On the vanishing of the additive measures of strain and rotation for finite deformations

Explicit formulae for the finite strain and rotation measures are given, in the cases when either one of the infinitesimal tensors of strain and rotation vanishes. Conversely, when the finite strain or rotation measure vanishes, explicit formulae for the infinitesimal tensors of strain and rotation are also obtained.     

On the cross-effects of coupled macroscopic transport equations in porous media

In this paper, the derivation of macroscopic transport equations for this cases of simultaneous heat and water, chemical and water or electrical and water fluxes in porous media is presented. Based on themicro-macro passage using the method of homogenization of periodic structures, it is shown that the resulting macroscopic equations reveal zero-valued cross-coupling effects for the case of heat and water transport as well as chemical and water transport. In the case of electrical and water transport, a nonsymmetrical coupling was found.Notations b mobility - c concentration of a chemical - D rate of deformation tensor - D molecular diffusion coefficient - D _ij ^eff macroscopic (or effective) diffusion tensor - electric field - E ₀ initial electric field - k _ij molecular tensor - j, j ^*, current densities - K _ij macroscopic permeability tensor - l characteristic length of the ERV or the periodic cell - L characteristic macroscopic length - L _ijkl coupled flows coefficients - n _i unit outward vector normal to - p pressure - q _t ,q _t ⁺ , heat fluxes - q _c ,q _c ⁺ , chemical fluxes - s specific entropy or the entropy density - S entropy per unit volume - t time variable - t _ij local tensor - T absolute temperature - v _i velocity - V ₀ initial electric potential - V electric potential - x macroscopic (or slow) space variable - y microscopic (or fast) space variable - _i local vectorial field - _i local vectorial field - electric charge density on the solid surface - , bulk and shear viscosities of the fluid - _ij local tensor - _ij local tensor - _i local vector - _ij molecular conductivity tensor - _ij ^eff effective conductivity tensor - homogenization parameter - fluid density - ₀ ion-conductivity of fluid - _ij dielectric tensor - _i ¹ , _i ² , _i ³ local vectors - ⁴ local scalar - _S solid volume in the periodic cell - _L volume of pores in the periodic cell - boundary between _S and _L - ^s rate of entropy production per unit volume - total volume of the periodic cell - _l volume of pores in the cell On leave from the Politechnika Gdanska; ul. Majakowskiego 11/12, 80-952, Gdask, Poland.     

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.     

An experimental study of the behavior of transient isolated convection in a rigidly rotating fluid

The temperature field of starting thermal plumes were measured in a rotating annulus with various rotation rates and buoyancies. The experiments revealed many details of the internal structure of these convective phenomena and also significant horizontal displacements from their source. Measurements show an increase in the maximum temperature observed in the thermal caps with increasing rotation and a more rapid cooling of the buoyancy source.List of symbols D angle relating inward centripetal acceleration to buoyant acceleration, defined by tan D = R/g - g gravitational acceleration - P total pressure of ambient fluid - R radial coordinate measured from rotation axis - R ₀ distance from rotation axis to buoyancy source - u velocity of fluid parcel along the radial direction - velocity of fluid parcel along the azimuthal direction - w velocity of fluid parcel along the axial direction - z axial coordinate, measured upward from the plane containing the buoyancy source - density of a buoyant parcel of fluid - ₀ density of the ambient fluid - azimuthal angle measured from the radial line passing through the buoyancy source - rotation rate of the R––z coordinate system in radians/second     

Transport in ordered and disordered porous media V: Geometrical results for two-dimensional systems

In this paper we continue the geometrical studies of computer generated two-phase systems that were presented in Part IV. In order to reduce the computational time associated with the previous three-dimensional studies, the calculations presented in this work are restricted to two dimensions. This allows us to explore more thoroughly the influence of the size of the averaging volume and to learn something about the use of anon-representative region in the determination of averaged quantities.

