeffizienten _h wird für kleine Drücke theoretisch begründet und an sieben Stoffen nachgeprüft. Die WärmeleitfÄhigkeit wird als Funktion von c_v für drei Stoffe dargestellt.

---

Some relations between transport coefficients and thermodynamical properties

A formerly given relation [1] between viscosity and isenthalpic Joule-Thomson-coefficient _h is proved theoretically for small pressures and checked with seven substances. The heat conductivity is presented as a function ofc_v for three substances.

---

Bezeichnungen B ^* dimensionsloser zweiter Virialkoeffizient - B ₁ ^* Ableitung vonB ^* nachT ^*.B ₁ ^*=T ^* dB ^*/dT ^* - c _v isochore spezifische WärmekapazitÄt - C _p ^o isobare molare WärmekapazitÄt fürp 0 - h spezifische Enthalpie - k Boltzmann-Konstante.k=R/N _A - M molare Masse - N _A Avogadro-Konstante - p Druck - R molare Gaskonstante - R _i spezifische Gaskonstante des Stoffesi - it Celsius-Temperatur - T Kelvin-Temperatur - T ^* dimensionslose Temperatur.T ^*=kT/ - _h isenthalper Joule-Thomson-Koeffizient. _h=(T/p)_h - , Konstanten der Potentialfunktion - ViskositÄt - WärmeleitfÄhigkeit - (^2,2)^* dimensionsloses Sto\integral     

Correlations between porous medium flow data and turbulent tube flow results for aqueous hydroxypropylguar solutions (Part 2)

Turbulent tube flow and the flow through a porous medium of aqueous hydroxypropylguar (HPG) solutions in concentrations from 100 wppm to 5000 wppm is investigated. Taking the rheological flow curves into account reveals that the effectiveness in turbulent tube flow and the efficiency for the flow through a porous medium both start at the same onset wall shear stress of 1.3 Pa. The similarity of the curves = ( _w ) and = ( _w ), respectively, leads to a simple linear relation / =k, where the constantk or proportionality depends uponc. This offers the possibility to deduce (for turbulent tube flow) from (for flow through a porous medium). In conjunction with rheological data, will reveal whether, and if yes to what extent, drag reduction will take place (even at high concentrations).The relation of our treatment to the model-based Deborah number concept is shown and a scale-up formula for the onset in turbulent tube flow is deduced as well.     

New concepts in relative permeabilities at high capillary numbers for surfactant flooding

Flooding oil reservoirs with surfactant solutions can increase the amount of oil that can be recovered. Macroscopic modelling of the process requires relative permeabilities to be functions of saturation and capillary number. With only limited experimental data, relative permeabilities have usually been assumed to be linear functions of saturation at high capillary numbers. The experimental data is reviewed, some of which suggest that this assumption is not necessarily correct. The basis for the assumption is therefore reviewed and it is concluded that the linear model corresponds to microscopically segregated flow in the porous medium. Based on new but equally plausible complementary assumptions about the flow pattern, a mixed flow model is derived. These models are then shown to be limiting cases of a droplet model which represents the mixing scale within the porous medium and gives a physical basis for interpolating between the models. The models are based on physical concepts of flow in a porous medium and so the approach described here represents a significant improvement in the understanding of high capillary number flow. This is shown by the fact that fewer parameters are needed to describe experimental data.Notation A total cross-sectional area assigned to capillary bundle - A ⁽ⁱ⁾ physical cross-sectional area of tube i - c ⁽ⁱ⁾ ordered configurational label for droplets in tube i - c configuration label for tube i (order not considered) - D defined by Equation (26) - E(...) expectation value with respect to the trinomial distribution - S _r ⁽⁾ fractional flow of phase - k absolute permeability - k _r relative permeability of phase - k _r ⁰ endpoint relative permeability of phase - L capillary tube length in bundle model - m ⁽ⁱ⁾ number of droplets of phase a occupying tube i - n exponent for phase a in Equation (2) - N number of droplets in bundle model - N _c capillary number - p pressure - p(c') probability of configuration c - Q ⁽ⁱ⁾ total volume flow rate in tube i - S saturation of phase - S flowing saturation of phase - S _r residual saturation of phase - S _r ⁽⁾ saturations when fractional flow of phase is 1 in the case of varying residual saturations for three-phase flow ( ) - t _c residence time for droplet configuration c - v ⁽ⁱ⁾ total fluid velocity in bundle tube i - , phase label - p pressure differential across capillary bundle - ⁽ⁱ⁾ tube conductivity defined by Equation (7) - viscosity of phase - interfacial tension - gradient operator - ... average over tube droplet configurations     

