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     

Indentation of the semi-infinite elastic solid by a concave rigid punch

A solution is obtained for the relationship between load, displacement and inner contact radius for an axisymmetric, spherically concave, rigid punch, indenting an elastic half-space. Analytic approximations are developed for the limiting cases in which the ratio of the inner and outer radii of the annular contact region is respectively small and close to unity. These approximations overlap well at intermediate values. The same method is applied to the conically concave punch and to a punch with a central hole. , , . , . . .     

Marangoni-convection in a rotating liquid container

In a partially filled and constantly spinning container in zerogravity condition there arises under the action of an axial temperature gradient a thermo-capillary convection. This so-called Marangoni convection has been treated analytically for a directly imposed temperature gradient upon the free liquid surface and also for a constant but different temperature at the upper and lower disc wall. The streamfunction and circulation have been obtained, from which the velocity distribution could be determined.

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Marangoni-Konvektion in einer in einem Behälter rotierenden Flüssigkeit

Zusammenfassung Durch das Vorhandensein eines axialen Temperaturgradienten ergibt sich in einem mit konstanter Geschwindigkeit rotierenden teilweise mit Flüssigkeit gefüllten Behälter eine thermalkapillare Korrelation. Diese sogenannte Marangoni-Konvektion wird analytisch behandelt für eine lineare axiale und eine beliebige axiale Temperaturverteilung auf der Flüssigkeitsoberfläche. Stromfunktion und Zirkulation werden analytisch bestimmt. Daraus ergeben sich die Geschwindigkeitsverteilungen in radialer, zirkumferentialer und axialer Richtung.

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Nomenclature a radius of cylindrical container - b radius to free liquid surface - h height of container - I _m, K_m Modified Besselfunktions of first and second kind and orderm - k _j roots of bi-cubic equation (24 b) - k=b/a diameter ratio of location of free liquid surface and container wall - r, , z polar cylindrical coordinates - T(r, z) temperature distribution of liquid - u, v, w radial-, circumferential-, and axial velocity of the liquid, resp. - thermal expansion coefficient - dynamic viscosity of liquid - =/ kinematic viscosity - density of liquid - surface tension of liquid - _r , _rz shear stresses - (r, z) circulation - (r, z) stream function - ₀ speed of spin of container about axis of symmetry     

Gasdynamics of a plasma in a multiple-mirror magnetic trap with “point” mirrors

Equations are derived for the gasdynamics of a dense plasma confined by a multiple-mirror magnetic field. The limiting cases of large and small mean free paths have been analyzed earlier: ₀ and k, where is the length of an individual mirror machine, ₀ is the size of the mirror, and k is the mirror ratio. The present work is devoted to a study of the intermediate range of mean free paths ₀ k. It is shown that in this region of the parameters the process of expansion of the plasma has a diffusional nature, and the coefficients of transfer of the plasma along the magnetic field are calculated.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 14–19, November–December, 1974.The authors thank D. D. Ryutov for the statement of the problem and interest in the work.     

Parameter identification in a soil with constant diffusivity

We consider infiltration into a soil that is assumed to have hydraulic conductivity of the form K = K = K_se^h and water content of the form = K – _r. Here h denotes capillary pressure head while K_s, , and _r represent soil specific parameters. These assumptions linearize the flow equation and permit a closed form solution that displays the roles of all the parameters appearing in the hydraulic function K and . We assume K_s and _r to be known. A measurement of diffusivity fixes the product of and resulting in a parameter identification problem for one parameter. We show that this parameter identification problem, in some cases, has a unique solution. We also show that, in some cases, this parameter identification problem can have multiple solutions, or no solution. In addition it is shown that solutions to the parameter identification problem can be very sensitive to small changes in the problem data.     

