Lp-estimates for the nonlinear spatially homogeneous Boltzmann equation

This paper studies L^p-estimates for solutions of the nonlinear, spatially homogeneous Boltzmann equation. The molecular forces considered include inverse k^th-power forces with k > 5 and angular cut-off.The main conclusions are the following. Let f be the unique solution of the Boltzmann equation with f(v,t)(1 + ¦v²¦)^(s ₁ ^{+ /p)/2} L¹, when the initial value f ₀ satisfies f ₀(v) 0, f ₀(v) (1 + ¦v¦²)^(s ₁ ^{+ /p)/2} L¹, for some s₁ 2 + /p, and f ₀(v) (1 + ¦v¦²)^s/2 L^p. If s 2/p and 1 < p < , then f(v, t)(1 + ¦v¦²)^{(s s} ₁)^/2 L^p, t > 0. If s >2 and 3/(1+ ) < p < , thenf(v,t) (1 + ¦v¦²)^(s(s ₁ ^{+ 3/p))/2} L^p, t > 0. If s >2 + 2C₀/C₁ and 3/(l + ) < p < , then f(v,t)(1 + ¦v¦²)^s/2 L^p, t > 0. Here 1/p + 1/p = 1, x y = min (x, y), and C₀, C₁, 0 < 1, are positive constants related to the molecular forces under consideration; = (k – 5)/ (k – 1) for k^th-power forces.Some weaker conclusions follow when 1 < p 3/ (1 + ).In the proofs some previously known L-estimates are extended. The results for L^p, 1 < p < , are based on these L-estimates coupled with nonlinear interpolation.     

Similar solutions and the asymptotics of filtration equations

In this paper we consider the asymptotic behavior of solutions of the quasilinear equation of filtration as t. We prove that similar solutions of the equation u _t = (u )_xx asymptotically represent solutions of the Cauchy problem for the full equation u _t = [(u)]_xx if (u) is close to u for small u.     

A Strongly Degenerate Quasilinear Equation: the Parabolic Case

We prove the existence and uniqueness of entropy solutions of the Neumann problem for the quasilinear parabolic equation u_t=÷ a(u, Du), where a(z,)=f(z,), and f is a convex function of with linear growth as ||||, satisfying other additional assumptions. In particular, this class includes the case where f(z,)=(z)(), >0, and is a convex function with linear growth as ||||.     

The theory of a vibrating-rod densimeter

The paper presents a theory of a device for the accurate determination of the density of fluids over a wide range of thermodynamic states. The instrument is based upon the measurement of the characteristics of the resonance of a circular section tube, or rod, performing steady, transverse oscillations in the fluid. The theory developed accounts for the fluid motion external to the rod as well as the mechanical motion of the rod and is valid over a defined range of conditions. A complete set of working equations and corrections is obtained for the instrument which, together with the limits of the validity of the theory, prescribe the parameters of a practical design capable of high accuracy.Nomenclature A, B, C, D constants in equation (60) - A _j , B _j constants in equation (18) - a _j ⁺ , a _j ^– wavenumbers given by equation (19) - C _f drag coefficient defined in equation (64) - C _f ^/0 , C _f ^/1 components of C _f in series expansion in powers of - c speed of sound - D _b drag force of fluid b - D ₀ coefficient of internal damping - E extensional modulus - force per unit length - F _j ⁺ , F _j ^– constants in equation (24) - f, g functions of defined in equations (56) - G modulus of rigidity - I second moment of area - K constant in equation (90) - k, k constants defined in equations (9) - L half-length of oscillator - Ma Mach number - m _a mass per unit length of fluid a - m _b added mass per unit length of fluid b - m _s mass per unit length of solid - n _j eigenvalue defined in equation (17) - P power (energy per cycle) - P _a , P _b power in fluids a and b - p pressure - R radius of rod or outer radius of tube - R _c radius of container - R _i inner radius of tube - r radial coordinate - T tension - T _visc temperature rise due to heat generation by viscous dissipation - t time - v _r , v radial and angular velocity components - y lateral displacement - z axial coordinate - dimensionless tension - _a dimensionless mass of fluid a - _b dimensionless added mass of fluid b - _b dimensionless drag of fluid b - dimensionless parameter associated with - ₀ dimensionless coefficient of internal damping - dimensionless half-width of resonance curve - dimensionless frequency difference defined in equation (87) - spatial resolution of amplitude - R, , , _s , increments in R, , , _s , - dimensionless amplitude of oscillation - dimensionless axial coordinate - ratio of to - _a , _b ratios of to for fluids a and b - angular coordinate - parameter arising from distortion of initially plane cross-sections - _f thermal conductivity of fluid - dimensionless parameter associated with - viscosity of fluid - _a , _b viscosity of fluids a and b - dimensionless displacement - _j jth component of - density of fluid - _a , _b density of fluids a and b - _s density of tube or rod material - density of fluid calculated on assumption that * - dimensionless radial coordinate - * dimensionless radius of container - dimensionless times - _{rr rr}, _r radial normal and shear stress components - spatial component of defined in equation (13) - _j jth component of - dimensionless streamfunction - ⁰, ¹ components of in series expansion in powers of - phase angle - _r phase difference - _ra , _rb phase difference for fluids a and b - streamfunction - _j jth component defined in equation (22) - dimensionless frequency (based on ) - _a , _b dimensionless frequency in fluids a and b - _s dimensionless frequency (based on _s ) - angular frequency - ₀ resonant frequency in absence of fluid and internal damping - _r resonant frequency in absence of internal fluid - _ra , _rb resonant frequencies in fluids a and b - dimensionless frequency - dimensionless frequency when _a vanishes - dimensionless frequencies when _a vanishes in fluids a and b - dimensionless resonant frequency when _a , _b, _b and ₀ vanish - dimensionless resonant frequency when _a , _b and _b vanish - dimensionless resonant frequency when _b and _b vanish - dimensionless frequencies at which amplitude is half that at resonance     

