On the forced motion of elastic solids

Summary This note develops a method for the solution of the elastokinetic boundary value problem for time dependent surface tractions and/or displacements, as well as body forces which are functions of time and space. The method of Williams is extended to resolve three-dimensional problems of elastodynamics by classical mathematical techniques.Nomenclature x _i position vector - t time - u _i displacement vector - _ij stress tensor - F _i vector characterizing body force per unit volume - stress vector acting on surface S with unit outer normal v _i - density - , Lamé's constants - _ij Kronecker delta     

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.     

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      

The dynamics of solid-solid phase transitions 2. Incoherent interfaces

Incoherent phase transitions are more difficult to treat than their coherent counterparts. The interface, which appears as a single surface in the deformed configuration, is represented in its undeformed state by a separate surface in each phase. This leads to a rich but detailed kinematics, one in which defects such as vacancies and dislocations are generated by the moving interface. In this paper we develop a complete theory of incoherent phase transitions in the presence of deformation and mass transport, with phase interface structured by energy and stress. The final results are a complete set of interface conditions for an evolving incoherent interface.Frequently used symbols A_i,C_i generic subsurface of S_t - B_i undeformed phase-i region - C configurational bulk stress, Eshelby tensor - F deformation gradient - G inverse deformation gradient - H relative deformation gradient - J bulk Jacobian of the deformation - ¯K, K_i total (twice the mean) curvature of and S_i - Lin (U, V) linear transformations from U into V - Lin⁺ linear transformations of ³ with positive determinant - Orth⁺ rotations of ³ - Q^a external bulk mass supply of species a - ¯S bulk Cauchy stress tensor - S bulk Piola-Kirchhoff stress tensor - S_i undeformed phase i interface - U_i relative velocity of S_i - Unim⁺ linear transformations of ³ with unit determinant - ¯V, V_i normal velocity of and S_i - intrinsic edge velocity of S and A _i S - W_i volume flow across the phase-i interface - X material point - b external body force - e internal bulk configurational force - f_i external interfacial force (configurational) - ¯g external interfacial force (deformational) - grad, div spatial gradient and divergence - gradient and divergence on - h relative deformation - h^a, diffusive mass flux of species a and list of mass fluxes - ¯m outward unit normal to a spatial control volume - ¯n, n_i unit normal to and S_i - n subspace of ³ orthogonal to n - ¯q^a external interfacial mass supply of species a - s ......... - ¯v, v_i compatible velocity fields of and S_i - ¯w, w_i compatible edge velocity fields for and A_i - x spatial point - y_i deformation or motion of phase i - y^. material velocity - generic subsurfaces of - , _i deformed body and deformed phase-i region - () energy supplied to by mass transport - symmetry group of the lattice - _i, surface jacobians - lattice - () power expended on - spatial control volume - S deformed phase interface - lattice point density - interfacial power density - , A total surface stress - C configurational surface stress for phase 1 (material) - ¯C_i configurational surface stress (spatial) - F_i tangential deformation gradient - G_i inverse tangential deformation gradient - H incoherency tensor - ¯1(x), 1_i(X) inclusions of ¯n(x) and n _i (X) into ³ - K configurational surface stress for phase 2 (material) - ¯L, l_i curvature tensor of and S_i - ¯P(x), P_i(X) projections of ³ onto ¯n(x) and n_i (X) - ¯S, S deformational surface stress (spatial and material) - ¯a, a normal part of total surface stress - c normal part of configurational surface stress for phase 1 (material) - e_i internal interfacial configurational force - ¯v, v_i unit normal to and A _i - (x),_i(X) projections of ³ onto ¯n(x) and n _i (X) - _i normal internal force (material) - bulk free energy - slip velocity - _i=(–1)ⁱ _i ......... - ^a, chemical potential of species a and list of potentials - ^a, bulk molar density of species a and list of molar densities - _i normal internal force (spatial) - surface tension - , _i effective shear - referential-to-spatial transform of field - interfacial energy - grand canonical potential - l unit tensor in ³ - x, vector and tensor product in ³ - (...)^., _t(...) material and spatial time derivative - , Div material gradient and divergence - gradient and divergence on S_i - (...), (...) normal time derivative following and S_i - (...) limit of a bulk field asx ,x_i - [...],...> jump and average of a bulk field across the interface - (...)_ext extension of a surface tensor to ³ - tangential part of a vector (tensor) on and S_i     