Flow in porous media II: The governing equations for immiscible,two-phase flow

The Stokes flow of two immiscible fluids through a rigid porous medium is analyzed using the method of volume averaging. The volume-averaged momentum equations, in terms of averaged quantities and spatial deviations, are identical in form to that obtained for single phase flow; however, the solution of the closure problem gives rise to additional terms not found in the traditional treatment of two-phase flow. Qualitative arguments suggest that the nontraditional terms may be important when / is of order one, and order of magnitude analysis indicates that they may be significant in terms of the motion of a fluid at very low volume fractions. The theory contains features that could give rise to hysteresis effects, but in the present form it is restricted to static contact line phenomena.Roman Letters (, = , , and ) A interfacial area of the- interface contained within the macroscopic system, 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 the averaging volume, 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 - g gravity vector, m²/s - H mean curvature of the- interface, m^–1 - H area average of the mean curvature, m^–1 - H – H , deviation of the mean curvature, m^–1 - I unit tensor - K Darcy's law permeability tensor, m² - K permeability tensor for the-phase, m² - K viscous drag tensor for the-phase equation of motion - K viscous drag tensor for the-phase equation of motion - L characteristic length scale for volume averaged quantities, m - characteristic length scale for the-phase, m - n unit normal vector pointing from the-phase toward the-phase (n = –n ) - p _c p – P , capillary pressure, N/m² - p pressure in the-phase, N/m² - p intrinsic phase average pressure for the-phase, N/m² - p – p , spatial deviation of the pressure in the-phase, N/m² - r ₀ radius of the averaging volume, m - t time, s - v velocity vector for the-phase, m/s - v phase average velocity vector for the-phase, m/s - v intrinsic phase average velocity vector for the-phase, m/s - v – v , spatial deviation of the velocity vector for the-phase, m/s - V averaging volume, m³ - V volume of the-phase contained within the averaging volume, m³ Greek Letters V /V, volume fraction of the-phase - mass density of the-phase, kg/m³ - viscosity of the-phase, Nt/m² - surface tension of the- interface, N/m - viscous stress tensor for the-phase, N/m² - / kinematic viscosity, m²/s     

Uniqueness and the maximum principle for quasilinear elliptic equations with quadratic growth conditions

In this paper, we show that the maximum principle holds for quasilinear elliptic equations with quadratic growth under general structure conditions.Two typical particular cases of our results are the following. On one hand, we prove that the equation (1) {ie77-01} where {ie77-02} and {ie77-03} satisfies the maximum principle for solutions in H ¹()L(), i.e., that two solutions u ₁, u ₂H¹() L() of (1) such that u ₁u₂ on , satisfy u ₁u₂ in . This implies in particular the uniqueness of the solution of (1) in H ₀ ¹ ()L().On the other hand, we prove that the equation (2) {ie77-04} where fH^–1() and g(u)>0, g(0)=0, satisfies the maximum principle for solutions uH¹() such that g(u)¦Du|{²L¹(). Again this implies the uniqueness of the solution of (2) in the class uH ₀ ¹ () with g(u)¦Du|{²L¹().In both cases, the method of proof consists in making a certain change of function u=(v) in equation (1) or (2), and in proving that the transformed equation, which is of the form (3) {ie77-05}satisfies a certain structure condition, which using ((v₁ -v ₂)⁺)ⁿ for some n>0 as a test function, allows us to prove the maximum principle.     

Convective heat transfer within fibrous insulation slabs

A numerical study of convective heat flow within a fibrous insulating slab is presented. The material is treated as an anisotropic porous medium and the variation of properties with temperature is taken into account. Good agreement is obtained with available experimental data for the same geometry.

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Zusammenfassung Für den konvektiven Wärmestrom in einem faserförmigen Isolierstoff wird eine numerische Berechnung angegeben. Der Stoff wird als anisotropes poröses Medium mit temperaturabhängigen Stoffwerten angesehen. Die Übereinstimmung mit verfügbaren Versuchswerten ist gut.