Rheology of molten polymers

Summary TheCross equation describes the flow of pseudoplastic liquids in terms of an upper and a lower Newtonian viscosity corresponding to infinite and zero shear, and ₀, and of a third material constant related to the mechanism of rupture of linkages between particles in the intermediate, non-Newtonian flow regime, Calculation of of bulk polymers is important, since it cannot be determined experimentally. The equation was applied to the melt flow data of two low density polyethylenes at three temperatures.Using data in the non-Newtonian region covering 3 decades of shear rate to extrapolate to the zero-shear viscosity resulted in errors amounting to about onethird of the measured ₀ values. The extrapolated upper Newtonian viscosity was found to be independent of temperature within the precision of the data, indicating that it has a small activation energy.The ₀ values were from 100 to 1,400 times larger than the values at the corresponding temperatures.The values of were large compared to the values found for colloidal dispersions and polymer solutions, but decreased with increasing temperature. This shows that shear is the main factor in reducing chain entanglements, but that the contribution of Brownian motion becomes greater at higher temperatures.

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Zusammenfassung Die Gleichung vonCross beschreibt das Fließverhalten von pseudoplastischen Flüssigkeiten durch drei Konstante: Die obereNewtonsche Viskosität (bei sehr hohen Schergeschwindigkeiten), die untereNewtonsche Viskosität ₀ (bei Scherspannung Null), und eine Materialkonstante, die vom Brechen der Bindungen zwischen Partikeln im nicht-Newtonschen Fließbereich abhängt. Die Berechnung von ist wichtig für unverdünnte Polymere, wo man sie nicht messen kann.Die Gleichung wurde auf das Fließverhalten der Schmelzen von zwei handelsüblichen Hochdruckpolyäthylenen bei drei Temperaturen angewandt. Die Werte von ₀, durch Extrapolation von gemessenen scheinbaren Viskositäten im Schergeschwindigkeitsbereich von 10 bis 4000 sec^–1 errechnet, wichen bis 30% von den gemessenen ₀-Werten ab. Die Aktivierungsenergie der war so klein, daß die-Werte bei den drei Temperaturen innerhalb der Genauigkeit der Extrapolation anscheinend gleich waren. Die ₀-Werte waren 100 bis 1400 mal größer als die-Werte.Im Verhältnis zu kolloidalen Dispersionen und verdünnten Polymerlösungen war das der Schmelzen groß, nahm aber mit steigender Temperatur ab. Deshalb wird die Verhakung der Molekülketten hauptsächlich durch Scherbeanspruchung vermindert, aber der Beitrag derBrownschen Bewegung nimmt mit steigender Temperatur zu.