Rotationsviskosimetrie mit kugelförmigen Spindeln

Zusammenfassung Die Rheodynamik der stationären viskometrischen Drehströmung um eine rotierende Kugel wird mit Methoden der Variationsrechnung untersucht. Neben iterativen numerischen Lösungsmethoden, die zu exakten Resultaten führen, wird auch eine approximative Ein-Gradienten-Lösung konstruiert, die durch Quadraturen dargestellt wird. Ausgehend von dieser analytischen Approximation werden einfache Methoden zur Auswertung von Experimentaldaten vorgeschlagen, die mit Hilfe von Eintauch-Rotationsviskosimetern mit kugelförmigen Meßspindeln gewonnen wurden.

---

Summary The rotational viscometric flow around a rotating sphere has been studied by variational methods. The exact numerical, as well as an approximate analytical solutions are given. Employing the analytical approximation, a simple method of evaluating viscometric data from immersional (portable) viscometers with a rotating sphere is proposed.

---

A Achsenschnitt durch den Bereich der Strömung - B - b, c anpaßbare empirische Konstanten - C Kalibrierungsoperator - D Schergeschwindigkeit der viskosimetrischen Strömung - D _ij Komponenten des Deformationsgeschwindigkeitstensors - D _I, _I Stoffkonstanten der VF des Ellis-Modells - g metrischer Koeffizient - H() Funktional der Ein-Gradienten-Approximation, Gl. [27] - J[] energetisches Potential - J _a[] Ein-Gradienten-Approximation fürJ - K Konsistenzkoeffizient, Parameter der VF des Potenzmodells - m Parameter des Ellis-Modells - M Drehmoment - n Parameter des Potenzmodells - n, n Differentialindices der VF, Gl. [20c, d] - n*,n** Differentialindices der RC, Gl. [9], [13] - r, , z polare Zylinderkoordinaten - R Spindelhalbmesser - rheometrischer Operator - S Spindeloberfläche - U(D) energetische Funktion nachBird, Gl. [20e] - v _i physikalische Komponenten der Geschwindigkeit - Z() transformierte VF, Gl. [20f] - (n) durch Gl. [35] definierte Funktion - k Verhältnis der Radien von Spindel und Wand - ( durch Gl. [43] definierte Funktion - natürliche (Radial-)Koordinate - Schubspannung der viskosimetrischen Strömung - _ij Komponenten des Spannungstensors - _S() Spannungsprofil an der Spindeloberfläche - _M Maximalspannung an der Spindeloberfläche - mittlere Spannung an der Spindeloberfläche, Gln. [3], [22] - natürliche (Meridional-) Koordinate - Winkelgeschwindigkeit in der Flüssigkeit - Winkelgeschwindigkeit der Spindelrotation - ( rheometrische Charakteristik Mit 4 Abbildungen und 3 Tabellen     

Nonstationary vibration of a rotating shaft with nonlinear spring characteristics during acceleration through a critical speed (A critical speed of a summed-and-differential harmonic oscillation)

Nonstationary vibration of a flexible rotating shaft with nonlinear spring characteristics during acceleration through a critical speed of a summed-and-differential harmonic oscillation was investigated. In numerical simulations, we investigated the influence of the angular acceleration , the initial angular position of the unbalance _n and the initial rotating speed on the maximum amplitude. We also performed experiments with various angular accelerations. The following results were obtained: (1) the maximum amplitude depends not only on but also on _n and : (2) when the initial angular position _n changes. the maximum amplitude varies between two values. The upper and lower bounds of the maximum amplitude do not change monotonously for the angular acceleration: (3) In order to always pass the critical speed with finite amplitude during acceleration. the value of must exceed a certain critical value.Nomenclature O-xyz rectangular coordinate system - , ₁, ₁ inclination angle of rotor and its projections to thexy- andyz-planes - I _r polar moment of inertia of rotor - I diametral moment of inertia of rotor - i _r ratio ofI _r toI - dynamic unbalance of rotor - directional angle of fromx-axis - c damping coefficient - spring constant of shaft - N _nt ,N _nt nonlinear terms in restoring forees in ₁ and ₁ directions - ₄ representative angle - a small quantity - V. V _u .V _N potential energy and its components corresponding to linear and nonlinear terms in the restoring forees - directional angle - _n coefficients of asymmetrical nonlinear terms - _n coefficients of symmetrical nonlinear terms - coefficients of asymmetrical nonlinear terms experessed in polar coordinates - coefficients of symmetrical nonlinear terms expressed in polar coordinates - rotating speed of shaft - t time - _n initial angular position of att=0 - p natural frequency - p ₁.p _t natural frequencies of forward and backward precessions - , ₁, ₁ total phases of harmonic, forward precession and backward precession components in summed-and-differential harmonic oscillation - , ₁, ₁ phases of harmonic, forward precession and backward precession components in summed-and-differential harmonic oscillation - P, R _t ,R _b amplitudes of harmonic, forward precession and backward precession components in summed-and-differential harmonic oscillation - difference between phases ( = _f – _u) - acceleration of rotor - initial rotating speed - t _t ,r _b amplitudes of nonstationary oscillation during acceleration - (r _t )_max, (r _b )_max maximum amplitudes of nonstationary oscillation during acceleration - (r ₁ ¹ )_max, (r _b ¹ )_max maximum value of angular acceleration of non-passable case - ₀ critical value over which the rotor can always pass the critical speed - p ₁,p ₂,p ₃,p ₄ natural frequencies of experimental apparatus     