Nomenclature

Roman Letters A interfacial area of the– interface associated with the local closure problem, m² - a _i i=1, 2, gaussian probability distribution used to locate the position of particles - l unit tensor - characteristic length for the-phase particles, m - ⁰ reference characteristic length for the-phase particles, m - characteristic length for the-phase, m - _i i=1,2,3 lattice vectors, m - m convolution product weighting function - m _V special convolution product weighting function associated with a unit cell - n _i i=1, 2 integers used to locate the position of particles - n unit normal vector pointing from the-phase toward the-phase - r _p position vector locating the centroid of a particle, m - r gaussian probability distribution used to determine the size of a particle, m - r ₀ characteristic length of an averaging region, m - V averaging volume, m³ - V volume of the-phase contained in the averaging volume,V, m³ - x position of the centroid of an averaging area, m - x ₀ reference position of the centroid of an averaging area, m - y position vector locating points in the-phase relative to the centroid, m Greek Letters V /V, volume average porosity - _{a i} standard deviation ofa _i - _r standard deviation ofr - intrinsic phase average of      

On the prediction of riblet performance with engineering turbulence models

The paper reports the outcome of a numerical study of fully developed flow through a plane channel composed of ribleted surfaces adopting a two-equation turbulence model to describe turbulent mixing. Three families of riblets have been examined: idealized blade-type, V-groove and a novel U-form that, according to computations, achieves a superior performance to that of the commercial V-groove configuration. The maximum drag reduction attained for any particular geometry is broadly in accord with experiment though this optimum occurs for considerably larger riblet heights than measurements indicate. Further explorations bring out a substantial sensitivity in the level of drag reduction to the channel Reynolds number below values of 15 000 as well as to the thickness of the blade riblet. The latter is in accord with the trends of very recent, independent experimental studies.Possible shortcomings in the model of turbulence are discussed particularly with reference to the absence of any turbulence-driven secondary motions when an isotropic turbulent viscosity is adopted. For illustration, results are obtained for the case where a stress transport turbulence model is adopted above the riblet crests, an elaboration that leads to the formation of a plausible secondary motion sweeping high momentum fluid towards the wall close to the riblet and thereby raising momentum transport.Nomenclature c _f Skin friction coefficient - c _f Skin friction coefficient in smooth channel at the same Reynolds number - k Turbulent kinetic energy - K ⁺ k/ _w - h Riblet height - S Riblet width - H Half height of channel - Re Reynolds number = volume flow/unit width/ - Modified turbulent Reynolds number - R _t turbulent Reynolds numberk ²/ - P _k Shear production rate ofk, _t (U _i /x _j + U _j /x _i ) U _i /x _j - dP/dz Streamwise static pressure gradient - U _i Mean velocity vector (tensor notation) - U Friction velocity, _w/ where _w=–H dP/dz - W Mean velocity - W _b Bulk mean velocity through channel - y ⁺ yU /v. Unless otherwise stated, origin is at wall on trough plane of symmetry - Kinematic viscosity - _t Turbulent kinematic viscosity - Turbulence energy dissipation rate - Modified dissipation rate – 2(k ^1/2/x _j )² - Density - _k , Effective turbulent Prandtl numbers for diffusion ofk and      

Flow birefringence of polymer melts: Application to the investigation of time dependent rheological properties

Summary Transient stresses including normal stresses, which are developed in a polymer melt by a suddenly imposed constant rate of shear, are investigated by mechanical measurement and, indirectly, with the aid of the flow birefringence technique. For the latter purpose use is made of the so-called stress-optical law, which is carefully checked.It appears that the essentially linear model of the rubberlike liquid, as proposed byLodge, is capable of describing the behaviour of polymer melts rather well, if the applied total shear does not exceed unity. In order to describe also steady state values of the stresses successfully, one should extend measurements to extremely low shear rates.These statements are verified with the aid of a method which was originally designed bySchwarzl andStruik for the practical calculation of interrelations between linear viscoelastic functions. In the present paper dynamic shear moduli are used as reference functions.