     

On the closure problem for Darcy's law

In a previous derivation of Darcy's law, the closure problem was presented in terms of an integro-differential equation for a second-order tensor. In this paper, we show that the closure problem can be transformed to a set of Stokes-like equations and we compare solutions of these equations with experimental data. The computational advantages of the transformed closure problem are considerable.Roman Letters 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 _e area of entrances and exits for the-phase contained within the averaging volume, m² - B second-order tensor used to respresent the velocity deviation - b vector used to represent the pressure deviation, m^–1 - C second-order tensor related to the permeability tensor, m^–2 - D second-order tensor used to represent the velocity deviation, m² - d vector used to represent the pressure deviation, m - g gravity vector, m/s² - I unit tensor - K C ^–1,–D, Darcy's law permeability tensor, m² - L characteristic length scale for volume averaged quantities, m - characteristic length scale for the-phase, m - l _i i=1, 2, 3, lattice vectors, m - 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 - p pressure in the-phase, N/m ² - p intrinsic phase average pressure, N/m² - p –p , spatial deviation of the pressure in the-phase, N/m² - r position vector locating points in the-phase, m - r ₀ radius of the averaging volume, m - t time, s - v velocity vector in the-phase, m/s - v intrinsic phase average velocity in the-phase, m/s - v phase average or Darcy velocity in the \-phase, m/s - v –v , spatial deviation of the velocity in the-phase m/s - V averaging volume, m³ - V volume of the-phase contained in 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     

Translatory accelerating motion of a circular disk in a viscous fluid

The exact solution of the equation of motion of a circular disk accelerated along its axis of symmetry due to an arbitrarily applied force in an otherwise still, incompressible, viscous fluid of infinite extent is obtained. The fluid resistance considered in this paper is the Stokes-flow drag which consists of the added mass effect, steady state drag, and the effect of the history of the motion. The solutions for the velocity and displacement of the circular disk are presented in explicit forms for the cases of constant and impulsive forcing functions. The importance of the effect of the history of the motion is discussed.Nomenclature a radius of the circular disk - b one half of the thickness of the circular disk - C dimensionless form of C ₁ - C ₁ magnitude of the constant force - D fluid drag force - f(t) externally applied force - F() dimensionaless form of applied force - F ₀ initial value of F - g gravitational acceleration - H() Heaviside step function - k magnitude of impulsive force - K dimensionless form of k - M a dimensionless parameter equals to (1+37#x03C0;_s/4_f) - S displacement of disk - t time - t ₁ time of application of impulsive force - u velocity of the disk - V dimensionless velocity - V ₀ initial velocity of V - V _t terminal velocity - parameter in (13) - parameter in (13) - (t) Dirac delta function - ratio of b/a - () function given in (5) - dynamical viscosity of the fluid - kinematic viscosity of the fluid - _f fluid density - _s mass density of the circular disk - dimensionless time - _i dimensionless form of t _i - dummy variable - dummy variable     

Hyperbolic phenomena in a strongly degenerate parabolic equation

We consider the equation u _t =((u) (u _x )) _x , where >0 and where is a strictly increasing function with lim _s = <. We solve the associated Cauchy problem for an increasing initial function, and discuss to what extent the solution behaves qualitatively like solutions of the first-order conservation law u _t = ((u)) _x . Equations of this type arise, for example, in the theory of phase transitions where the corresponding free-energy functional has a linear growth rate with respect to the gradient.     

The finite element solutions of laminar flow and combined convection of air in a staggered or an in-line tube-bank

A finite element method is used to solve the full Navier-Stokes and energy equations for the problems of laminar combined convection from three isothermal heat horizontal cylinders in staggered tube-bank and four isothermal heat horizontal cylinders in in-line tube-bank. The variations of surface shear stress, pressure and Nusselt number are obtained over the entire cylinder surface including the zone beyond the separation point. The predicted values of total, pressure and friction drag coefficients, average Nusselt number and the plots of velocity flow fields and isotherms are also presented.