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Nomenclature C_p specific heat of the gas at the mean temperature - Da Darcy number=k_y/H² - Gr^* modified Grashof number=gTHk_y/²= (Grashof number) × (Darcy number) - H thickness of the specimen - P gas pressure - Pr^* modified Prandtl number= C_p/_x - Ra^* modified Rayleigh number=Gr^* Pr^* - R_p ratio of permeabilities=k_y/k_x - R_k ratio of conductivities= _y/_x - T absolute temperature of the gas - t₁ absolute temperature of the hot face - T₂ absolute temperature of the cold face - T_m mean temperature of the gas=(T₁+T₂)/2 - k_x specific permeability of the porous medium along the x-direction - k_y specific permeability of the porous medium along the y-direction - p T/T_m - q exponent - r exponent - u gas velocity along the x-direction - v gas velocity along the y-direction - X_* distance along the x-direction - y_* distance along the y-direction - T temperature difference=t₁–T₂ - thermal coefficient of expansion of the gas - _m thermal coefficient of expansion of the gas at the mean temperature - _* T–T_m - dimensionless temperature= ^*/T - _a apparent thermal conductivity of the porous medium along the x-direction - _al local apparent thermal conductivity of the porous medium along the x-direction - _x thermal conductivity of the porous medium along the x-direction in the absence of convection - _y thermal conductivity of the porous medium along the y-direction in the absence of convection - dynamic viscosity of the gas - _m dynamic viscosity of the gas at the mean temperature - kinematic viscosity of the gas - _m kinematic viscosity of the gas at the mean temperature - density of the gas - _m density of the gas at the mean temperature - ^* stream function at any point - dimensionless stream function= ^*/( _m/_m)     

Finite-difference analysis of MHD Stokes problem for a vertical infinite plate in dissipative fluid

Finite-difference solution of MHD flow past an impulsively started vertical infinite plate in an electrically conducting fluid has been presented on taking into account the viscous dissipative heat. Results for velocity and temperature are shown graphically whereas the numerical values of the skin-friction and the rate of heat transfer are entered in the table. The results are discussed in terms of the parameters M (the Hartmann number), G (the Grashof number, G>0, cooling of the plate by free convection, G<0, heating of the plate by free convection currents), E (the Eckert number) and P (the Prandtl number).Nomenclature B ₀ applied magnetic field - c _p specific heat at constant pressure - g acceleration due to gravity - k thermal conductivity - t time - T temperature of the fluid near the plate - T temperature of the fluid far away from the plate - U ₀ velocity of the plate - u velocity of the fluid - coefficient of volume expansion - kinematic viscosity - scalar electrical conductivity - coefficient of viscosity - density of the fluid     

Untersuchung des lokalen Stoffüberganges am laminaren und turbulenten Rieselfilm

Zusammenfassung Der lokale Stoffübergang wurde in Abhängigkeit von der Meßlänge, dem Startort und der Zulaufhöhe gemessen. Der Gültigkeitsbereich der Theorie von Nusselt wird ermittelt. Die Reynolds-Zahl nahm Werte zwischen 3,86 und 2496 an. Die örtlich wirkende Hydrodynamik ist entscheidend für das Anwachsen der örtlichen Sherwood-Zahl. Die Genauigkeit aller Versuchsergebnisse kann auf ± 5% abgeschätzt werden.

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Investigation of the local mass transfer of a laminar and turbulent falling liquid film

The local mass transfer was measured as a function of the measuring length, the starting point and the liquid height above the ring-slot. The range of the Reynolds number was 3,86 Re 2496. The validity of the Nusselt theory and the range of it is shown. The local hydrodynamic is the most important factor of the increase of the local Sherwood number. The accuracy of the measurements is ± 5%.

