Optical co-ordinates in general relativity 2 — the problem of distance

Summary Let denote the congruence of null geodesics associated with a given optical observer inV ₄. We prove that determines a unique collection of vector fieldsM₍₎ ( =1, 2, 3) and ₍₀₎ overV ₄, satisfying a weak version of Killing's conditions.This allows a natural interpretation of these fields as the infinitesimal generators of spatial rotations and temporal translation relative to the given observer. We prove also that the definition of the fieldsM₍₎ and ₍₀₎ is mathematically equivalent to the choice of a distinguished affine parameter f along the curves of, playing the role of a retarded distance from the observer.The relation between f and other possible definitions of distance is discussed.

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Sommario Sia la congruenza di geodetiche nulle associata ad un osservatore ottico assegnato nello spazio-tempoV ₄. Dimostriamo che determina un'unica collezione di campi vettorialiM₍₎ ( =1, 2, 3) e ₍₀₎ inV ₄ che soddisfano una versione in forma debole delle equazioni di Killing. Ciò suggerisce una naturale interpretazione di questi campi come generatori infinitesimi di rotazioni spaziali e traslazioni temporali relative all'osservatore assegnato. Dimostriamo anche che la definizione dei campiM₍₎, ₍₀₎ è matematicamente equivalente alla scelta di un parametro affine privilegiato f lungo le curve di, che gioca il ruolo di distanza ritardata dall'osservatore. Successivamente si esaminano i legami tra f ed altre possibili definizioni di distanza in grande.

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Work performed in the sphere of activity of: Gruppo Nazionale per la Fisica Matematica del CNR.     

Approximate relationships for the determination of the pressure on the surface of a sphere or cylinder with an arbitrary mach number in the free stream

Numerical calculations have been made [1–4] of the pressure distribution over the surface of a sphere or cylinder during transverse flow in the range 0 /2, where is the angle reckoned from the stagnation point along the meridional plane, and on the basis of these results simple analytical equations have been proposed in order to determine the pressure for arbitrary Mach numbers M in the free stream. The gas is assumed to be ideal and perfect.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 185–188, March–April, 1985.     

Exact Solutions of the Hydrodynamic Equations Derived from Partially Invariant Solutions

The paper proposes a heuristic approach to constructing exact solutions of the hydrodynamic equations based on the specificity of these equations. A number of systems of hydrodynamic equations possess the following structure: they contain a reduced system of n equations and an additional equation for an extra function w. In this case, the reduced system, in which w = 0, admits a Lie group G. Taking a certain partially invariant solution of the reduced system with respect to this group as a seed:rdquo; solution, we can find a solution of the entire system, in which the functional dependence of the invariant part of the seed solution on the invariants of the group G has the previous form. Implementation of the algorithm proposed is exemplified by constructing new exact solutions of the equations of rotationally symmetric motion of an ideal incompressible liquid and the equations of concentrational convection in a plane boundary layer and thermal convection in a rotating layer of a viscous liquid.     

Electrode layers in flows of an inviscid plasma with good conductivity

The structure of the electromagnetic electrode layers that are produced in flows across a magnetic field by a completely ionized and inviscid plasma with good conductivity and a high magnetic Reynolds number is examined in a linear approximation. Flow past a corrugated wall and flow in a plane channel of slowly varying cross section with segmented electrodes are taken as specific examples. The possibility is demonstrated of the formation of nondissipative electrode layers with thicknesses on the order of the Debye distance or electron Larmor radius and of dissipative layers with thicknesses on the order of the skin thickness, as calculated from the diffusion rate in a magnetic field [2].In plasma flow in a transverse magnetic field, near the walls, along with the gasdynamie boundary layers, which owe their formation to viscosity, thermal conductivity, etc. (because of the presence of electromagnetic fields, their structures may vary considerably from that of ordinary gasdynamic layers), proper electromagnetic boundary layers may also be produced. An example of such layers is the Debye layer in which the quasi-neutrality of the plasma is upset. No less important, in a number of cases, is the quasi-neutral electromagnetic boundary layer, in which there is an abrupt change in the frozen-in parameter k=B/p (B is the magnetic field and p is the density of the medium). This layer plays a special role when we must explicitly allow for the Hall effect and the related formation of a longitudinal electric field (in the direction of the veloeiryv of the medium). We will call this the magnetic layer. The magnetic boundary layer can be dissipative as well as noudissipative (see below). The dissipative magnetic layer has been examined in a number of papers: for an incompressible medium with a given motion law in [1], for a compressible medium with good conductivity in [2], and with poor conductivity in [3]. In the present paper, particular attention will be devoted to nondissipative magnetic boundary layers.     