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Die Finite-Elemente-Lösung von laminarer Strömung und kombinierter Konvektion von Luft in einer versetzten oder fluchtenden Rohranordnung

Zusammenfassung Eine Methode der finiten Elemente wird zur Lösung der vollständigen Navier-Stokes- und der Energiegleichung für die Probleme der laminaren kombinierten Konvektion an drei isothermen geheizten horizontalen Zylindern in versetzter Rohranordnung sowie für vier isotherme geheizte horizontale Zylinder in fluchtender Anordnung verwendet.Die Veränderung der Wandschubspannung, des Druckes und der Nusselt-Zahl werden für die gesamte Zylinderoberfläche, einschließlich des Bereiches nach dem Ablösepunkt, bestimmt. Die Werte des gesamten Widerstandsbeiwertes aufgrund von Druck und Reibung, die durchschnittliche Nusselt-Zahl und die Diagramme des Geschwindigkeitsfeldes und der Isothermen werden ebenfalls aufgezeigt.

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Nomenclature C specifie heat - C _D total drag coefficient - C _f friction drag coefficient - C _p pressure drag coefficient - D diameter of cylinder,L=2R ₀ - G, g gravitational acceleration - Gr Grashof number, g(T_w–T )D ³/v ² - h local heat transfer coefficient - K thermal conductivity - L spacing between the centers of cylinder - M _l shape function - N _i shape function - Nu, local and average Nusselt numbers - P dimensionless pressure, p^*/u ² - p ^*,p pressure, free stream pressure - Pe Peclet number,RePr - Pr Prandtl number, c/K - Ra Rayleigh number,Gr Pr - Re Reynolds number,Du /v - R ₀ radius of cylinder - T temperature - T _w temperature on cylinder surface with fixed value - T free stream temperature - v dimensionless x-direction component of velocity,v ^*/u - u ^* x-direction component of velocity - u free stream velocity - v dimensionless Y-direction component of velocity,v ^*/u - v ^* Y-direction component of velocity - X x-direction axis - x dimensionless x-direction coordinate,x ^*/D - x^* x-direction coordinate - Y Y-direction axis - y dimensionless Y-direction coordinate,y ^*/D - y ^* Y-direction coordinate Greek symbols coefficient of volumetric thermal expansion - plane angle - dynamic viscosity - kinematic viscosity, / - density of fluid - _w dimensionless surface shear stress, _* ^w /u ² - skw/* surface shear stress - dimensionless temperature,      

Nonlinear rheological behavior of a concentrated spherical silica suspension

Linear and nonlinear viscoelastic properties were examined for a 50 wt% suspension of spherical silica particles (with radius of 40 nm) in a viscous medium, 2.27/1 (wt/wt) ethylene glycol/glycerol mixture. The effective volume fraction of the particles evaluated from zero-shear viscosities of the suspension and medium was 0.53. At a quiescent state the particles had a liquid-like, isotropic spatial distribution in the medium. Dynamic moduli G^* obtained for small oscillatory strain (in the linear viscoelastic regime) exhibited a relaxation process that reflected the equilibrium Brownian motion of those particles. In the stress relaxation experiments, the linear relaxation modulus G(t) was obtained for small step strain (0.2) while the nonlinear relaxation modulus G(t, ) characterizing strong stress damping behavior was obtained for large (>0.2). G(t, ) obeyed the time-strain separability at long time scales, and the damping function h() (–G(t, )/G(t)) was determined. Steady flow measurements revealed shear-thinning of the steady state viscosity () for small shear rates (< ^–1; = linear viscoelastic relaxation time) and shear-thickening for larger (>^–1). Corresponding changes were observed also for the viscosity growth and decay functions on start up and cessation of flow, + (t, ) and _– (t, ). In the shear-thinning regime, the and dependence of ₊(t,) and _–(t,) as well as the dependence of () were well described by a BKZ-type constitutive equation using the G(t) and h() data. On the other hand, this equation completely failed in describing the behavior in the shear-thickening regime. These applicabilities of the BKZ equation were utilized to discuss the shearthinning and shear-thickening mechanisms in relation to shear effects on the structure (spatial distribution) and motion of the suspended particles.Dedicated to the memory of Prof. Dale S. Parson     

A technique for evaluation of three-dimensional behavior in turbulent boundary layers using computer augmented hydrogen bubble-wire flow visualization