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Bezeichnungen a Temperaturleitfähigkeit m²/s=/(c_p) - c Konzentration, c=¯c + c kmol/m³ - c_i0 Konzentration im Flüssigkeitskern kmol/m³ - D Diffusionskoeffizient m²/s - EL-NR Elektrodennummer - Fa Faraday-Konstante A s/kgäq=96,5·10⁶ - g Erdbeschleunigung m/s² - i_G Grenzstromdichte A/m² - u Geschwindigkeit in x-Richtung, u= + u - U Umfang des Rohres m - v Geschwindigkeit in y-Rich- m/stung, v=¯v + v - V^* Volumenstrom m³/s - x Lauflänge, Koordinate in m Strömungsrichtung - x_M Meßlänge für den Stoff-Übergang m - x_ST Startort für den Stoff-Übergang m - y Wegkoordinate senkrecht zur Rohroberfläche m - z Wertigkeit der Elektro-denreaktion kgäq/kmol - ZH Zulaufhöhe m - Wärmeübergangskoeffizient W/m²C - Stoffübergangskoeffizient m/s - Filmdicke m - Wärmeleitfähigkeit W/(mC) - kinematische Viskosität m²/s - Re=u/=V^*/U Reynolds-Zahl - Pr=/a=c_p/ Prandtl-Zahl - Sc=/D Schmidt-Zahl - Nu= / Nusselt-Zahl - Sh= /D Sherwood-Zahl - SHL lokale Sherwood-Zahl - SHM mittlere Sherwood-Zahl - - zeitlich gemittelt - örtlich gemittelt Die Durchführung der Arbeit am Institut für Verfahrens — und Kältetechnik der ETH Zürich bei Prof. Dr. P. Grassmann wurde ermöglicht durch Zuschüsse der Kommission zur Förderung der wissenschaftlichen Forschung und meiner Eltern.     

Calculation of heat transfer accompanying hypersonic three-dimensional flow of air over slender blunt-nosed cones

The effective length method [1, 2] has been used to make systematic calculations of the heat transfer for laminar and turbulent boundary layers on slender blunt-nosed cones at small angles of attack ( + 5° in a separationless hypersonic air stream dissociating in equilibrium (half-angles of the cones 0 20°, angles of attack 0 15°, Mach numbers 5 M 25). The parameters of the gas at the outer edge of the boundary layer were taken equal to the inviscid parameters on the surface of the cones. Analysis of the results leads to simple approximate dependences for the heat transfer coefficients.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 173–177, September–October, 1981.     

A mixing length model for turbulent flow in constant cross section ducts

In this paper, a study is made of the damping influence of the wall on turbulent fluid flow. By considering the oscillation of the whole of the boundary, van Driest's original hypothesis has been extended to obtain the wall damping factor in flow in a duct of constant cross section. The damping factor is used in conjunction with mixing length expressions to obtain the velocity field. Particular examples considered are plane parallel flow and axisymmetric flow in a pipe and in an annulus.

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Ein Modell für die Mischungslänge von turbulenten Strömungen in Rohren mit konstantem Querschnitt

Zusammenfassung In dieser Arbeit wurde der dämpfende Wandeinfluß in turbulenten Strömungen untersucht. Unter Berücksichtigung der Schwingungen in der gesamten Grenzschicht wurde die ursprüngliche Theorie von van Driest erweitert und ein Dämpfungsfaktor an der Wand in Rohrströmungen mit konstantem Querschnitt ermittelt. Dieser Dämpfungsfaktor diente in Verbindung mit Ausdrücken für die Mischungslänge zur Bestimmung des Geschwindigkeitsfeldes. Ausgewählte Beispiele waren die ebene Parallelströmung sowie die Zylinderströmung in einem Rohr und einem Ringspalt.

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Nomenclature A, A^*(=Au/v) Parameter defined in text - b, b^*(=bu/v) semi-width of parallel plate channel - c(= 1/A) parameter defined in text - E[, /2] complete elliptic integral of the second kind - d damping factor - F, G, H functions - l, l^*(=/v) mixing length - M_O, _O functions - r, r^*(=ru/v) radius - A real part of function - R, S, T, U functions - u, u^*(=u/u) velocity in flow direction Z - friction velocity - x, y, z co-ordinates (z in flow direction) - y^*(=yu/v) non-dimensional wall distance - fluid density - , _eff kinematic viscosity, effective kinematic viscosity - phase angle, or polar coordinate angle - shear stress - (=r/r_W) radius ratio - angular velocity Suffixes w wall value - far from a wall     

An analytical solution for the velocity and shear rate distribution of non-ideal Bingham fluids in concentric cylinder viscometers