Asymptotic solution to the problem of hypersonic flow over bodies in the neighborhood of the separation point of a thin shock layer

A study is made of the problem of hypersonic flow of an inviscid perfect gas over a convex body with continuously varying curvature. The solution is sought in the framework of the asymptotic theory of a strongly compressed gas [1–4] in the limit M when the specific heat ratio tends to 1. Under these assumptions, the disturbed flow is situated in a thin shock layer between the body and the shock wave. At the point where the pressure found by the Newton-Buseman formula vanishes there is separation of the flow and formation of a free layer next to the shock wave [1–4]. The singularity of the asymptotic expansions with respect to the parameter ₁ = ( –1)/( + 1) associated with separation of the strongly compressed layer has been investigated previously by various methods [3–9]. Local solutions to the problem valid in the neighborhood of the singularity have been obtained for some simple bodies [3–7]. Other solutions [7, 9] eliminate the singularity but do not give the transition solution entirely. In the present paper, an asymptotic solution describing the transition from the attached to the free layer is constructed for a fairly large class of flows.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 99–105, January–February, 1982.     

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

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On the Quasi‐Steadiness Hypothesis as Applied to Gas Exhaustion from a Receiver

A semiempirical method of determining the stabilization time for a quasisteady mode of gas exhaustion from a receiver after sudden opening of the nozzle and the time evolution of the real flow rate at the stage of the transitional process are considered. The numerical solution of the equations of exhaustion gas dynamics in a twodimensional formulation and the results of model experiments demonstrated that the method can be used to estimate the conditions of applicability of the quasisteadiness hypothesis and to determine the discharge coefficient of the nozzle with controlled accuracy.     

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.     

Some problems concerning nonstationary currents in dense plasma systems confined by walls

Nonstationary currents are examined in a dense magnetized plasma with 1, in which energy release and heat loss by thermal conduction and radiation are possible. Solutions are found in two limiting cases: ¦f¦ ¦ div (T)¦ and ¦f¦ ¦ div(T)¦ (f is the radiation intensity, is the coefficient of heat conduction, and T is the temperature). In the first case a solution was obtained of some problems of the cooling and heating of a plasma illustrated in part by the evolution in time of the temperature profile in the boundary layer. In the second case an isomorphic solution was found for an arbitrary dependence of the coefficient of heat conduction on the temperature, pressure, and magnetic field.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 3–8, January–February, 1972.The author is grateful to G. I. Budker for formulating the problem.     

Instability of rarefaction shocks in systems of conservation laws

It is well-known that rarefaction shocks are unstable solutions of nonlinear hyperbolic conservation laws. Indeed, for scalar equations rarefaction shocks are unstable in the class of smooth solutions, but for systems one can only say in general that rarefaction shocks are unstable in the larger class of weak solutions. (Here unstable refers to a lack of continuous dependence upon perturbations of the initial data.) Since stability in the class of weak solutions is not well understood, ([T, TE]), entropy considerations have played a leading role in ruling out shocks that violate the laws of physics. However, for non-strictly hyperbolic systems the analogy with the equations of gas dynamics breaks down, and general entropy or admissibility criteria for the variety of shocks which appear, (see, e.g., [IMPT]), are not known. In this paper we address the question of when the instability of a shock can be demonstrated within the class of smooth solutions alone. We show by elementary constructions that this occurs whenever there exists an alternative solution to the Riemann problem with the same shock data which consists entirely of rarefaction waves and contact discontinuities with at least one non-zero rarefaction wave. We show that for 2×2 strictly hyperbolic, genuinely nonlinear systems the condition is both necessary and sufficient. We show too that for the full 3×3 (Euler) equations of gas dynamics with polytropic equations of state, rarefaction shocks of moderate strength are unstable in the class of smooth solutions if and only if the adiabatic gas constant satisfies 1 < < 5/3 (see Theorem 8). More precisely, there is a constant y _*, 0 < y _* < 1, depending only on , such that if y _* p _lp _rp _l for 1-shocks, and if y _* p _rP _lp _r for 3-shocks (where p _r and p _l denote the pressures on both sides of the rarefaction shock), then the shock is unstable if and only if 1 < < 5/3. Thus for such shocks, the theory of the Riemann problem for polytropic gases in the range 1 < < 5/3 can be rigorously developed with a knowledge of the smooth solutions alone by using stability under smoothing as an admissibility criterion, rather than by using the classical entropy inequalities.     