Asymptotic solution of the thin viscous shock layer equations at low Reynolds numbers for a cold surface

Hypersonic three-dimensional viscous rarefied gas flow past blunt bodies in the neighborhood of the stagnation line is considered. The question of the applicability of the gasdynamic thin viscous shock layer model [1] is investigated for the transition flow regime from continuum to free-molecular flow. It is shown that for a power-law temperature dependence of the viscosity coefficient T the quantity (Re)^1/(1+), where = ( – 1)/2 and is the specific heat ratio, is an important determining parameter of the hypersonic flow at low Reynolds numbers. In the case of a cold surface approximate asymptotic solutions of the thin viscous shock layer equations are obtained for noslip conditions on the surface and generalized Rankine-Hugoniot relations on the shock wave at low Reynolds numbers. These solutions give simple analytic expressions for the thermal conductivity and friction coefficients as functions of the determining flow parameters. As the Reynolds number tends to zero, the values of the thermal conductivity and friction coefficients determined by this solution tend to their values in free-molecular flow for an accommodation coefficient equal to unity. This tending of the thermal conductivity and friction coefficients to the free-molecular limit takes place for both two-and three-dimensional flows. The asymptotic solutions are compared with numerical calculations and experimental data.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, 2004, pp. 159–170. Original Russian Text Copyright © 2004 by Brykina.     

Thermodynamically complete equations of state for α-, ε-, and γ-iron

The numerical model of phase transition in iron in stress waves described in [1] contains equations of state with a limited range of applicability. They do not consider thermal excitation of conduction electrons and the presence of and — -triple point on the phase equilibrium curve, the effect of which should appear in shock loading of porous or preheated specimens. The present study will offer thermodynamically complete equations of state for the -, -, -phases of iron, free of these shortcomings.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 112–114, May–June, 1986.     

Stationary travelling waves on a film flowing down an inclined plane

At small flow rates, the study of long-wavelength perturbations reduces to the solution of an approximate nonlinear equation that describes the change in the film thickness [1–3]. Steady waves can be obtained analytically only for values of the wave numbers close to the wave number _n that is neutral in accordance with the linear theory [1, 2]. Periodic solutions were constructed numerically for the finite interval of wave numbers 0.5_{n n} in [4]. In the present paper, these solutions are found in almost the complete range of wave numbers 0 _n that are unstable in the linear theory. In particular, soliton solutions of this equation are obtained. The results were partly published in [5].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 142–146, July–August, 1980.     

Determination of permeability tensors for two-phase flow in homogeneous porous media: Theory

In this paper we continue previous studies of the closure problem for two-phase flow in homogeneous porous media, and we show how the closure problem can be transformed to a pair of Stokes-like boundary-value problems in terms of pressures that have units of length and velocities that have units of length squared. These are essentially geometrical boundary value problems that are used to calculate the four permeability tensors that appear in the volume averaged Stokes' equations. To determine the geometry associated with the closure problem, one needs to solve the physical problem; however, the closure problem can be solved using the same algorithm used to solve the physical problem, thus the entire procedure can be accomplished with a single numerical code.Nomenclature a a vector that maps V onto , m^-1. - A a tensor that maps V onto . - A area of the - interface contained within the macroscopic region, m². - A area of the -phase entrances and exits contained within the macroscopic region, m². - A area of the - interface contained within the averaging volume, m². - A area of the -phase entrances and exits contained within the averaging volume, m². - Bo Bond number (= (=(–)g²/). - Ca capillary number (= v/). - g gravitational acceleration, m/s². - H mean curvature, m^-1. - I unit tensor. - permeability tensor for the -phase, m². - viscous drag tensor that maps V onto V. - ^* dominant permeability tensor that maps onto v , m². - ^* coupling permeability tensor that maps onto v , m². - characteristic length scale for the -phase, m. - l characteristic length scale representing both and , m. - L characteristic length scale for volume averaged quantities, m. - n unit normal vector directed from the -phase toward the -phase. - n unit normal vector representing both n and n . - n unit normal vector representing both n and n . - P pressure in the -phase, N/m². - p superficial average pressure in the -phase, N/m². - p intrinsic average pressure in the -phase, N/m². - p –p , spatial deviation pressure for the -phase, N/m². - r ₀ radius of the averaging volume, m. - r position vector, m. - t time, s. - v fluid velocity in the -phase, m/s. - v superficial average velocity in the -phase, m/s. - v intrinsic average velocity in the -phase, m/s. - v –v , spatial deviation velocity in the -phase, m/s. - V volume of the -phase contained within the averaging volmue, m³. - averaging volume, m³. Greek Symbols V /, volume fraction of the -phase. - viscosity of the -phase, Ns/m². - density of the -phase, kg/m³. - surface tension, N/m. - (v +v ^T ), viscous stress tensor for the -phase, N/m².     