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Zusammenfassung Mit der Zeit anwachsende Spannungen, darunter auch Normalspannungen, wie sie sich nach dem plötzlichen Anlegen einer konstanten Schergeschwindigkeit in einer Polymerschmelze entwickeln, werden mit Hilfe mechanischer Messungen und indirekt mit Hilfe der Strömungsdoppelbrechung untersucht. Für den letzteren Zweck wird das sogenannte spannungsoptische Gesetz herangezogen, dessen Gültigkeit sorgfältig überprüft wird.Es ergibt sich, daß das im Wesen lineare Modell der gummiartigen Flüssigkeit, wie es vonLodge vorgeschlagen wurde, sich recht gut zur Beschreibung des Verhaltens von Polymerschmelzen eignet, solange der im ganzen angelegte Schub den Wert Eins nicht überschreitet. Um auch stationäre Werte der Spannungen in die Beschreibung erfolgreich einzubeziehen, sollte man die Messungen bis zu extrem niedrigen Schergeschwindigkeiten ausdehnen.Die gemachten Feststellungen werden mit Hilfe einer Methode verifiziert, die vonSchwarzl undStruik ursprünglich für die praktische Berechnung von Beziehungen zwischen Zustandsfunktionen entwickelt wurde, die dem linear viskoelastischen Verhalten entsprechen. In der vorliegenden Veröffentlichung dienen die dynamischen Schubmoduln als Bezugsfunktionen.

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a _T shift factor - B _ij Finger deformation tensor - C stress-optical coefficient, (m²/N) - f (p _jl ) undetermined scalar function - G shear modulus, (N/m²) - G(t) time dependent shear modulus, (N/m²) - G() shear storage modulus, (N/m²) - G() shear loss modulus, (N/m²) - G _r reduced shear storage modulus, (N/m²) - G _r reduced shear loss modulus, (N/m²) - H() shear relaxation time spectrum, (N/m²) - k Boltzmann constant, (Nm/°K) - n _ik refractive index tensor - p undetermined hydrostatic pressure, (N/m²) - p _ij ,p _ik stress tensor, (N/m²) - p ₂₁ shear stress, (N/m²) - p ₁₁ –p ₂₂ first normal stress difference, (N/m²) - p ₂₂ –p ₃₃ second normal stress difference, (N/m²) - q shear rate, (s^–1) - t, t time, (s) - T absolute temperature, (°K) - T ₀ reference temperature, (°K) - x the ratiot/ - x position vector of a material point after deformation, (m) - x position vector of a material point before deformation, (m) - ₀, ₁ constants in eq. [37] - ₀, ₁ constants in eq. [37] - shear deformation - (t, t) time dependent shear deformation - _ij unity tensor - n flow birefringence in the 1–2 plane - (q) non-Newtonian shear viscosity, (N s/m²) - ^* () complex dynamic viscosity, (N s/m²) - | ^* ()| absolute value of complex dynamic viscosity, (N s/m²) - () real part of complex dynamic viscosity, (N s/m²) - () imaginary part of complex dynamic viscosity, (N s/m²) - (t — t) memory function, (N/m² · s) - v number of effective chains per unit of volume, (m^–3) - temperature dependent density, (kg/m³) - ₀ density at reference temperatureT ₀, (kg/m³) - relaxation time, (s) - integration variable, (s) - (x) approximate intensity function - ₁ (x) error function - extinction angle - _m orientation angle of the stress ellipsoid - circular frequency, (s^–1) - 1 direction of flow - 2 direction of the velocity gradient - 3 indifferent direction - t time dependence The present investigation has been carried out under the auspices of the Netherlands Organization for the Advancement of Pure Research (Z. W. O.).North Atlantic Treaty Organization Science Post Doctoral Fellow.Research Fellow, Delft University of Technology.With 11 figures and 2 tables     

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.