Der Spannungs- und Verschiebungszustand drehsymmetrischer Membrane mit beliebig gekrümmter Meridiankurve

Zusammenfassung Die Einführung von Zylinderkoordinaten (x, r, ) in die Gleichgewichtsbedingungen der Schnittkräfte bzw. in die Beziehungen zwischen Verzerrung und Verschiebungen am differentialen Schalenabschnitt ermöglicht die Berechnung des Spannungs- und Verschiebungszustandes von drehsymmetrischen Membranen mit beliebig gekrümmter Meridiankurve auf die Integration einer einfachen, linearen partiellen Differentialgleichung zweiter Ordnung für eine charakteristische FunktionF bzw. zurückzuführen. Eine geschlossene Lösung und damit eine Darstellung der Schnittkräfte und Verschiebungen durch explizite Formeln ist bei harmonischer Belastung cosn für zwei Funktionsgruppen=x ² und=x ^–3 möglich. Im Sonderfall der drehsymmetrischen und der antimetrischen Belastung mitn=0 undn=1 gelten die Gleichungen der Schnitt- und Verschiebungsgrößen für eine beliebige Meridianfunktion=(). Die Betrachtungen der Randbedingungen offener Schalen bei harmonischer Belastung geben über die infinitesimalen Deformationen einer drehsymmetrischen Membran mit überall negativer Krümmung Aufschluß.     

Formation of an eddy current in the thermal acceleration of a completely ionized plasma

The steady-state plane slowly varying flow of a completely ionized nonviscous quasi-neutral plasma in a shaped channel with continuous metal walls is considered. The Hall effect is taken into account. It is shown that for 1, where is the plasma parameter ( = 8p/B², p is the gas-kinetic pressure of the plasma, and B is the magnetic field strength), the acceleration of the plasma is necessarily accompanied by the appearance of natural electromagnetic fields and an electric current, the distribution of which for small discharge voltages has an eddy-current form. The eddy currents disappear when the discharge voltage is increased. The acceleration of a plasma with isothermal electrons is investigated in detail.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 30–34, November–December, 1971.     

Transport in ordered and disordered porous media IV: Computer generated porous media for three-dimensional systems

In the method of volume averaging, the difference between ordered and disordered porous media appears at two distinct points in the analysis, i.e. in the process of spatial smoothing and in the closure problem. In theclosure problem, the use of spatially periodic boundary conditions isconsistent with ordered porous media and the fields under consideration when the length-scale constraint,r ₀L is satisfied. For disordered porous media, spatially periodic boundary conditions are an approximation in need of further study.In theprocess of spatial smoothing, average quantities must be removed from area and volume integrals in order to extractlocal transport equations fromnonlocal equations. This leads to a series of geometrical integrals that need to be evaluated. In Part II we indicated that these integrals were constants for ordered porous media provided that the weighting function used in the averaging process contained thecellular average. We also indicated that these integrals were constrained by certain order of magnitude estimates for disordered porous media. In this paper we verify these characteristics of the geometrical integrals, and we examine their values for pseudo-periodic and uniformly random systems through the use of computer generated porous media.

