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.     

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.     

Stability of stationary traveling waves on the surface of a vertical film of viscous fluid

The stability of stationary traveling waves of the first and second families with respect to infinitesimal perturbations of arbitrary wavelength is subjected to a detailed numerical investigation. The existence of a unique region of stability of the first family is established for wave numbers (₁, ₁) corresponding to the optimal wave regime. There are several regions of stability of the second family ( _k , _k),k=2,3,..., lying close to the local flow rate maxima. In the regions of instability the growth rates of perturbations of the first family are several times greater than for the second family. This difference increases with increase in the Reynolds number. The calculations make it possible to explain a number of experimental observations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 33–41, May–June, 1989.The authors are grateful to V. Ya. Shkadov for his constant interest, and to A. G. Kulikovskii, A. A. Barmin and their seminar participants for useful discussions and suggestions.     

Solutions for the anisotropic elastic wedge at critical wedge angles

The classical solution for an isotropic elastic wedge loaded by uniform tractions on the sides of the wedge becomes infinite everywhere in the wedge when the wedge angle 2 equals , 2 or 2_* where tan 2_* = 2_*. When the wedge is loaded by a concentrated couple at the wedge apex the solution also becomes infinite at 2 = 2_*. A similar situation occurs when the wedge is anisotropic except that 2_* is governed by a different equation and depends on material properties. Solutions which do not become infinite everywhere in the wedge are available for isotropic elastic wedges. In this paper we present solutions for the anisotropic elastic wedge at critical wedge angles. The main feature of the solutions obtained here is that they are in a real form even though Stroh's complex formalism is employed.     

Numerical analysis of the branching of couette flow between concentric cylinders for various wave numbers

The character of stability loss of the circular Couette flow, when the Reynolds number R passes through the critical value R₀, is investigated within a broad range of variation of the wave numbers. The Lyapunov-Schmidt method is used [1, 2]; the boundary-value problems for ordinary differential equations arising in the case of its realization are solved numerically on a computer. It is shown that the branching character substantially depends on the wave number . For all a, excluding a certain interval (₁, ₂), the usual postcritical branching takes place: at a small supercriticality the circular flow loses stability and is softly excited into a secondary stationary flow — stable Taylor vortices. For wave numbers from the interval (₁,₂) a hard excitation of Taylor vortices takes place: at a small subcriticality R=R₀–² the secondary mode is unstable and merges with the Couette flow for 0; however, for a small supercriticality in the neighborhood of a circular flow there exist no stationary modes which are different.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 132–135, May–June, 1976.     

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.     

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.     

Absolute determination of the isochromatic parameter by load-stepping photoelasticity

A new approach to full-field automated photoelasticity is presented in which a circular polariscope is used to enable the isochromatic phase value () to be determined unambiguously and without input of a known isochromatic value obtained using an auxiliary technique. Values of cos are obtained from light-field and dark-field images for three loads of small incremental steps. Using a relatively straight-forward procedure, ramped phase maps for are produced which can be unwrapped using conventional techniques. The resulting distribution of is then found absolutely using information provided by which is the incremental change in the isochromatic phase value between the load steps. The results obtained for disk-in-compression tests presented here in comparison with theoretical solutions demonstrate that the technique is both simple to use and very accurate. A similar approach may be adopted using three wavelengths instead of three load steps.     

Attractors for the Two-Dimensional Navier–Stokes-α Model: An α-Dependence Study

The two-dimensional Navier–Stokes- model is considered on the torus and on the sphere. Upper and lower bounds for the dimension of the global attractors are given. The dependence of the dimension of the global attractors on is studied. Special attention is given for the limiting cases when 0, that is, when the Navier–Stokes- model tends to the Navier–Stokes equations, and to the case when .     

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.     

Parameter determination for thermodynamical models of the pseudoelastic behaviour of shape memory alloy by resistivity measurements and infrared thermography