Contribution to the theory of the high compression ratio gas ejector with cylindrical mixing chamber

Ways of improving the operation of a gas ejector with a high compression ratio are investigated. The conditions for obtaining the maximal compression ratio at the critical operating regime of the gas ejector are studied theoretically and experimentally with account for mixing of the supersonic injecting and subsonic ejected streams ahead of the choking section. The principles for the rational utilization of the effect of stream mixing in the ejector ahead of the choking section are indicated; the use of these principles permits a several-fold increase of the compression ratio of the supersonic ejector. A theory is given for the critical regime of the gas ejector with uniformly perforated nozzle, and the hydraulic parameters of the required wall perforationss are determined. It is shown that perforation as a hydraulic factor can improve significantly the parameters of the sonic ejector in the critical regime.The foundations of modem gas ejector theory were developed by Khristianovich [1, 2]. In these studies he established the relationship between the parameters of the flow at the end of the mixing chamber (section 3, p ₀ is the total pressure, is the reduced velocity) and the parameters of the ejecting (section 1, p ₀ ,) and the ejected (p₀₁,₁) flows with account for compressibility for the ejector with a cylindrical mixing chamber (Fig. 1a). The ejector theory [1, 2] (see also [3, 4]) is given in the hydraulic approximation: the flow at the end of the mixing chamber is assumed uniform, flow friction on the mixing chamber walls is neglected. The use of the gasdynamic functions [5–9] made it possible to obtain computational equations for the ejector in a convenient form and to extend them to the case of mixing of gases with different thermophysical properties. We note that for subsonic velocities of the ejecting and ejected flows the system of ejector equations [1, 2] is supplemented by the condition of equality of the static pressures p=P₁ at the stream contact section 1.The results of extensive experimental studies of subsonic ejectors are in good agreement with the results of this theory.For sonic or supersonic velocity of the ejecting gas (=1) the condition p=p₁ is not satisfied in the general case. Fundamental for the development of ejector theory was the establishment by Millionshchikov and Ryabinkov in 1948 of the existence of a critical operating regime of the supersonic ejector [7, 10]. They showed that the limiting operating regimes of the gas ejector for high pressure differentials ==p ₀ /p₀₁ are determined by the conditions for the choking of the ejected jet by the expanding supersonic ejecting flow. With the occurrence of the critical regime the velocity of the ejected jet at the choking section (section 2, Fig. 1a) reaches the speed of sound (=1); this limits the further increase of the pressure ratio and the ejector compression ratio =p ₀ /p ₀ for a given ejection coefficient k (k is the ratio of the ejected and ejecting gas flow rates). The relationships between these flow parameters at sections 2 and 1 supplement the system of ejector equations and permit determining its critical characteristics.Millionshchikov and Ryabinkov showed that for moderate values of the pressure ratio good agreement of the theoretical and experimental ejector characteristics are given by the assumption of constant static pressure p₂=const at section 2 (Fig. 1a).The limit of the applicability of the theory based on the condition p₂= = const, was studied experimentally by Lyzhin [10].The theory of the critical regime of the gas ejector was developed in 1953 in studies of Nikol'skii, Shustov, Vasil'ev, Taganov, and Mezhirov [10, 11]. Nikol'skii showed that the condition of constant static pressure at the choking section is not in agreement with the momentum equation.For a more rigorous theoretical determination of the critical ejector regime he proposed joining between sections 1 and 2 (Fig. 1a) the calculation of the ejecting jet using the method of characteristics and the hydraulic calculation of the ejected jet; example calculations were made by Nikol'skii and Shustov. Taganov and Mezhirov suggested a method for calculating the ejector critical regime using a linear distribution of the pressure in the supersonic ejecting jet (at the choking section 2).A simple and successful method for calculating the ejector critical regime was given by Vasil'ev, who used the hydraulic representation of the ejecting and ejected flows in the choking section; both flows are assumed uniform at section 2, the static pressures in these flows in the general case are different and are determined by the momentum equation. A similar theory for the ejector critical regime was developed independently in [12, 13], and the theory with account for the supersonic ejecting flow (ahead of the choking section) was developed using the method of characteristics in [14].It should be noted that the results of the calculations of the critical characteristics of the ejectors using all three of these methods were practically indentical and in good agreement with experiment for large and moderate values of the ejection coefficients. We emphasize that in the theories of the ejector critical regime the flow mixing between sections 1 and 2 is neglected.The critical regime theory imposes significant limitations on the possible characteristics of the gas ejector, first of all, on the achievable compression ratio =p ₀ /p ₀ . Thus, from the data of [10], even for a pressure ratio =1000 the maximal theoretical value of the compression ratio for the supersonic ejector does not exceed 40 (see in Fig. 2 the limiting ejector characteristics based on the critical regime theory); for the sonic air ejector (=1) the theoretical value of 3.5 (see Fig. 9b on p. 26). Therefore it is important to analyze the methods for influencing the critical regime parameters in order to determine ways to improve the operation of the gas ejector with a high compression ratio.     