Nomenclature

Roman Letters A interfacial area of the- interface associated with the local closure problem, m² - A _e area of entrances and exits for the-phase contained within the averaging system, m² - a _i i=1, 2, 3 gaussian probability distribution used to locate the position of particles - I unit tensor - L general characteristic length for volume averaged quantities, m - L characteristic length for , m - L characteristic length for , m - characteristic length for the -phase particles, m - ⁰ reference characteristic length for the-phase particles, m - characteristic length for the-phase, m - _i i=1, 2, 3 lattice vectors, m - m convolution product weighting function - m _v special convolution product weighting function associated with the traditional volume average - n _i i=1, 2, 3 integers used to locate the position of particles - n unit normal vector pointing from the-phase toward the-phase - n _e outwardly directed unit normal vector at the entrances and exits of the-phase - r _p position vector locating the centroid of a particle, m - r gaussian probability distribution used to determine the size of a particle, m - r ₀ characteristic length of an averaging region, m - r position vector, m - r _m support of the weighting functionm, m - averaging volume, m³ - V volume of the-phase contained in the averaging volume,, m³ - x positional vector locating the centroid of an averaging volume, m - x ₀ reference position vector associated with the centroid of an averaging volume, m - y position vector locating points relative to the centroid, m - y position vector locating points in the-phase relative to the centroid, m Greek Letters indicator function for the-phase - Dirac distribution associated with the- interface - V /V, volume average porosity - /L, small parameter in the method of spatial homogenization - standard deviation ofa _i - _r standard deviation ofr - _r intrinsic phase average of      

Application of the method of integral diffusion to investigation of turbulent transfer

A number of authors have critically examined semiempirical mixing length theories [1]. A defect of these theories is connected with the fact that the magnitude of the mixing length, which is assumed to be small in constructing the theory, turns out in experiments to be comparable with the characteristic dimensions of the flow region. Thus, the concept of volume convection [2–4] or integral diffusion [5], which is understood to be a transfer mechanism in which the friction stress is not expressed in terms of the velocity gradient, is introduced along with the concept of gradient diffusion. In addition, there are a number of experimental papers [6] in which it is shown that the turbulent friction stress cannot be equal to zero at the place in the flow where the derivative of the velocity is equal to zero. Mixing length theory does not describe this effect.It is possible to generalize mixing length theory [7–9] in a way which eliminates these defects. Flow of an incompressible fluid is considered.     

On forced vibration of anisotropic cylinders

Summary The dynamic response of a circular cylinder with thick walls of transverse curvilinear isotropy subjected to a uniformly distributed pressure varying periodically with time is analyzed by means of the Laplace transformation, and the exact solution is obtained in closed form. The previously obtained solutions for forced vibrations with isotropy, and free vibrations with transverse curvilinear isotropy are included as special cases of the general results reported here.Nomenclature t time - r, , z cylindrical coordinates - _ii components of normal strain - _ii components of normal stress - u radial displacement - c _ij elastic constant - mass density - c ² c ₁₁/ - ² c ₂₂/c ₁₁ - a, b inner, outer radius of the cylinder - , A, B constants - forced angular frequency - function defined by (9) - p, real, complex variables - constant defined by (14) - real number - , Lamé elastic constants - J (x) Bessel function of first kind - Y (x) Bessel function of second kind - I (x) modified Bessel function of first kind - K (x) modified Bessel function of second kind     

Flow around stepped bodies with strong compression in a shock layer

This article discusses plane and axisymmetric flows of a nonviscous ideal gas around bodies of stepped form, forming with a Mach number M= and an adiabatic indexN1. The greatest amount of attention is paid to the case where there is no Newtonian free layer, but the shock layer is detached at great distances from the nose of the body.Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 4, pp. 104–112, July–August, 1973.     