Flow about a circular cylinder with a single large-scale surface perturbation

An experimental study of the flow around a cylinder with a single straight perturbation was conducted in a wind tunnel. With this bluff body, positioned in a uniform crossflow, the vortex shedding frequency and other flow characteristics could be manipulated.The Strouhal number has been shown to be a function of the perturbation angular position, _p , as well as the perturbation size and Reynolds number. As much as a 50% change in Strouhal number could be achieved, simply by changing _p by 1°. The perturbation size compared to the local boundary layer thickness, , was varied from approximately 1 to about 20 . The Reynolds number was varied from 10,000 to 40,000. For perturbation sizes approximately 5 to 20 and Reynolds numbers of 20,000 to 40,000, a consistent Strouhal number variation with _p was observed.A detailed investigation of the characteristic Strouhal number variation has shown that varying _p had a significant influence on the boundary layer separation and transition to turbulence. These significant changes occurring in the boundary layer have been shown to cause variations in the spacing between the shear layers, base pressure, drag, lift, and the longitudinal spacing between the vortices in the vortex street.List of Symbols a longitudinal spacing of vortices in the vortex street - C _d drag coefficient - C _dc drag coefficient corrected for blockage effect - C _l lift coefficient - C _p pressure coefficient, p _i –p /q - C _pb base pressure coefficient - C _pbc base pressure coefficient corrected for blockage effect - d perturbation diameter - d ^* spacing between the shear layers; defined as conditionally averaged spacing between points in the shear layers corresponding to 0.99u _max/U - D cylinder diameter; diameter of the circumscribing circle for a cable - f _v vortex shedding frequency - H wind tunnel test section cross-sectional width - L spanwise length of the cylinder - p _i tap pressure - p free stream static pressure - q free stream dynamic pressure - Re Reynolds number based on cylinder diameter - rms root-mean-square - S Strouhal number, f _v D/U - S _max maximum value of S - S _min minimum value of S - t time - u _c vortex convection velocity - u _max maximum velocity in the shear layer - U free stream velocity - U _c free stream velocity, corrected for blockage effect - x streamwise dimension referenced from the back of the cylinder - z lateral wake dimension, i.e., perpendicular to the free stream velocity vector and cylinder axis, referenced from the cylinder axis - x spacing between two hot wire probes aligned streamwise - phase difference between two hot wire probes aligned streamwise - boundary layer thickness - angle from stagnation point in degrees - _p perturbation angular position - _{b p} where S drops back to about the S of a cylinder - _c critical angle, angular position where S drops steeply with 1° change in - _{m p} where S was minimum - _{r p} after S recovers from drop in value - _{t p} where S starts to increase from about the S of a cylinder     

The effect of surface mass transfer on buoyancy induced flow in a variable porosity medium adjacent to a vertical heated plate

The effect of surface mass transfer on buoyancy induced flow in a variable porosity medium adjacent to a heated vertical plate is studied for high Rayleigh numbers. Similarity solutions are obtained within the frame work of boundary layer theory for a power law variation in surface temperature,T _Wx and surface injectionv _Wx(–1/2). The analysis incorporates the expression connecting porosity and permeability and also the expression connecting porosity and effective thermal diffusivity. The influence of thermal dispersion on the flow and heat transfer characteristics are also analysed in detail. The results of the present analysis document the fact that variable porosity enhances heat transfer rate and the magnitude of velocity near the wall. The governing equations are solved using an implicit finite difference scheme for both the Darcy flow model and Forchheimer flow model, the latter analysis being confined to an isothermal surface and an impermeable vertical plate. The influence of the intertial terms in the Forchheimer model is to decrease the heat transfer and flow rates and the influence of thermal dispersion is to increase the heat transfer rate.