     

Viscoelastic swirling flow with free surface in cylindrical chambers

The flow of a viscoelastic liquid driven by the steadily rotating bottom cover of a cylindrical cup is investigated. The flow field and the shape of the free surface are determined at the lowest significant orders of the regular domain perturbation in terms of the angular velocity of the bottom cap. The meridional field superposed on a primary azimuthal field shows a structure of multiple cells. The velocity field and the shape of the free surface are strongly effected by the cylinder aspect ratio and the elasticity of the liquid. The use of this flow configuration as a free surface rheometer to determine the first two Rivlin-Ericksen constants is shown to be promising.Nomenclature R, ,Z Coordinates in the physical domain D - , , Coordinates in the rest stateD ₀ - r, ,z Dimensionless coordinates in the rest stateD ₀ - Angular velocity - Zero shear viscosity - Surface tension coefficient - Density - Dimensionless surface tension parameter - ₁, ₂ The first two Rivlin-Ericksen constants - Stream function - Dimensionless second order meridional stream function - ^* Dimensionless second normal stress function - 2 Dimensionless sum of the first and second normal stress functions - N ₁,N ₂ The first and second normal stress functions - n Unit normal vector - D Stretching tensor - A _n nth order Rivlin-Ericksen tensor - S Extra-stress - u Velocity field - U Dimensionless second order meridional velocity field - V Dimensionless first order azimuthal velocity field - p Pressure - Modified pressure field - P Dimensionless second order pressure field - J Mean curvature - a Cylinder radius - d Liquid depth at rest - D Dimensionless liquid depth at rest - h Free surface height - H Dimensionless free surface height at the second order     

Analytical evaluation of stress-strain test data

An analytical model for deducing the actual stress-strain properties from laboratory test results is discussed. As an illustration, an elastic bilinear material is used for unconfined cylindrical compression test conditions, as simulated with a finite element analysis. The results obtained are applicable for assisting in evaluating measured strength and stiffness properties of some clay soils, concrete test cylinders, concrete cores, and rock cores.The quantitative results of this study can be used for interpreting measured stress-strain data for unconfined compression test conditions. The error in measured results is shown to be influenced by Poisson's ratio, length-to-diameter ratio of the specimen, end condition, and ratio of inelastic modulus to initial elastic modulus. Curves for adjusting the measured results to the theoretical results are presented.Nomenclature D specimen diameter - E _i initial elastic stiffness modulus - E _y elastic stiffness modulus beyond the yield stress, plastic or inelastic modulus - L specimen length - axial strain - _av average strain - _g gage length strain - _y yield strain - Poisson's ratio - compressive stress - _av average stress - _t theoretical compressive stress - _y yield stress - _ym measured stress at the yield strain     

A refined theory for laminated beams: Part I—A new high order approach

This paper presents a new displacement-based one-dimensional model for the analysis of multilayered composite beams. The kinematic restriction of cross sections rigid in their own planes is introduced. The axial displacements over the cross sections are represented in terms of explicitly defined piecewise polynomial warping functions with discontinuous derivatives at the interlaminae, whereas the amplitude of the displacements along the beam axis is established by means of a variational formulation. It is proved that the proposed representation of the axial displacements yields the exact solution of the interior domain problem for a beam subjected to a transverse load varying according to a polynomial law. It is shown that two or three coordinate functions are sufficient to yield continuous distributions of equilibrated stresses except for small neighborhoods of the constrained cross sections, where a higher number of warping functions could be used in order to obtain a better accuracy. The numerical results show excellent agreement with plane stress finite element and plane strain exact solutions.