An analytical solution is presented for the calculation of the flow field in a concentric cylinder viscometer of non-ideal Bingham-fluids, described by the Worrall-Tuliani rheological model. The obtained shear rate distribution is a function of the a priori unknown rheological parameters. It is shown that by applying an iterative procedure experimental data can be processed in order to obtain the proper shear rate correction and the four rheological parameters of the Worrall-Tuliani model as well as the yield surface radius. A comparison with Krieger's correction method is made. Rheometrical data for dense cohesive sediment suspensions have been reviewed in the light of this new method. For these suspensions velocity profiles over the gap are computed and the shear layer thicknesses were found to be comparable to visual observations. It can be concluded that at low rotation speeds the actually sheared layer is too narrow to fullfill the gap width requirement for granular suspensions and slip appears to be unavoidable, even when the material is sheared within itself. The only way to obtain meaningfull measurements in a concentric cylinder viscometer at low shear rates seems to be by increasing the radii of the viscometer. Some dimensioning criteria are presented.Notation A, B Integration constants - C Dimensionless rotation speed = µ/_y - c = 2µ - d = ₀ ²–2c_y - f() = (–₀)²+2c(–_y) - r Radius - r _b Bob radius - r _c Cup radius - r _y Yield radius - r ₀ Stationary surface radius - r Rotating Stationary radius - Y ₀ Shear rate parameter = /µ Greek letters Shear rate - = (r _y /r _b )²– 1 - µ Bingham viscosity - µ₀ Initial differential viscosity - µ µ₀-µ - Rotation speed - Angular velocity - Shear stress - _b Bob shear stress - _B Bingham stress - _y (True) yield stress - ₀ Stress parameter = _B +µ Y ₀ - _B - _y      

Stress analysis of the cylinder subjected to arbitrarily distributed axisymmetrical loads

Summary Stress analysis has been carried out for a finite cylinder subjected to arbitrarily distributed axisymmetrical surface loads. Direct stress _x in the axial direction is assumed to be of the form _x = ₀+r ₁ +r ₂ where ₀ to ₂ are functions of x. Using the equations of equilibrium and compatibility the other direct stresses and the shearing stress are expressed by ₁ and ₂. Fundamental equations governing ₁ and ₂ are introduced using the variational principle of complementary energy. From the results of the present analysis it is evident that the boundary conditions can be satisfied completely even for the case where the external forces are specified in complicated form, and that more accurate solutions can easily be obtained by introducing additional terms in _x.

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Spannungsanalyse für den Zylinder unter axialsymmetrischer Last in beliebiger Verteilung

Übersicht Für einen endlichen Zylinder unter axialsymmetrischer Oberflächenlast in beliebiger Verteilung werden die Spannungen ermittelt. Die Normalspannung in Axialrichtung wird in der Form _x = ₀+r ₁ +r ₂ angesetzt mit ₀, ₁, ₂ als Funktionen von x. Mit Hilfe der Gleichgewichtsund Verträglichkeitsbedingungen werden die anderen Normalspannungen und die Schubspannung durch ₁ und ₂ ausgedrückt. Über das Variationsprinzip für die Komplementärenergie werden die grundlegenden Gleichungen für ₁ und ₂ eingeführt. Die Ergebnisse zeigen, daß die Randbedingungen selbst für komplizierte Belastungsarten vollständig erfüllbar sind und mit zusätzlichen Termen in _x mühelos noch genauere Lösungen bestimmt werden können.

     

The measurement of the material parameters of viscoelastic fluids using a rotating sphere and a rheogoniometer

Summary In this work, measurement of the flow field around a rotating sphere has been used to obtain the material parameters of a second-order Rivlin-Ericksen fluid. Experiments were carried out with a Laser-Doppler anemometer to obtain the velocity distribution and usingGiesekus' analysis, the material parameters for the second-order fluid were obtained.

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Zusammenfassung In dieser Untersuchung wird die Ausmessung des Strömungsfeldes um eine rotierende Kugel dazu verwendet, um die Stoffparameter einer Rivlin-Ericksen-Flüssigkeit zweiter Ordnung zu erhalten. Die Experimente zur Bestimmung der Geschwindigkeitsverteilung werden mit einem Laser-Doppler-Anemometer durchgeführt, und zur Auswertung der Parameter der Flüssigkeit zweiter Ordnung wird eine Analyse vonGiesekus benutzt.