---

Der Effekt des Oberflächenstoffaustausches bei auftriebsinduzierter Strömung in einem variablen porösen Medium, das an eine vertikale, beheizte Platte angrenzt

Zusammenfassung Es wird der Effekt des Oberflächenstoffaustausches in auftriebsinduzierter Strömung in einem variablen porösen Medium, das an eine vertikale, beheizte Platte angrenzt, für große Reynoldszahlen untersucht. Ähnliche Lösungen werden im Rahmen der Grenzschicht-Theorie, durch Variation des Potenzansatzes der Oberflächentemperatur,T _Wx , und der Oberflächengeschwindigkeit,v _Wx(–1/2), erreicht. Die Analyse vereinigt sowohl den Ausdruck, der Porösität und Permeabilität verbindet, als auch den Ausdruck, der Porösität und Wärmeleitfähigkeit miteinander verbindet. Der Einfluß der Temperaturverteilung auf Strömung und Wärmeübergangskennzahlen wird ebenfalls im Detail analysiert. Als Ergebnis der vorliegenden Untersuchung ergibt sich die Tatsache, daß variable Porösität Wärmeübertragungsrate und Betrag der Geschwindigkeit in Wandnähe steigert. Die bestimmenden Gleichungen, sowohl für das Darcysche Strömungsmodell als auch für das Forchheimersche Strömungsmodell, werden mit Hilfe eines implizierten Differenzenschemas gelöst. Die Berechnung wird für die beiden Fälle, isotherme Oberfläche und undurchlässige vertikale Platte, angewandt. Der Einfluß der Terme für die Trägheitskräfte im Forchheimerschen Modell senkt Wärmeübergangs- und Durchgangsrate, wogegen die Wärmeübergangsrate durch den Einfluß der Temperaturverteilung erhöht wird.

---

Nomenclature a constant defined by Eq. (12) - A constant defined by Eq. (12) - B constant defined by Eq. (3) - b _s/_f ratio of thermal conductivities - C constant defined by Eq. (1) - C _P specific heat of the convective fluid - d particle diameter - f dimensionless function defined by Eq. (14) - f _w lateral mass flux parameter - g acceleration due to gravity - k ₀ mean permeability of the mediumk ₀= ₀ ³ d ²/150 (1– ₀)² k ₀=1.75d/(1– ₀) 150 (Inertia parameter) - L length of the source or sink - m mass transfer - n constant defined in Eq. (12) - k (y) permeability of the porous medium - k (y) interial coefficient in the Ergun expression - Gr modified Grashof numberGr=(g k ₀ k ₀ (T _w–))/ ² - R _a Rayleigh number (g k ₀ x T _w–)/ - R _ad modified Rayleigh number (g k ₀ d|T _w–|)/ - N _u Nusselt number - s x/d - Q overall heat transfer rate - T temperature - T _w surface temperature - T ambient fluid temperature - u velocity in vertical direction - v velocity in horizontal direction - x vertical coordinate - y horizontal coordinate Greek symbols ₀ mean thermal diffusivity _f/ C_p - coefficient of thermal expansion - constant defined in Eq. (4) - ratio of particle to bed diameter - _e effective thermal conductivity - f thermal conductivity of fluid - _s thermal conductivity of solid - dimensionless similarity variable in Eq. (13) - value of at the edge of the boundary layer - constant defined in Eq. (1) - _e effective molecular thermal diffusivity - (y) porosity of the medium - ₀ mean porosity of the medium - viscosity of the fluid - ₀ density of the convective fluid - stream function - w condition at the wall - condition at infinity     

Reflection, transmission and excitation of SH-surface waves by a discontinuity in mass-loading on a semi-infinite elastic medium