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Sommario In questo lavoro viene presentato un nuovo modello monodimensionale per l'analisi di travi composite multistrato. Viene introdotta l'ipotesi di indeformabilita delle sezioni nel proprio piano mentre gli spostamenti assiali nella sezione sono rappresentati facendo uso di funzioni ingobbamento definite sull'intera altezza e con derivata discontinua all'in erlamina. Infine, l'ampiezza degli spostamenti lungo l'asse della trave è determinata facendo uso di una formulazione variazionale. Si mostra come la rappresentazione degli spostamenti assiali proposta sia in grado di fornire la soluzione esatta, all'interno del dominio, per una trave soggetta ad un carico trasversale variabile con legge nolinomiale. Due o tre funzioni coordinate sono sufficienti a fornire distribuzioni di sforzi che verificano l'equilibrio anche all'interlamina, a meno di zone rislrette in vicinanza di sezioni vincolate. I risultati numerici mostrano un eccellente accordo con soluzioni agli elementi finiti in stato piano di tensione e con soluzioni esatte in stato piano di deformazione.

     

Application of the Chapman-Enskog method to the case of a binary two-temperature gas mixture

An interesting property of the flows of a binary mixture of neutral gases for which the molecular mass ratio =m/M1 is that within the limits of the applicability of continuum mechanics the components of the mixture may have different temperatures. The process of establishing the Maxwellian equilibrium state in such a mixture divides into several stages, which are characterized by relaxation times _i which differ in order of magnitude. First the state of the light component reaches equilibrium, then the heavy component, after which equilibrium between the components is established [1]. In the simplest case the relaxation times differ from one another by a factor of ^*.Here the mixture component temperature difference relaxation time _T /, where is the relaxation time for the light component. If 1, 1, so that _T ~1, then for the characteristic hydrodynamic time scale t~1 the relative temperature difference will be of order unity. In the absence of strong external force fields the component velocity difference is negligibly small, since its relaxation time _vt1.In the case of a fully ionized plasma the Chapman-Enskog method is quite easily extended to the case of the two-temperature mixture [3], since the Landau collision integral is used, which decomposes directly with respect to . In the Boltzmann cross collision integral, the quantity appears in the formulas relating the velocities before and after collision, which hinders the decomposition of this integral with respect to , which is necessary for calculating the relaxation terms in the equations for temperatures differing from zero in the Euler approximation [4] (the transport coefficients are calculated considerably more simply, since for their determination it is sufficient to account for only the first (Lorentzian [5]) terms of the decomposition of the cross collision integrals with respect to ). This led to the use in [4] for obtaining the equations of the considered continuum mixture of a specially constructed model kinetic equation (of the Bhatnagar-Krook type) which has an undetermined degree of accuracy.In the following we use the Boltzmann equations to obtain the equations of motion of a two-temperature binary gas mixture in an approximation analogous to that of Navier-Stokes (for convenience we shall term this approximation the Navier-Stokes approximation) to determine the transport coefficients and the relaxation terms of the equations for the temperatures. The equations in the Burnett approximation, and so on, may be obtained similarly, although this derivation is not useful in practice.     

Application of Rayleigh-Ritz method to forced convection turbulent heat transfer

An analytical model to predict heat transfer rates to an incompressible fluid in turbulent flow, with fully developed velocity profile, between a heated plate and a parallel, insulated plate is developed. The model employs van Driest's mixing length expression near the wall, a constant eddy diffusivitiy in the core and a constant turbulent Prandtl number. An approximate solution obtained by employing Rayleigh-Ritz method is shown to compare well with the exact solution obtained by numerical integration of the differential equations. The results are compared with the available experimental data and analytical solutions.

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Anwendung der Rayleigh-Ritz-Methode auf die Wärmeübertragung bei erzwungener turbulenter Strömung

Zusammenfassung Es wird ein analytisches Modell zur Berechnung der Wärmeübertragung an ein inkompressibles Fluid in turbulenter Strömung mit voll ausgebildetem Geschwindigkeitsprofil zwischen einer beheizten Platte und einer dazu parallelen isolierten Platte angegeben. Das Modell verwendet van Driest's Ausdruck für die wandnahe Mischungslänge, eine konstante Wirbeldiffusivität im Kern und eine konstante turbulente PrandtlZahl. Eine Näherungslösung nach der Rayleigh-Ritz-Methode läßt sich gut mit der exakten Lösung vergleichen, die durch numerische Integration der Differentialgleichungen erhalten wurde. Die Ergebnisse werden mit verfügbaren Versuchswerten und analytischen Lösungen verglichen.