Two thermodynamical models of pseudoelastic behaviour of shape memory alloys have been formulated. The first corresponds to the ideal reversible case. The second takes into account the hysteresis loop characteristic of this shape memory alloys.Two totally independent techniques are used during a loading-unloading tensile test to determine the whole set of model parameters, namely resistivity and infrared thermography measurements. In the ideal case, there is no difficulty in identifying parameters.Infrared thermography measurements are well adapted for observing the phase transformation thermal effects.Notations 1 austenite 2 martensite - ₍₎ Macroscopic infinitesimal strain tensor of phase - ₍₂₎ ^f Traceless strain tensor associated with the formation of martensite phase - Macroscopic infiniesimal strain tensor - Macroscopic infinitesimal strain tensor deviator - _f Trace - Equivalent strain - ^pe Macroscopic pseudoelastic strain tensor - x Distortion due to parent (austenite =1)product (martensite =2) phase transformation (traceless symmetric second order tensor) - M Total mass of a system - M() Total mass of phase - V Total volume of a system - V() Total volume of phase - z=M(2)/M Weight fraction of martensite - 1-z=M(1)/M Weight fraction of austenite - u ₀ ^* () Specific internal energy of phase (=1,2) - s ₀ ^* () Specific internal entropy of phase - Specific configurational energy - Specific configurational entropy - ₀ ^f (T) Driving force for temperature-induced martensitic transformation at stress free state ( ₀ ^f T) = T ^*–Ts ^*) - Kirchhoff stress tensor - Kirchhoff stress tensor deviator - Equivalent stress - Cauchy stress tensor - Mass density - K Bulk moduli (K ₀=K) - L Elastic moduli tensor (order 4) - E Young modulus - Energetic shear (₀ = ) - Poisson coefficient - M _s ^o (M _F ^o ) Martensite start (finish) temperature at stress free state - A _s ^o (A _F ^o ) Austenite start (finish) temperature at stress free state - C _v Specific heat at constant volume - k Conductivity - Pseudoelastic strain obtained in tensile test after complete phase transformation (AM) (unidimensional test) - ₀ Thermal expansion tensor - r Resistivity - 1MPa 10⁶ N/m ² - ⁽⁾ Specific free energy of phase - _n Specific free energy at non equilibrium (R model) - _n ^eq Specific free energy at equilibrium (R model) - _n ^v Volumic part of ^eq - Specific free energy at non equilibrium (R _L model) - _conf Specific coherency energy (R _L model) - _c Specific free energy at constrained equilibria (R _L model) - _it (T) Coherency term (R _L model)     

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.     

Strömungstechnische Gesichtspunkte der Flußbelastung durch Abwasser aus Kläranlagen

Zusammenfassung Für ein im Durchlauf betriebenes System bestehend aus einem Fluß (Vorfluter) und den angeschlossenen Kläranlagen wird eine Methode zur Bestimmung der Vorfluterbelastung durch die eingeleiteten Klärwässer angegeben. Die Methode erfaßt mit Rücksicht auf die Anwendung des Verursacherprinzips im Gewässerschutz die Belastung durch jede Kläranlage für sich, und zwar in Abhängigkeit von der Wasserführung, den Emissionsraten der betreffenden Kläranlage und dem Selbstreinigungsvermögen von den organischen Stoffen aus der betreffenden Kläranlage. Die abhängigen Veränderlichen sind mit der Fließgeschwindigkeit gewichtete Mittelwerte von Schmutzstoffdichten über den Vorfluterquerschnitt. Im Falle konstanter Vorflutertemperatur und zeitunabhängiger Struktur der Klärwässer ergeben sich beispielsweise für die abhängigen Veränderlichen einfache analytische Darstellungen, welche sich als spezielle Formen des -Theorems erweisen. Es wird gezeigt, bei einem unendlich langen Vorfluter mit konstantem Volumenstrom stromabwärts der Klärwassereinleitungen stimmen die erwähnten gewichteten Mittelwerte mit den entsprechenden ungewichteten stromabwärts der Klärwassereinleitungen überein. Die entwickelte Methode kann leicht erweitert werden, um den Sauerstoffschwund im Vorfluter durch jede Kläranlage für sich zu bestimmen.

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Fluid mechanical aspects of river pollution by effluents from waste treatment plants

The pollution of a river by effluent inflows from waste treatment plants is modeled under steady-state conditions. With respect to modern policies of environmental protection the method describes the river pollution by each plant separately, depending on the flow conditions, the emission rates of the plant and the microbiological decomposition of the biodegradable matter from the plant. Each dependent variable is a weighted cross-sectional mean of a density of organic matter. If the water temperature is constant and the composition of each effluent is independent of time the method gives simple analytic expressions for the dependent variables, which prove to be special versions of the -theorem. It is shown for an infinitely long river of constant volume rate of flow downstream of the effluent inflows: the weighted means mentioned agree with the corresponding nonweighted downstream of the effluent inflows. The present paper can easily the extended to determine the oxygen deficit in the river due to each plant.