The scattering of an SH-wave by a discontinuity in mass-loading on a semi-infinite elastic medium is investigated theoretically. The incident wave is either a plane body wave or a plane SH-surface wave. The problem is reduced to a Wiener-Hopf problem for the scattered wave. In this problem the amplitude spectral density of the particle displacement occurs as unknown function. Special attention is given to the numerical values of the surface wave contributions to the scattered field.Nomenclature x ₁, x ₂, x ₃ Cartesian coordinates - , polar coordinates in x ₁, x ₃-plane - volume mass density - surface mass density of mass-loading - , Lamé constants - U scalar wave function, defined by (2.1) - c _S speed of propagation of uniform shear waves in bulk medium (c _S=(/)^1/2) - angular frequency - t time - k _S wave number of uniform shear waves (k _S=/c _S) - reduced specific acoustic impedance of mass-loading (=k _S /) - k _m wave number of SH-surface wave (k _m=k _S(1+ ²)^1/2) - _1,2,3 partial differentiation with respect to x _1,2,3 - ⁱ angle between x ₃-axis and direction of propagation of incident body wave - ⁱ wave number in horizontal direction of incident body wave ( ⁱ=k _S sin( ⁱ)) - ⁱ wave number in vertical direction of incident body wave ( ⁱ=k _S cos( ⁱ)) - C _1,2 complex amplitude of surface wave excited by a body wave - R reflection factor of surface wave, when surface wave is incident - T transmission factor of surface wave, when surface wave is incident - S particle displacement vector The research presented in this paper has been carried out with partial financial support from the Delfts Hogeschoolfonds.     

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

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

Nomenclature

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

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

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

---

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

---

---

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

Measurement of the turbulent diffusivity in the near field of variable density jets using conditional velocimetry

Pulsed laser Mie scattering and laser Doppler velocimetry (LDV), both conditioned on the origin of the seed particles, have been successively performed in turbulent jets with variable density. In the early stages of the jet developments, significant differences are measured between the ensemble average LDV data obtained by jet seeding and those obtained by seeding the ambient air. Careful analysis of the marker statistics shows that this difference is a quantitative measure of the turbulent mixing. The good agreement with gradient–diffusion modelling suggests the validity of a general diffusion equation where the velocities involved are expressed in terms of ensemble conditional Favre averages. This operator accounts for all events (including intermittent ones) and for variations in the density of the marked fluid whose velocity is still specified by the binary origin of the marker.List of symbols D_L laminar diffusivity, m²/s - D_T turbulent diffusivity, m²/s - d diameter of the jet nozzle, m - F_r Froude number - J diffusion vector, m/s - k global sensitivity of the detection system for one particle (signal level) - N_P number of seed particles in the probe volume - N_P,i number of seed particles in sample i - N_P⁽ⁱ⁾ value of N_P in channel i - N_B number of Doppler bursts - count rate of bursts, s^–1 - N_v number of validated Doppler bursts - count rate of validated bursts, s^–1 - N_id number of ideal particles - N_id* number of marked ideal particles - P* probability that an ideal particle be marked by a seed particle - P(z) probability density function for z, m³/kg - probability to have k seed particles in the probe volume - probability of having k seed particle conditioned on a given value of z - r radial coordinate, m - R =⁽¹⁾/⁽²⁾, density ratio - S₁ local signal level with jet seeding - S₁⁽¹⁾ reference signal level in pure stream 1 with jet seeding - s₁ = S_1/S₁⁽¹⁾, normalized signal - v_c volumic capacity of the probe volume, m³ - V velocity vector, m/s - V_x axial velocity component, m/s - V_r radial velocity component, m/s - V_P particulate velocity vector, m/s - V_Pj velocity vector of particle j, m/s - V_Pij velocity vector of the jth particle in sample i, m/s - V_i velocity vector of the marked flow for realization i, m/s - V_1,i velocity vector of the flow such it is marked in realization i by particles issuing only from stream 1, m/s - x axial coordinate, m - Y_i local mass fraction of species i - Z mixture fraction:local mass fraction of jet fluid - Z_i mixture fraction for realization iGreek local density, kg/m³ - _i local density for realization i, kg/m³ - ⁽¹⁾ density in stream 1 (density of the jet fluid), kg/m³ - ₁ time of flight of jet seed particles to reach the probe volume, s - _B duration of a Doppler burst, sAverages <A> ensemble average of A - Ā time average of A - Favre average, , ( ) the present notation is only due to printing problems - A Favre fluctuation,      

Product inhibition in packed-bed immobilized enzyme reactors for complex processes: Modelling of two-substrate processes