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Nomenclature A⁺ dimensionless constant in van Driest formula - a⁺ dimensionless distance from the wall after which the eddy diffusivity of momentum is constant - b half-gap of passage - b⁺ dimensionless half-gap=bu^*/ - C_f skin friction coefficient - C_p constant pressure specific heat - d hydraulic mean diameter defined as 4xarea/perimeter=4b - h convective heat transfer coefficient - K⁺ dimensionless constant in van Driest formula - k fluid thermal conductivity - m mass flow rate of fluid - Nu Nusselt number hd/k - P pressure - Pr Prandtl number=/ - Pr_t turbulent Prandtl number=_m/ - q_w heat flux at wall - Re Reynolds number=v_md/ - T Temperature - u⁺ dimensionless velocity=V_x/u^* - u^* friction velocity= - V_x axial velocity - x axial distance from the entrance - x⁺ dimensionless distance=x/d - y distance from the heated wall - y⁺ dimensionless distance=yu^*/ Greek Symbols thermal molecular diffusivity - function equal to (_H+)/ - boundary layer thickness - _H eddy diffusivity of heat - _m eddy diffusivity of momentum - _m0 uniform eddy diffusivity of momentum in the core - dimensionless temperature - T-T_i/q_wd/k uniform heat flux - T-T_w/T_i-T_w uniform temperature - fluid kinematic viscosity - fluid density - fluid shearing stress - bulk mean temperature—fully developed region - fully developed transverse temperature profile Suffixes 1 fully developed - 2 in the entrance region - i at the inlet - m bulk mean value - w at the heated wall     

Rheological behaviour of two-phase flow of solid particles in a gas

The present investigation was concerned with the rheological behaviour of dilute suspensions of solid particles in a gas in a vertical cocurrent flow moving upwards. Starting from the experimentally determined dependence of the pressure drop on the concentration of solid particles and the Reynolds number of the carrier medium in the steady flow region, the rheological parameters were estimated using pseudo-shear diagrams. Air was the carrier medium and the dispersed phase was one of six fractions of polypropylene powder and five fractions of glass ballotini. The results show that the investigated two-phase systems have pseudoplastic character which becomes more pronounced with increases in concentration, equivalent diameter and density of solid particles in the flowing suspension. C _d coefficient of particle resistance - d _e equivalent diameter of particles - D column diameter - Fr Froude number - g gravitational acceleration - K rheological parameter - L length - n rheological parameter - p _t pressure drop due to friction - p _m total pressure drop - p _ag pressure drop due to acceleration of the gas phase - p _as pressure drop due to acceleration of the solid phase - p _g hydrostatic pressure of the gas phase - p _s specific effective weight of the dispersed phase - r radius - Re Reynolds number - Re _p Reynolds number of a particle - Re _G generalized Reynolds number - Re _G1 generalized Reynolds number relating to the end of the laminar flow region - Re _G2 generalized Reynolds number relating to the beginning of the turbulent flow region - w _z axial component of velocity - u _t steady free-fall velocity of a single particle - w average velocity - w _g average velocity of the gas phase - w _s average velocity of the dispersed phase of solid particles - relative mass fraction of solid particles - x _s volume fraction of solid particles - _g coefficient of pressure drop due to friction - µ dynamic viscosity - _g density of the gas phase - _m density of the suspension - _s density of solid particles - _ds density of the dispersed phase - _w shear stress at the wall     

The investigation on the properties and structures of starting vortex flow past a backward-facing step by WBIV technique