A system is described which allows the recreation of the three-dimensional motion and deformation of a single hydrogen bubble time-line in time and space. By digitally interfacing dualview video sequences of a bubble time-line with a computer-aided display system, the Lagrangian motion of the bubble-line can be displayed in any viewing perspective desired. The u and v velocity history of the bubble-line can be rapidly established and displayed for any spanwise location on the recreated pattern. The application of the system to the study of turbulent boundary layer structure in the near-wall region is demonstrated.List of Symbols Reynolds number based on momentum thickness u /v - t⁺ nondimensional time - u shear velocity - u local streamwise velocity, x-direction - u ⁺ nondimensional streamwise velocity - v local normal velocity, -direction - x ⁺ nondimensional coordinate in streamwise direction - ⁺ nondimensional coordinate normal to wall - ₊ ^wire wire nondimensional location of hydrogen bubble-wire normal to wall - z ⁺ nondimensional spanwise coordinate - momentum thickness - v kinematic viscosity - _W wall shear stress     

Gap Estimates for Double-Well Schrödinger Operators and Quantum Stabilization of Anharmonic Crystals

The spectrum of the Schrödinger operator of a one-dimensional quantum anharmonic oscillator of mass m is studied. This spectrum consists of simple (nondegenerate) eigenvalues E _n , $$n\in {\mathbb N}_0$$ such that n ^– E _n + as n + with a certain > 1. The gap parameter =min _n (E _n – E _n-1) is in the center of the study. It is proven that this parameter is a continuous function of m; its small mass and large mass asymptotics are found. The influence of the dependence of on m on the stability of systems of interacting quantum anharmonic oscillators is briefly discussed.     

Supersonic nonequilibrium flow past segmental bodies of a mixture simulating the atmosphere of venus

Calculations of the flow of the mixture 0.94 CO₂+0.05 N₂+0.01 Ar past the forward portion of segmentai bodies are presented. The temperature, pressure, and concentration distributions are given as a function of the pressure ahead of the shock wave and the body velocity. Analysis of the concentration distribution makes it possible to formulate a simplified model for the chemical reaction kinetics in the shock layer that reflects the primary flow characteristics. The density distributions are used to verify the validity of the binary similarity law throughout the shock layer region calculated.The flow of a CO₂+N₂+Ar gas mixture of varying composition past a spherical nose was examined in [1]. The basic flow properties in the shock layer were studied, particularly flow dependence on the free-stream CO₂ and N₂ concentration.New revised data on the properties of the Venusian atmosphere have appeared in the literature [2, 3] One is the dominant CO₂ concentration. This finding permits more rigorous formulation of the problem of blunt body motion in the Venus atmosphere, and attention can be concentrated on revising the CO₂ thermodynamic and kinetic properties that must be used in the calculation.The problem of supersonic nonequilibrium flow past a blunt body is solved within the framework of the problem formulation of [4].Notation V body velocity - shock wave standoff - universal gas constant - ratio of frozen specific heats - hRt/m enthalpy per unit mass undisturbed stream P pressure - density - T temperature - m molecular weight - c_p specific heat at constant pressure - (X) concentration of component X (number of particles in unit mass) - R body radius of curvature at the stagnation point - _j rate of j-th chemical reaction shock layer P V ² pressure - density - TT temperature - mm molecular weight Translated from Izv. AN SSSR. Mekhanika Zhidkosti i Gaza, Vol. 5, No. 2, pp. 67–72, March–April, 1970.The author thanks V. P. Stulov for guidance in this study.     

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.     

Green's functions,bi-linear forms,and completeness of the eigenfunctions for the elastostatic strip and wedge

The bi-harmonic Green's functionG(r,r) for the infinite strip region -1y1, -<x<, with the boundary conditionsG=G/y ony=±1, is obtained in integral form. It is shown thatG has an elegant bi-linear series representation in terms of the (Papkovich-Fadle) eigenfunctions for the strip. This representation is then used to show that any function bi-harmonic in arectangle, and satisfying the same boundary conditions asG, has a unique representation in the rectangle as an infinite sum of these eigenfunctions. For the case of the semi-infinite strip, we investigate conditions on sufficient to ensure that is exponentially small asx. In particular it is proved that this is so, solely under the condition that be bounded asx.A corresponding pattern of results is established for the wedge of general angle. The Green's function is obtained in integral form and expressed as a bilinear series of the (Williams) eigenfunctions. These eigenfunctions are proved to be complete for all functions bi-harmonic in anannular sector (and satisfying the same boundary conditions as the Green's function). As an application it is proved that if an elastostatic field exists in a corner region with free-free boundaries, and with either (i) the total strain energy bounded, or (ii) the displacement field bounded, then this field has a unique representation as a sum of those Williams eigenfunctions whichindividually posess the properties (i), (ii).The methods used here extend to all other linear homogeneous boundary conditions for these geometries.On leave of absence at the University of British Columbia, Vancouver, B.C. Canada, during 1977–79. This work was supported in part by N.R.C. grants Nos. A9259 and A9117.     