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Bezeichnungen a Anzahl der Kläranlagen - D(t_b) Kennzahl, Einführung in 4.3 - ^eA Emissionsrate der abbaubaren or ganischen Verschmutzung aus der -ten Kläranlage - ^eU Emissionsrate der nichtabbaubaren organischen Verschmutzung aus der -ten Kläranlage - Vorfluterquerschnitt, Einführung in Gl. (4) - F Flächeninhalt von - dF Betrag eines Flächenelements, Einführung in Gl. (6) - J_A Diffusionsstromdichten, Einführung in Gl. (2) bzw. Gl. (3) - L Anzahl der Stromstrecken - M Gesamtmasse der abbaubaren or- ganischen Verschmutzung in den N Teilchen, Einführung in Gl. (17) - N Anzahl der verschmutzten Flußwasserteilchen, welche die -te Nahfeldvermischungszone während des Zeitintervalles t_a t_b für immer verlassen - P(x, t, x, t_c) Teilchendichte, Einführung in Gl. (11) und Gl.(12) - Q Selbstreinigungsvermögen, Einführung in Gl.(26) - t Zeitpunkt, Einführung in Gl.(11) - t, t_b Intervallgrenzen, Einführung in 4.1 - t_c Zeitpunkt, Einführung in Gl.(11) - t Zeitdifferenz, Einführung im Anschluß an Gl.(10) - t^* charakteristische Zeit, Einführung in 4.3 - Strömungsgeschwindigkeit Komponente von ¯b in Richtung der zu Tal weisenden Oberflächennormalen eines Vorfluterquerschnitts, Einführung in Gl. (5) und Gl. (6) - Volumenstrom, Einführung in Gl. (7) - x Ortsvektor - x Ortsvektor eines bestimmten markierten Teilchens zur Zeit t_c, Einführung in Gl.(11) - x längs der Stromachse gemessene Längenkoordinate - x x-Koordinate des Vorfluterquerschnitts durch x - x,x₊₁ x-Koordinaten der Vorfluterquerschnitte, welche die -te Stromstrecke stromaufwärts bzw. stromabwärts begrenzen. Einführung in 4.2. - transformierte Variable, Einführung in Gl.(65) - Zeitvariable - (t_b) Kennzahl, Einführung in 4.3. - Masse der abbaubaren organischen Verschmutzung in dem markierten Teilchen, Einführung in Gl.(14) - , Integrationsvariablen, Einführung in Gl.(38) bzw. Gl.(28) - _A durch die -te Kläranlage bedingte Dichte der abbaubaren organischen Verschmutzung - _U durch die -te Kläranlage bedingte Dichte der nichtabbaubaren organischen Verschmutzung - Mittelwerte von bzw· , Einführung in Gl.(31) bzw. Gl.(8) - _m -Wert zu einem Maximum, Einführung in Gl.(31) - Verhältnis zweier Mittelwerte, Einführung in Gl.(64) - stochastischer Mittelwert einer Zufallsgröße Y - Y Schwankung einer Zufallsgröße Y um den stochastisehen Mittelwert - Mittlung über den Vorfluterquerschnitt Der saubere Vorfluter sei definiert durch Standardwerte für Mindestanforderungen an die Flußwasserqualität. Vorschläge für solche Standardwerte werden in jüngster Zeit unter Berücksichtigung des Umweltschutzes ausführlich diskutiert ([1]; [2], S.- K 13 -).     

On the Ellipticity of the Middle Surface of a Shell and its Application to the Asymptotic Analysis of ‘Membrane’ Shells

We consider a surface S = (), where ² is a bounded, connected, open set with a smooth boundary and : ³ is a smooth map; let () denote the components of the two-dimensional linearized strain tensor of S and let 0 with length 0 > 0. We assume the the norm ,|| ()||0, in the space V0() = { H¹() × H¹() × L²(); = 0 on 0 } is equivalent to the usual product norm on this space. We then establish that this assumption implies that the surface S is uniformly elliptic and that we necessarily have 0 = .     

Saint-Venant's principle in nonlinear plane elasticity with sufficiently small strains

A homogeneous, isotropic cylinder in an equilibrium state of plane strain, whose cross-section is a rectangle R : [0 < y ₁ < 2L; 0 < y ₂ < h] with h/L 1, is considered. There are no body forces and the long sides are stress free. At y ₁ = 0 and y ₁ = 2L, there are arbitrary loadings, each statically equivalent to a uniformly distributed tensile or compressive stress c. Within the theory of nonlinear elasticity and with the strains and strain gradients assumed to be sufficiently small (but with no such assumptions on the displacement gradients), it is proved that if (,=1,2) represents the Cauchy stress tensor and the Kronecker delta, then |–c₁₁| decays exponentially to zero in R with distance from the nearer end, and the decay constant depends only upon the material but is independent of L.     