An experimental investigation of a starting vortex flow around a backward-facing step was conducted in a water channel. The properties and structures of the flow were investigated by qualitative flow visualization using the hydrogen bubble method and by quantitative velocity and vorticity measurements using White-light Bubble Image Velocimetry (WBIV) — a newly developed PIV method. Some invariant properties and 4-stage structures of starting vortex flow were observed.List of symbols a flow acceleration during starting stage - h height of backward-facing step - d _v dimensionless vortex size - t time - t dimensionless time - U free uniform velocity - u, v streamwise and spanwise velocity components respectively - Re Reynolds number based on a and h - x, y streamwise and spanwise coordinates respectively in flow field - x _c , y _c dimensionless vortex center position - vorticity - ov dimensionless vorticity - _max maximum vorticity - ov _max dimensionless maximum vorticity - circulation - dimensionless circulation - kinematic viscosity This work was supported by the CNSF Grant 1939 100-1-3     

Application of ‘Operational Quadrature Methods’ in Time Domain Boundary Element Methods

The usual time domain Boundary Element Method (BEM) contains fundamentalsolutions which are convoluted with time-dependent boundary data andintegrated over the boundary surface. Here, a new approach for theevaluation of the convolution integrals, the so-called OperationalQuadrature Methods developed by Lubich, is presented. In thisformulation, the convolution integral is numerically approximated by aquadrature formula whose weights are determined using the Laplacetransform of the fundamental solution and a linear multisep method. Tostudy the behaviour of the method, the numerical convolution of afundamental solution with a unit step function is compared with theanalytical result. Then, a time domain Boundary Element formulationapplying the Operational Quadrature Methods is derived. For thisformulation only the fundamental solutions in Laplace domain arenecessary. The properties of the new formulation are studied with anumerical example.     

Distribution of eddy viscosity and mixing length in a non-Newtonian flow of a solid particle-gas suspension

The present work concerns the study of the radial distribution of eddy viscosity and mixing length and their dependence upon the Reynolds number and the concentration of the solid phase in a non-Newtonian flow of a suspension of solid particles in a gas. The investigated systems have a pseudoplastic character and the deviation from Newtonian behaviour increases with an increase in the concentration of the dispersed phase. Relations are presented for the eddy viscosity and mixing length in the flow of pseudoplastic fluids. From the analysis of results it follows that the mixing length and eddy viscosity increases with an increase in the Reynolds number. In contrast, an increase in the concentration of the solid phase and consequently of the pseudoplasticity causes a decrease in the investigated quantities. The radial distribution of the mixing length and the eddy viscosity is characterized by a maximum, after which the investigated quantities vary only slightly. This enables the area of the core of the turbulent flow to be defined. E Non-dimensional eddy viscosity - K fluid consistency defined by Ostwald-De Waele's formula (power law) - K fluid consistency, eq. (12) - L mixing length - L _t non-dimensional mixing length - N ⁺ position parameter, eq. (3) - n power-law index - n ⁺ Reichardt's position parameter, eq. (4) - n slope of the dependence ln _w = f[ln (8w/D)] - R pipe radius - r radial distance from the pipe axis - Re=D W /µ Reynolds number - U ⁺ non-dimensional local mean axial velocity, eq. (2) - u ^* = w (/8)^0.5 friction velocity - non-dimensional local mean axial velocity - local mean axial velocity - u turbulence velocity component - mean axial velocity at pipe axis - w average velocity over cross-section of pipe - loading ratio of solid to air, i.e. ratio of mass flow rates of solids to air - Y _m ⁺ =Rⁿ u^2–n /K 8^n–1 non-dimensional distance of pipe centre from the wall - y distance from pipe wall - y ⁺ non-dimensional distance from pipe wall, eq. (5) - friction factor - µ _L laminar part of viscosity - µ _t eddy viscosity - density - shear stress - _w shear stress on the wall     

Asymptotics of the homogenized moduli for the elastic chess-board composite

We find the asymptotic behavior of the homogenized coefficients of elasticity for the chess-board structure. In the chess board white and black cells are isotropic and have Lamé constants (, ,) and (, ) respectively. We assume that the black cells are soft, so 0. It turns out that the Poisson ratio for this composite tends to zero with .     

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.