Flow in porous media III: Deformable media

Stokes flow in a deformable medium is considered in terms of an isotropic, linearly elastic solid matrix. The analysis is restricted to steady forms of the momentum equations and small deformation of the solid phase. Darcy's law can be used to determine the motion of the fluid phase; however, the determination of the Darcy's law permeability tensor represents part of the closure problem in which the position of the fluid-solid interface must be determined.Roman Letters A interfacial area of the- interface contained within the macroscopic system, m² - A interfacial area of the- interface contained within the averaging volume, m² - A _e area of entrances and exits for the-phase contained within the macroscopic system, m² - A ^* interfacial area of the- interface contained within a unit cell, m² - A _e ^* area of entrances and exits for the-phase contained within a unit cell, m² - E Young's modulus for the-phase, N/m² - e i unit base vectors (i = 1, 2, 3) - g gravity vector, m²/s - H height of elastic, porous bed, m - k unit base vector (=e ₃) - characteristic length scale for the-phase, m - L characteristic length scale for volume-averaged quantities, m - n unit normal vector pointing from the-phase toward the-phase (n = -n ) - p pressure in the-phase, N/m² - P p – g·r, N/m² - r ₀ radius of the averaging volume, m - r position vector, m - t time, s - T total stress tensor in the-phase, N/m² - T ⁰ hydrostatic stress tensor for the-phase, N/m² - u displacement vector for the-phase, m - V averaging volume, m₃ - V volume of the-phase contained within the averaging volume, m³ - v velocity vector for the-phase, m/s Greek Letters V /V, volume fraction of the-phase - mass density of the-phase, kg/m³ - shear coefficient of viscosity for the-phase, Nt/m² - first Lamé coefficient for the-phase, N/m² - second Lamé coefficient for the-phase, N/m² - bulk coefficient of viscosity for the-phase, Nt/m² - T –T ⁰ , a deviatoric stress tensor for the-phase, N/m²     

Some conditions for the existence of a “Pinch” structure of the current sheet in plasma

In pulsed plasma accelerators of various configurations it is frequently possible to observe an instability of the current sheet characterized by separation into individual narrow channels (local pinches). In this paper the flow of plasma over these channels is qualitatively investigated. It is shown that a large part of the discharge current can flow through a series of narrow channels only when they completely decelerate the plasma flow. This places upper and lower bounds with respect to pressure on the region of existence of the pinch structure, if it is assumed that deceleration by the magnetic field predominates over viscous deceleration. The upper bound is obtained from the condition of deceleration of the plasma by the magnetic field near the pinches, the lower bound from the condition that the ratio of specific heats of the gas must be small. As the initial gas pressure decreases, so does the plasma deceleration time, but the characteristic times of the excitation and ionization processes increase; therefore the condition 1 ceases to be satisfied. The results are compared with experiment. The induced currents in the plasma near the pinches are also a reason for the stability of the network of current filaments.According to [1] a pulsed high-current discharge in a low-pressure gas develops as follows. After breakdown and the attainment of a certain uniform value of the gas conductivity the current continues to increase only in a surface layer (skin effect), which, if the particle density is sufficient, acts as an impermeable piston on the gas in front of it (snowplow model). Clearly, to judge from the collapse time of the current cylinder [2], this gives a good picture of the situation in the Z pinch. However, in coaxial and rail guns the gas is by no means always completely raked up by the current sheet (see, for example, [3]). This is also indicated by the observed broad spectrum of plasmoid velocities.It has been noted in high-speed photographs of pulsed discharges of very different configurations (Z and pinches, coaxial and rail plasma guns) that the current sheet is often divided into a series of channels [4–12], through which the main current flows. This follows from their high acceleration as compared with the rest of the plasma [4, 5], from the arrangement of the cathode spots along the electrodes, and from the large value of the plasma density in the channels [11]. This effect is well reproduced from discharge to discharge.Experiments with the Z pinch [7] have shown that the division of the current sheet into pinches has almost no effect on its velocity, which is in good agreement with calculations based on the snowplow model, although the diameter of the channels (determined, it is true, from the luminescence distribution and not the current density) is much less than the distance between them. On the other hand, in rail guns the pinches only slightly entrain the gas filling the tube [11]. The surface layer pinch instability is observed in the pressure range from several mm to several Hg [9]. Both pressure limits depend heavily on the nature of the gas in which the discharge takes place, the upper limit being approximately inversely proportional to the molecular weight of the gas [9], In a rail gun at low initial gas pressures (p < <20 Hg) pinches appear at some optimal rate of gas release from the electrodes and the walls [12].A uniform transverse magnetic field has only a slight influence on the pinches, since it is smaller in magnitude than the magnetic field of the pinches themselves [11]. A transverse magnetic field nonuniform along the length of the pinches causes them to decay, if the magnitude of the external field is comparable with the pinch field. Pinch decay was observed by the author in a rail gun when one of the electrodes was cut out so that on a certain section of the rail its transverse dimension was reduced to the diameter of the pinch, which was much less than the distance between electrodes (this setup was described in [11]).The instability of a plane current sheet in a plasma has been demonstrated on several occasions [13, 14]. Under experimental conditions the pinches move through the plasma. Therefore we will consider the case when a considerable part of the discharge current flows through a series of narrow channels (assuming that for this case the conductivity of the pinches is infinite).In conclusion, the author thanks A. K. Musin for his helpful suggestions.     