Mixed convection flow in narrow vertical ducts

The mixed convection flow in a vertical duct is analysed under the assumption that , the ratio of the duct width to the length over which the wall is heated, is small. It is assumed that a fully developed Poiseuille flow has already been set up in the duct before heat from the wall causes this to be changed by the action of the buoyancy forces, as measured by a buoyancy parameter . An analytical solution is derived for the case when the Reynolds numberRe, based on the duct width, is of 0 (1). This is extended to the case whenRe is 0 (^–1) by numerical integrations of the governing equations for a range of values of representing both aiding and opposing flows. The limiting cases, || 1 andR=Re of 0 (1), andR and both large, with of 0 (R ^1/3) are considered further. Finally, the free convection limit, large with R of 0 (1), is discussed.

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Mischkonvektion in engen senkrechten Rohren

Zusammenfassung Mischkonvektion in einem senkrechten Rohr wird unter der Voraussetzung untersucht, daß das Verhältnis der Rohrbreite zur Länge, über welche die Wand beheizt wird, klein ist. Es wird angenommen, daß sich bereits eine voll entwickelte Poiseuille-Strömung in dem Rohr eingestellt hat, bevor Antriebskräfte, gemessen mit dem Auftriebsparameter , aufgrund der Wandbeheizung die Strömung verändern. Es wird eine analytische Lösung für den Fall erhalten, daß die mit der Rohrbreite als charakteristische Länge gebildete Reynolds-ZahlRe konstant ist. Dies wird mittels einer numerischen Integration der wichtigsten Gleichungen auf den FallRe =f (^–1) sowohl für Gleich- als auch für Gegenstrom ausgedehnt. Weiterhin werden die beiden Grenzfälle betrachtet, wenn || 1 undR=Re konstant ist, sowieR und beide groß mit proportionalR ^1/3. Schließlich wird der Grenzfall der freien Konvektion, großes mit konstantem R, diskutiert.

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Nomenclature g acceleration due to gravity - Gr Grashof number - G modified Grashof number - h duct width - l length of the heated section of the duct wall - p pressure - Pr Prandtl number - Q flow rate through the duct - Q ₀ heat transfer on the wally=0 - Q ₁ heat transfer on the wally=1 - Re Reynolds number - R modified Reynolds number - T temperature of the fluid - T ₀ ambient temperature - T applied temperature difference - u, velocity component in thex-direction - v, velocity component in they-direction - x, co-ordinate measuring distance along the duct - y, co-ordinate measuring distance across the duct - buoyancy parameter - ₀ modified buoyancy parameter, ₀=R ^–1/3 - coefficient of thermal expansion - ratio of duct width to heated length, =h/l - (non-dimensional) temperature - _w applied temperature on the wally=0 - kinematic viscosity - density of the fluid - ₀ shear stress on the wally=0 - ₁ shear stress on the wally=1 - stream function     

A generalized Norton-Hoff model and the Prandtl-Reuss law of plasticity

The aim of this article is to study the quasistatic evolution of a three-dimensional elastic-perfectly plastic solid which satisfies the Prandtl-Reuss law. The evolution of the field of stresses -which solves a time dependent variational inequality — and that of the field of displacements u, have been described in previous works [15], [26], [35], [36], [37] but it was not shown there that and u satisfy indeed the Prandtl-Reuss constitutive law. In this article we find and u in a class of functions which are sufficiently regular for the Prandtl-Reuss law to make sense and we prove that and u satisfy the constitutive law. This result is attained by considering the elastic-perfectly plastic model as the limit of a family of elastic-visco-plastic models like those of Norton and Hoff. The Norton-Hoff type models which we introduce depend on a viscosity parameter > 0; we study the perturbed models (i.e. > 0 fixed) and then we pass to the limit 0.Dedicated to James Serrin on the occasion of his 60^th Birthday     

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.     

Effect of the Nature of the Interaction between a Gas and a Surface on the Relations for the Knudsen Layer in the Case of Strong Evaporation

The effect of the temperature accommodation coefficient _T on the relations at the Knudsen layer edge is investigated for strong evaporation using the moment method. An explicit expression for the dimensionless density as a function of the temperature and the Mach number M is obtained for 0 < _T < 1. For _T = 0 the entire solution is obtained in explicit form. It is shown that for = 0 and a condensation coefficient << 1 the temperature outside the Knudsen layer changes sharply as M varies from 0 to a certain value much less than unity after which the temperature ceases to depend on . For the model of specular reflection of the molecules from the surface the density and the temperature outside the Knudsen layer are found in explicit form as functions of the Mach number.