An analytical model was developed for describing the performance of packed-bed enzymic reactors operating with two cosubstrates, and when one of the reaction products is inhibitory to the enzyme. To this aim, the compartmental analysis technique was used. The relevant equations obtained were solved numerically, and the effect of the main operational parameters on the reactor characteristics were studied.Notation C _infa,i ^sup* local concentration of products in the pores of stage i - C _j,i concentration of substrate j in the pores of stage i - D _infa ^sup* internal (pore) diffusion coefficient for the reaction product a - D _j internal (pore) diffusion coefficient of substrate j - J _infa,i ^sup* net flux of product a, taking place from the pores of stage i into the corresponding bulk phase - J _j,i net flux of substrate j, taking place from the bulk phase of stage i into the corresponding pores - K _b inhibition constant - K _m,1, K _m,2 Michaelis constants for substrate 1 and 2, respectively - K _q inhibition constant - n total number of elementary stages in the reactor - Q volumetric flow rate throughout the reactor - R _j,i, R _infa,i ^sup* local reaction rates in pores of stage i, in terms of concentration of substrate j and product a respectively - S _infa,i ^sup* , S _infa,i-1 ^sup* bulk concentration of the reaction product a, in the stages i and i — 1, respectively - S _j,0 concentration of substrate j in the reactor feed - S _j,i-1, S _j,i concentration of substrate j in the bulk phase leaving stages i — 1 and i, respectively - V total volume of the reactor - V _m maximal reaction rate in terms of volumetric units - y axial coordinate of the pores - y ₀ depth of the pores - ^* dimensionless parameter, defined in Equation (22) - ₁ dimensionless parameter, defined in Equation (6) - ₂ dimensionless parameter, defined in Equation (6) - ₁ dimensionless parameter, defined in Equation (6) - ₂ dimensionless parameter, defined in Equation (6) - ^* dimensionless parameter, defined in Equation (22) - ₁ dimensionless parameter, defined in Equation (6) - ₂ dimensionless parameter, defined in Equation (6) - ^* dimensionless parameter, defined in Equation (22) - ^* dimensionless parameter, defined in Equation (22) - volumetric packing density of catalytic particles (dimensionless) - porosity of the catalytic particles (dimensionless) - V _infi ^sup* dimensionless concentration of reaction product in pores of stage i, defined in Equation (17) - j,i dimensionless concentration of substrate j in pores of stage i; defined in Equation (6) - _j,i-1, _j.i dimensionless concentration of substrate j in the bulk phase of stage i; defined in Equation (6) - dimensionless position along the pore; defined in Equation (6)     

Messung des ersten Normalspannungskoeffizienten von Kunststoffschmelzen im Bogenspalt

Zusammenfassung Bei einer stationären Schichtenströmung in einem Bogenspalt (azimutale Druckströmung im Ringspalt) bildet sich zwischen Innen- und Außenwand eine Druckdifferenz aus, deren Größe ein Maß für den 1. Normalspannungskoeffizienten der elastischen Flüssigkeit im Spalt ist. Die Strömung läßt sich zur Messung des 1. Normalspannungskoeffizienten verwenden. Der Schergeschwindigkeitsbereich der Messung liegt, wie bei der Kapillarrheometrie zur Bestimmung der Viskosität, zwischen 1 und 1000 s^–1. Die Auswertung der Messungen ist wegen des inhomogenen Scherfeldes relativ kompliziert. In der Arbeit wird ein besonders wirkungsvolles numerisches Auswerteverfahren hergeleitet und auf bestehende Messungen angewendet. Eine Besonderheit des Auswerteverfahrens ist die Freiheit der Wahl des Approximationsansatzes für die Viskositätskurve, während analytische Verfahren meist an einen bestimmten Ansatz gebunden sind. Außerdem braucht, im Gegensatz zu anderen derartigen Verfahren, die Position des schubspannungsfreien Stromfadensr ₀ nicht bestimmt zu werden.

---

Summary The stress in steady viscometric flow of molten polymers is determined by the viscosity and by the two normal stress coefficients ₁ and ₂. The paper describes a method of measuring ₁ by means of steady circumferential shear flow in an annulus. The cylinders are stationary and the fluid flows due to a circumferential pressure gradient. The radial normal stresses at the outer and at the inner wall are different from each other. The pressure-differencep is a measure for the 1. normal stress coefficient of the viscoelastic fluid. Due to the inhomogeneous shear field, the evaluation of ₁ fromp measurements is quite complicated. A powerful numerical method of evaluation has been developed and applied to existing data. The method is not restricted to a special empirical formula for the flow curve (as an analytical method would be) and does not require the knowledge of the positionr ₀ of the stress-free stream line.