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Notations A ₁,A₂ Rivlin-Ericksen tensor - A ² Parameter used in eq. [12] - a Radius of the sphere - B Parameter used in eq. [12] - I Unit tensor - m ₀(₁ – ₂)/a², parameter used by ref. (8) - N ₁,N₂ First and second normal stress difference - p Isotropic pressure - Radial distance from the centre of the rotating body - S ₁,S₂ Stress tensor - v _r,v,v Velocity components in a spherical coordinate system - ₀,₁,₂ Material parameters used in eq. [2] - Shear rate - _a Apparent voscosity - ₀ Zero-shear viscosity - Angle measured from the axis of rotation - Fluid density - Stream function - Shear stress - Angular velocity With 3 figures     

Existence theorem for a minimum problem with free discontinuity set

We study the variational problem Where is an open set in ⁿ ,n2gL ^q () L (), 1q<+, O<, <+ andH _n–1 is the (n–1)-dimensional Hausdorff Measure.     

Method of mass-average quantities for the boundary layer in an outer flow with transverse nonuniformity

An engineering method is proposed for calculating the friction and heat transfer through a boundary layer in which a nonuniform distribution of the velocity, total enthalpy, and static enthalpy is specified across the streamlines at the initial section x₀. Such problems arise in the vortical interaction of the boundary layer with the high-entropy layer on slender blunt bodies, with sudden change of the boundary conditions for an already developed boundary layer (temperature jump, surface discontinuity), and in wake flow past a body, etc.Notation x, y longitudinal and transverse coordinates - u,, H, h gas velocity, stream function, total and static enthalpy - p,,, pressure, density, viscosity, Prandtl number - , q friction and thermal flux at the body surface - r(x), (x) body surface shape and boundary layer thickness - V, M freestream velocity and Mach number - u⁽⁰⁾(x₀,), H⁽⁰⁾(x₀,), h⁽⁰⁾(x₀,) parameter distributions at initial section - u⁽⁰⁾(x,), h⁽⁰⁾(x,), h⁽⁰⁾(x,) profiles of quantities in outer flow in absence of friction and heat transfer at the surface of the body The indices v=0, 1 relate to plane and axisymmetric flows - , w, b, relate to quantities at the outer edge of the inner boundary layer, at the body surface in viscid and nonviscous flows, and in the freestream, respectively. The author wishes to thank O. I. Gubanov, V. A. Kaprov, I. N. Murzinov, and A. N, Rumynskii for discussions and assistance in this study.     

The contribution of internal degrees of freedom to the non-Newtonian rheology of model polymer fluids

We report non-equilibrium molecular dynamics simulations of rigid and non-rigid dumbbell fluids to determine the contribution of internal degrees of freedom to strain-rate-dependent shear viscosity. The model adopted for non-rigid molecules is a modification of the finitely extensible nonlinear elastic (FENE) dumbbell commonly used in kinetic theories of polymer solutions. We consider model polymer melts — that is, fluids composed of rigid dumbbells and of FENE dumbbells. We report the steady-state stress tensor and the transient stress response to an applied Couerte strain field for several strain rates. We find that the rheological properties of the rigid and FENE dumbbells are qualitatively and quantitatively similar. (The only exception to this is the zero strain rate shear viscosity.) Except at high strain rates, the average conformation of the FENE dumbbells in a Couette strain field is found to be very similar to that of FENE dumbbells in the absence of strain. The theological properties of the two dumbbell fluids are compared to those of a corresponding fluid of spheres which is shown to be the most non-Newtonian of the three fluids considered.Symbol Definition b dimensionless time constant relating vibration to other forms of motion - F force on center of mass of dumbbell - F _i force on bead i of dumbbell - F force between center of masses of dumbbells and - F _ij force between beads i and j - h vector connecting bead to center of mass of dumbbell - H dimensionless spring constant for dumbbells, in units of / ² - I moment of inertia of dumbbell - J general current induced by applied field - k _B Boltzmann's constant - L angular momentum - m mass of bead, (= m/2) - M mass of dumbbell, g - N number of dumbbells in simulation cell - P translational momentum of center of mass of dumbbell - P pressure tensor - P _xy xy component of pressure tensor - Q separation of beads in dumbbell - Q _eq equilibrium extension of FENE dumbbell and fixed extension of rigid dumbbell - Q ₀ maximum extension of dumbbell - r _ij vector connecting beads i and j - r position vector of center of mass dumbbell - R vector connecting centers of mass of two dumbbells - t time - t ^* dimensionless time, in units of m/ - T ^* dimensionless temperature, in units of /k - u potential energy - u velocity vector of flow field - u _x x component of velocity vector - V volume of simulation cell - X general applied field - strain rate, s^–1 - ^* dimensionless shear rate, in units of /m ² - general transport property - Lennard-Jones potential well depth - friction factor for Gaussian thermostat - shear viscosity, g/cms - ^* dimensionless shear viscosity, in units of m/ ² - ^* dimensionless number density, in units of ^–3 - Lennard-Jones separation of minimum energy - relaxation time of a fluid - angular velocity of dumbbell - orientation angle of dumbbell      