Critical phenomena and waves in gas-liquid systems

The results of the hydraulic studies of gas-liquid media, wave processes in two-phase media and critical phenomena are described. Some methodological foundations to describe these media and methods to obtain the basic similarity criteria for the hydraulics and gas-dynamics of bubble suspensions are discussed. A detailed consideration is given for the phase transition processes on interfaces and the interface stability. A relation has been revealed between the wave and critical phenomena in two-phase systems.Nomenclature a thermal diffusivity - Ar Archimedes number - B gas constant - C heat capacity - C _p heat capacity at constant pressure - C _v heat capacity at constant volume - c ₀ acoustic velocity in the mixture - c _l acoustic velocity in the liquid - C _f flow resistance coefficient - G mass rate of flow - g gravitational acceleration - L latent heat of evaporation - l initial perturbation width - M Mach number - Nu Nusselt number - P pressure - Pr Prandtl number - R bubble radius - (3P ₀/R ₀ ² _f )^–1 bubble resonance frequency square - T temperature - U medium motion velocity - W heavy phase velocity - W light phase velocity - We Weber number - heat release coefficient - dispersion coefficient - void fraction - adiabatic index - film thickness - dimensionless film thickness - kinematic viscosity coefficient - dynamical viscosity coefficient - dissipation coefficient in the mixture - dispersion parameter - _f liquid phase density - light phase density - heat conductivity - surface tension - frequency, ₀ ² =3P ₀/ _f R ₀ ²     

A three-parameter model describing the experimental relation between shear stress and shear rate for laminar flow of aqueous polymer solutions

Summary A three-parameter model is introduced to describe the shear rate — shear stress relation for dilute aqueous solutions of polyacrylamide (Separan AP-30) or polyethylenoxide (Polyox WSR-301) in the concentration range 50 wppm – 10,000 wppm. Solutions of both polymers show for a similar rheological behaviour. This behaviour can be described by an equation having three parameters i.e. zero-shear viscosity ₀, infinite-shear viscosity , and yield stress ₀, each depending on the polymer concentration. A good agreement is found between the values calculated with this three-parameter model and the experimental results obtained with a cone-and-plate rheogoniometer and those determined with a capillary-tube rheometer.

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Zusammenfassung Der Zusammenhang zwischen Schubspannung und Schergeschwindigkeit von strukturviskosen Flüssigkeiten wird durch ein Modell mit drei Parametern beschrieben. Mit verdünnten wäßrigen Polyacrylamid-(Separan AP-30) sowie Polyäthylenoxidlösungen (Polyox WSR-301) wird das Modell experimentell geprüft. Beide Polymerlösungen zeigen im untersuchten Schergeschwindigkeitsbereich von ein ähnliches rheologisches Verhalten. Dieses Verhalten kann mit drei konzentrationsabhängigen Größen, nämlich einer Null-Viskosität ₀, einer Grenz-Viskosität und einer Fließgrenze ₀ beschrieben werden. Die Ergebnisse von Experimenten mit einem Kegel-Platte-Rheogoniometer sowie einem Kapillarviskosimeter sind in guter Übereinstimmung mit den Werten, die mit dem Drei-Parameter-Modell berechnet worden sind.

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a Pa^–1 physical quantity defined by:a = {1 – ( / ₀)}/ ₀ - c l concentration (wppm) - D m capillary diameter - L m length of capillary tube - P Pa pressure drop - R m radius of capillary tube - u m s^–1 average velocity - v _r m s^–1 local axial velocity at a distancer from the axis of the tube - shear rate (–dv _r /dr) - local shear rate in capillary flow - s^–1 wall shear rate in capillary flow - Pa s dynamic viscosity - _a Pa s apparent viscosity defined by eq. [2] - ( _a ) Pa s apparent viscosity in capillary tube at a distanceR from the axis - ₀ Pa s zero-shear viscosity defined by eq. [4] - Pa s infinite-shear viscosity defined by eq. [5] - l ratior/R - kg m^– density - Pa shear stress - ₀ Pa yield stress - _r Pa local shear stress in capillary flow - _R Pa wall shear stress in capillary flow _R = (PR/2L) - _v m³ s^–1 volume rate of flow With 8 figures and 1 table     