---

a Pa s² Stoffparameter des Ansatzes des 1. Normalspannungskoeffizienten, s. Gl. [8] - AR _i — Koeffizient des Druckgefälles in-Richtung (Programm PFEIL) - AU _i — Koeffizient für Integration nach Simpson-Regel (Programm PFEIL) - b s² Stoffparameter des Ansatzes des 1. Normalspannungskoeffizienten - B _i — Koeffizient auf der rechten Seite des linearen Gleichungssystems (Programm PFEIL) - c — Exponent des Ansatzes des 1. Normalspannungskoeffizienten - CL _i CM _i CR _i — Koeffizienten der dimensionslosen Geschwindigkeit in dem linearen Gleichungssystem (Programm PFEIL) - F ₁,F ₂,F ₃ — Ableitungen der Summe der Fehlerquadrate nacha, b undc - G _k — Gewichtsfaktor - h m Spaltweite,r _a –r _i - H — dimensionslose Spaltweite, (r _a –r _i )/r _a - l m Länge des Bogenspaltes, 0,75(r _a +r _i ) - m — Exponent des Potenzansatzes, s. Gl. [13] - n — Dämpfungskonstante - N ₁ Pa 1. Normalspannungsdifferenz, _rr – - N ₂ Pa 2. Normalspannungsdifferenz - p Pa Druck - p Pa Druckgradient in-Richtung - P — dimensionsloser Druckgradient in-Richtung, s. Gl. [14] - p, p _k Pa Normalspannungsdifferenz zwischen Innen- und Außenwand im Bogenspalt, (– p + _rr ) _a – (–p + _rr ) _i - Q — Summe der Fehlerquadrate - r, R= r/r _a m, — Radiusvektor (Koordinate in Gradientenrichtung) - r ₀,R ₀=r ₀/r _a m, — Radius des neutralen Fadens - R — dimensionslose radiale Schrittweite - T, °C Temperatur bzw. Bezugstemperatur - v ms^–1 Geschwindigkeitskomponente in-Richtung - V ,V _,i — dimensionslose Geschwindigkeitskomponente in-Richtung - V _a ,V _k — dimensionslose Geschwindigkeit an der Außen- bzw. Innenwand - v _r ,v _z ms^–1 Geschwindigkeitskomponenten inr-undz-Richtung - ms ^–1 mittlere Geschwindigkeit in-Richtung - z m Koordinate in der indifferenten Richtung - K^–1 Temperaturkoeffizient der Viskosität - s^–1 Schergeschwindigkeit - s^–1 kritische Schergeschwindigkeit der Viskositätskurve, s. Gl. [13] - s^–1 Bezugsschergeschwindigkeit, - — dimensionslose Schergeschwindigkeit - — dimensionslose kritische Schergeschwindigkeit, - Pa s Viskosität - ₀ Pa s Nullviskosität - Pa s Bezugsviskosität, - — Radienverhältnis,r _i /r _a - ₁ Pa s ² 1. Normalspannungskoeffizient - Pa s² mittlerer 1. Normalspannungskoeffizient - ₂ Pa s² 2. Normalspannungskoeffizient - — Koordinate in Strömungsrichtung - Pa Spannung - a an der Außenwand - i, an der Innenwand - i laufender Index inr-Richtung - k Nummer des Meßpunktes - n Anzahl der Meßpunkte - n _i nord für Programm PFEIL - s _i süd für Programm PFEIL Mit 9 Abbildungen und 2 Tabellen     

Verbessertes Diagramm zur Berechnung von Wärmeübertragern

Zusammenfassung Die bekannten Wärmeübertragerdiagramme, in denen die dimensionslosen Temperaturänderungen beider Stoffströme auf den Koordinatenachsen aufgetragen sind, werden modifiziert, um die Nachteile der bisherhigen Darstellung zu vermeiden. Diese neuen Diagramme werden für einige einfache Stromführungen angegeben und mit anderen gebräuchlichen Diagrammen verglichen. Anhand von Beispielen wird die Anwendung erläutert.