General expressions for unsteady forces in a lattice of profiles

A large number of studies have been devoted to the unsteady flow of a viscid incompressible fluid past a lattice of thin profiles and the determination of the resulting aerodynamic forces and moments. For example, in the particular case of the motion of a lattice with stagger with zero phase shift of the oscillations between neighboring profiles, Haskind [1] determined the unsteady lift force and moment. Popescu [2] suggested expressions for the force and moment in the case when =0 and =0, using the method of conformal mapping. Samoilovich [3] obtained equations for the unsteady lift force and moment by the method of the acceleration potential for phase shift =0 and = of the oscillations between neighboring profiles. Musatov [4] used an electronic digital computer to calculate the overall unsteady aerodynamic characteristics of a grid by the vortex method, taking into account the amplitude of the oscillations and the initial circulation for =m (m1). Gorelov [5] determined the coefficients of the over-all unsteady aerodynamic force and moment of a profile in a lattice with the stagger and any value of =m. He used a method based on the unsteady flow past an isolated profile with subsequent account for the interference of the profiles in the lattice.In the following we find general expressions for the unsteady lift force and moment acting on a lattice moving in an incompressible fluid with the constant velocity U. These formulas generalize the known formulas for the isolated profile [6]. The profiles of a staggered grid (Section 1) are considered to be thin and slightly curved, and perform oscillations with a phase shift of the oscillations between neighboring profiles. The method of separation of singularities is used to obtain the solution in closed form. The coefficients of the expansion of the complex velocity in a series in the derivatives of a function are calculated. An integral equation relative to the unknown tangential velocity component in the wake is derived (Section 2), and its analytic solution is given (Section 3). For =0 the solution coincides with the solution obtained earlier in [7]. Expressions are obtained for the forces and moments (Section 4) in the form of four terms. The first two terms determine the force and moment for motion with constant circulation, and the last two determine these characteristics for motion with variable circulation. The suction force acting at the leading edges of the profiles is found in a general form. Particular cases of closely and widely spaced lattices are considered. Computational results are presented.     