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     

Aspect ratio effect on flow-induced forces on circular cylinders in a cross-flow

Finite-span circular cylinders with two different aspect ratios, placed in a cross-flow, are investigated experimentally at a cylinder Reynolds number of 46,000. Simultaneous measurements of the flow-induced unsteady forces on the cylinders and the stream velocity in the wake are carried out. These results together with mean drag measurements along the span and available literature data are used to evaluate the flow mechanisms responsible for the induced unsteady forces and the effect of aspect ratio on these forces. The coherence of vortex shedding along the span of the cylinder is partially destroyed by the separated flow emanating from the top and by the recirculating flow behind the cylinder. As a result, the fluctuating lift decreases drastically. Based on the data collected, it is conjectured that the fluctuating recirculating flow behind the cylinder is the flow mechanism responsible for the unsteady drag and causes it to increase beyond the fluctuating lift. The fluctuating recirculating flow is a direct consequence of the unsteady separated flow. The unsteady forces vary along the span, with lift increasing and drag decreasing towards the cylinder base. When the cylinder span is large compared to the wall boundary layer thickness, a submerged two-dimensional region exists near the base. As the span decreases, the submerged two-dimensional region becomes smaller and eventually vanishes. Altogether, these results show that fluctuating drag is the dominant unsteady force in finite-span cylinders placed in a cross-flow. Its characteristic frequency is larger than that of the vortex shedding frequency.List of symbols a span of active element on cylinder, = 2.5 cm - C _D local rms drag coefficient, 2D/ U ² da - C _L local rms lift coefficient, 2l/ U ² da - C _D local mean drag coefficient, 2D/ U ² da - C _D spanwise-averaged C _D for finite-span cylinder - (C _D ) _2D spanwise-averaged mean drag coefficient for two-dimensional cylinder - C _p pressured coefficient - -(C _p ) _b pressure coefficient at = - d diameter of cylinder, = 10.2 cm - D fluctuating component of instantaneous drag - D local rms of fluctuating drag - D local mean drag - E _D power spectrum of fluctuating drag, defined as - E _L power spectra of fluctuating lift, defined as - f _D dominant frequency of drag spectrum - f _L dominant frequency of lift spectrum - f _u dominant frequency of velocity spectrum - h span of cylinder - H height of test section, = 30.5 cm - L fluctuating component of instantaneous lift - L local rms of fluctuating lift - R _Du () cross-correlation function of streamwise velocity and local drag, - R _Lu () cross-correlation function of stream wise velocity and local lift, - Re Reynolds number, U d/y - S _L Strouhal number based on f _L ,f _L d/U - S _D Strouhal number based on f _D ,f _D d/U - S _u Strouhal number based on f _u , f _u d/U - t time - u fluctuating component of instantaneous streamwise velocity - U mean streamwise velocity - mean stream velocity upstream of cylinder - x streamwise distance measured from axis of cylinder - y transverse distance measured from axis of test section - z spanwise distance measured from cylinder base - angular position on cylinder circumference measured from forward stagnation - kinematic viscosity of air - density of air - time lag in cross-correlation function - _D normalized spectrum of fluctuating drag - _L normalized spectrum of fluctuating lift     

Flow in vicinity of blunt body stagnation point in diverging hypersonic stream

In the hypersonic thin shock layer approximation for a small ratio k of the densities before and after the normal shock wave the solution of [1] for the vicinity of the stagnation point of a smooth blunt body is extended to the case of nonuniform outer flow. It is shown that the effect of this nonuniformity can be taken into account with the aid of the effective shock wave radius of curvature R_*, whose introduction makes it possible to reduce to universal relations the data for different nonuniform outer flows with practically the same similarity criterion k. The results of the study are compared with numerical calculations of highly underexpanded jet flow past a sphere.Notations x, y a curvilinear coordinate system with axes directed respectively along and normal to the body surface with origin at the forward stagnation point - R radius of curvature of the meridional plane of the body surface - uV, vV., , p V ² respectively the velocity projections on the x, y axes, density, and pressure - and V freestream density and velocity The indices =0 and=1 apply to plane and axisymmetric flows Izv. AN SSSR, Mekhanika Zhidkosti i Gaza, Vol. 5, No. 3, pp. 102–105, 1970.