Supersonic flow past axisymmetric flared cones

Systematic data on the determination of the aerodynamic characteristics of axisymmetric bodies with a break in the generating line (Fig. 1a, b) in supersonic flow at zero angle of attack are presented in [1, 2, and others]. A characteristic feature of the flow past such bodies is the appearance of an extensive separation zone dec in the region of the break in the generator when the break angle exceeds some minimum value _min, which for a turbulent boundary layer depends basically on the Mach number M at the body surface ahead of the separation zone. In this case, compression waves which change into the oblique compression shocks dc and cc, emanate both from the beginning of the separation zone (point c) and from the end of it (point d). These shocks, intersecting at the point c, form the triple shock configuration acd and acc for which we introduce the notationac[c, d]. The maximum value (_max) of the generator break angle is limited by the possibility of the existence of an attached compression shock, dc. According to these data a change in the generator break angle for the range _minmax of the angle does not disrupt the nature of the flow in the separation zone, but only alters the size of this zone.We shall examine the flow past cones with values of the generator break angles (_max) for which the attached shock dc cannot exist.     

Dielectric properties of the composite system polyurethane — sodium chloride

Summary The dielectric properties of the composite system polyurethane-sodium chloride have been measured at frequencies between 10^–4 Hz and 3 · 10⁵ Hz in the temperature range from –150 °C up to +90 dgC. Three dielectric loss mechanisms have been found; they are indicated by ₁, ₂ and. The activation energy of the ₁-transition is 35 kcal/mole, that of the-transition 6.7 kcal/mole. The ₂-loss peak was only observed at frequencies of 10³ Hz and higher, forming one broad peak with the ₁-loss peak at lower frequencies. In the composite materials, the- and ₂-loss peaks measured at fixed frequencies were found at the same temperature. The ₂-loss peak, however, was shifted to a lower temperature, due to the sodium chloride filler. Comparison of experimental data of and tan with theoretical predictions concerning the dielectric properties of composite systems showed only partial agreement. The difference mainly consisted in. the temperature shift in the tan-peak of the ₁-transition.

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Zusammenfassung Die dielektrischen Eigenschaften des Verbundssystems Kochsalz-Polyurethankautschuk wurden im Frequenzgebiet zwischen 10^–4 Hz und 3.10⁵ Hz und im Temperaturbereich von –150 °C bis +90 °C gemessen. Es wurden drei dielektrische Verlustmaxima gefunden, die mit ₁, ₂ und angedeutet werden. Die Aktivierungsenergie des ₁-Überganges beträgt 35 kcal/Mol, die des-Überganges 6.7 kcal/Mol. Das ₂-Maximum konnte nur bei Frequenzen höher als 10³Hz vom ₁-Maximum gesondert erfaßt werden. Die Lage der ₂- und-Maxima war vom Füllgrad unabhängig. Das ₁-Maximum verschiebt sich mit steigendem Füllgrad zu niedrigeren Temperaturen. Die gemessenen Werte von und tan stimmen nur teilweise mit den Aussagen einer Theorie der dielektrischen Eigenschaften von Mischkörpern überein.