Zur Thermo-Gasdynamik der Strömung im zylindrischen Rohr

Zusammenfassung Bei höheren Gasgeschwindigkeiten in zylindrischen Rohren dürfen die Wärme-Austauschvorgänge — auch wenn sie technisch unbeabsichtigt sind — nicht mehr vernachlässigt werden. Bei Abkühlvorgängen kann thermisches Verblocken oder thermal choking auftreten; die Geschwindigkeiten nehmen im Unterschallbereich teils zu, teils ab, wobei die gewählte Darstellung eine einfache Übersicht erlaubt.     

Die Temperaturabhängigkeit der Oberflächenspannung von reinen Kältemitteln vom Tripelpunkt bis zum kritischen Punkt

Zusammenfassung Die Oberflächenspannung von sechs reinen Substanzen — SF₆, CCl₃F, CCl₂F₂, CClF₃, CBrF₃ und CHClF₂ — wurde mit Hilfe einer modifizierten Kapillarmethode gemessen. Die zur Berechnung der Oberflächenspannung erforderlichen Sättigungsdichten und wurden teils aus vorhandenen Zustandsgleichungen, teils aus ebenfalls gemessenen Brechungsindizes bestimmt. Die Temperaturabhängigkeit der Oberflächenspannung läßt sich durch einen erweiterten Ansatz nach van der Waals =_O (T_c-T)(1+...) darstellen, wobei bei einfachen Stoffen ein eingliedriger, bei polaren und assoziierenden Stoffen ein zweigliedriger Ansatz notwendig und ausreichend ist. Für den kritischen Exponenten der Oberflächenspannung wurde ein von der molekularen Substanz weitgehend unabhängiger Wert von =1.284±0.005 gefunden.

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Temperature dependence of surface tension of pure refrigerants from triple point up to the critical point

The surface tension of six fluids (SF₆, CCl₃F, CCl₂F₂, CClF₃, CBrF₃, CHClF₂) have been measured by means of a modified capillary rise method. The liquid vapor densities, which are needed to calculate the surface tension, have partly been determined by means of refractive indices simultaneously measured in the same apparatus. The temperature dependence of the surface tension is described by an extended van der Waals power law =_O(T_c-T)(1+...). For simple fluids one term and for polar and associating fluids two terms are necessary and sufficient. The critical exponent is found to be 1.284 ± 0.005 and nearly independent of the molecular structure.

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Formelzeichen a² Laplace-Koeffizient - a Parameter - B_O, B_on Koeffizient der Koexistenzkurve - g Erdbeschleunigung - H Höhe, kapillare Steighöhe - LL Lorentz-Lorenz-Funktion oder Refraktionskonstante - M molare Masse - M Zahl der Meßwerte - N Zahl der unbekannten Parameter - n Brechungsindex - p Druck - R,r Radius - s Entropie - SD Standardabweichung - T, t Temperatur - u innere Energie Griechische Formelzeichen Exponent des Laplace-Koeffizienten - Exponent der Koexistenzkurve - 2. Exponent der Oberflächenspannung - Wellenlänge des Lichts - Exponent der Oberflächenspannung - _D Dipolmoment - , Dichte der Flüssigkeit bzw. des Dampfes - Oberflächenspannung - reduzierte Temperatur (1-T/T_c) - ² gewichtete Varianz Indizes c kritischer Zustand - D Differenz - m Mittelwert - T Isotherme - t Zustand am Tripelpunkt - S Zustand am Schmelzpunkt - bezogen auf Oberfläche