Stochastic-Perturbation Analysis of a One-Dimensional Dispersion-Reaction Equation: Effects of Spatially-Varying Reaction Rates

We carry out a stochastic-perturbation analysis of a one-dimensional convection–dispersion-reaction equation for reversible first-order reactions. The Damköhler number, Da, is distributed randomly from a distribution that has an exponentially decaying correlation function, controlled by a correlation length, . Zeroth- and first-order approximations of the dispersion coefficient, D are computed from moments of the residence-time distribution obtained by solving a one-dimensional network model, in which each unit of the network represents a Darcy-level transport unit, and the solution of the transfer function in zeroth- and first-order approximations of the transport equation. In the zeroth-order approximation, the dispersion coefficient is calculated using the convection–dispersion-reaction equation with constant parameters, that is, perturbation corrections to the local equation are ignored. This zeroth-order dispersion coefficient is a linear function of the variance of the Damköhler number, (Da)². A similar result was reported in a two-dimensional network simulation. The zeroth-order approximation does not give accurate predictions of mixing or spreading of a plume when Damköhler numbers, Da 1 and its variance, (Da)² > 0.25 Da². On the other hand, the first-order theory leads to a dispersion coefficient that is independent of the reaction parameters and to equations that do accurately predict mixing and spreading for Damköhler numbers and variances in the range (Da)²/Da0.3     

Generalized Rouse model for a dilute solution of clustered polymers

A theory analogue to tha of Rouse is given, to describe the rheological behavior of dilute solutions consisting of clusters of crosslinked polymers. The frequency-dependent behavior of the dynamic moduli of these fluids differs substantially from that of the well-known Rouse-like fluid (GG^1/2). In our case the storage modulus becomes proportional to ^3/2, while the loss modulus is proportional to . The loss modulus dominates the dynamic behavior for frequencies smaller than the largest normal frequency of the clusters.     

Irrotational outflow of liquid from a stream through a lateral hole of finite depth

An algorithm is constructed for numerical determination of the flow parameters and coefficient of contraction of a jet in the case of irrotational lateral outflow of liquid from a semiinfinite stream through a nozzle of finite depth situated at an arbitrary angle to the mainstream flow. The solution is based on the use of N. E. Zhukovskii's method and the Schwarz-Christoffel formula. The results of calculations for a nozzle situated at an angle = /2 ± , where = /6, are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 162–164, January–February, 1977.     

Concentration dependence of the critical viscoelastic properties of gelatin at the gel point

Gelatin gel properties have been studied through the evolution of the storage [G()] and the loss [G()] moduli during gelation or melting near the gel point at several concentrations. The linear viscoelastic properties at the percolation threshold follow a power-law G()G() and correspond to the behavior described by a rheological constitutive equation known as the Gel Equation. The critical point is characterized by the relation: tan = G/G = cst = tan ( · /2) and it may be precisely located using the variations of tan versus the gelation or melting parameter (time or temperature) at several frequencies. The effect of concentration and of time-temperature gel history on its variations has been studied. On gelation, critical temperatures at each concentration were extrapolated to infinite gel times. On melting, critical temperatures were determined by heating step by step after a controlled period of aging. Phase diagrams [T = f(C)] were obtained for gelation and melting and the corresponding enthalpies were calculated using the Ferry-Eldridge relation. A detailed study of the variations of A with concentration and with gel history was carried out. The values of which were generally in the 0.60–0.72 range but could be as low as 0.20–0.30 in some experimental conditions, were compared with published and theoretical values.     

Representation of the rheological properties of polymer melts in terms of complex fluidity

In dynamic rheological experiments melt behavior is usually expressed in terms of complex viscosity ^* () or complex modulusG ^* (). In contrast, we attempted to use the complex fluidity ^* () = 1/µ ^* () to represent this behavior. The main interest is to simplify the complex-plane diagram and to simplify the determination of fundamental parameters such as the Newtonian viscosity or the parameter of relaxation-time distribution when a Cole-Cole type distribution can be applied. ^* () complex shear viscosity - () real part of the complex viscosity - () imaginary part of the complex viscosity - G ^* () complex shear modulus - G() storage modulus in shear - G() loss modulus in shear - J ^* () complex shear compliance - J() storage compliance in shear - J() loss compliance in shear - shear strain - rate of strain - angular frequency (rad/s) - shear stress - loss angle - ^* () complex shear fluidity - () real part of the complex fluidity - () imaginary part of the complex fluidity - ₀ zero-viscosity - ₀ average relaxation time - h parameter of relaxation-time distribution     

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.