---

Improved chart for heat exchanger design

The known heat exchanger charts with dimensionless temperature changes of both fluid streams as coordinate axes are modified to eliminate the disadvantages of the previous representation. Graphs are presented for some simple heat exchanger configurations. The new chart is compared to other usual charts. The application is illustrated by examples.

---

Formelzeichen A Austauschfläche - F Korrekturfaktor für die logarithmische mittlere Temperaturdifferenz; F=I_M/I_LM - k Wärmedurchgangskoeffizient - NTU Anzahl der Übertragungseinheiten (number of transfer units);NTU _i = k A/ _i - P dimensionslose Temperaturänderung - Pe Pecletzahl - R Kapazitätsstromverhältnis;R ₁ = ₁/ ₂;R ₂ = ₂/ ₁ - Kapazitätsstrom - I_M mittlere Temperaturdifferenz - I_LM logarithmische mittlere Temperaturdifferenz - dimensionslose mittlere Temperaturdifferenz - I_M Temperatur Indizes 1, 2 Stoffstrom 1, 2 - am Eintritt - am Austritt Herrn Prof. Dr.-Ing. K. Stephan zum 60. Geburtstag gewidmet     

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     

Diffuse approximation method for solving natural convection in porous media

The diffuse approximation is presented and applied to natural convection problems in porous media. A comparison with the control volume-based finite-element method shows that, overall, the diffuse approximation appears to be fairly attractive.Nomenclature H height of the cavities - I functional - K permeability - p(M _i ,M) line vector of monomials - p ^T p-transpose - M current point - Nu Nusselt number - Ri inner radius - Ro outer radius - Ra Rayleigh number - x, y cartesian coordinates - u, v velocity components - T temperature - M vector of estimated derivatives - _t thermal diffusivity - coefficient of thermal expansion - practical aperture of the weighting function - scalar field - (M, M _i ) weighting function - streamfunction - kinematic viscosity     

Model analysis of nonlinear viscoelastic behaviour by use of a single integral constitutive equation: Stresses and birefringence of a polystyrene melt in intermittent shear flows

Summary A single integral constitutive equation with strain dependent and factorized memory function is applied to describe the time dependence of the shear stress, the primary normal-stress difference, and, by using the stress-optical law, also the extinction angle and flow birefringence of a polystyrene melt in intermittent shear flows. The theoretical predictions are compared with measurements. The nonlinearity of the viscoelastic behaviour which is represented by the so called damping function, is approximated by a single exponential function with one parametern. The damping constantn as well as a discrete relaxation time spectrum of the melt can be determined from the frequency dependence of the loss and storage moduli.

---

Zusammenfassung Eine Zustandsgleichung vom Integraltyp mit einer deformationsabhängigen und faktorisierten Gedächtnisfunktion wird zur Beschreibung der Zeitabhängigkeit der Schubspannung, der ersten Normalspannungsdifferenz und, unter Verwendung des spannungsoptischen Gesetzes, auch des Auslöschungswinkels und der Strömungsdoppelbrechung einer Polystyrol-Schmelze bei Scherströmungen herangezogen. Die theoretischen Voraussagen werden mit Messungen verglichen. Die Nichtlinearität des viskoelastischen Verhaltens, repräsentiert durch die sogenannte Dämpfungsfunktion, wird durch eine einfache Exponentialfunktion mit nur einem Parametern angenähert. Die Dämpfungskonstanten kann, wie auch ein diskretes Relaxationszeitspektrum der Schmelze, aus der Frequenzabhängigkeit der Speicher- und Verlustmoduln bestimmt werden.

---

a _i weight factor of thei-th relaxation time - a _T shift factor - C stress-optical coefficient - n flow birefringence in the shear flow plane - shear relaxation modulus - G() shear storage modulus - () shear loss modulus - H() relaxation time spectrum - h( _t,t ² ) damping function - M _w weight-average molecular weight - M _n number-average molecular weight - n damping constant - p ₁₂ shear stress - p ₁₁ –p ₂₂ primary normal stress difference - t current time - t past time - extinction angle - ( — _i) delta function - time and shear rate dependent viscosity - | ^*| absolute value of the complex viscosity - shear rate - _t,t relative shear strain between the statest andt - memory function - angular frequency - relaxation time - _i i-th relaxation time of the line spectrum - time and shear rate dependent primary normal stress coefficient - s steady-state value - t time dependence - ° linear viscoelastic behaviour With 6 figures and 1 table     

A new interpretation of viscosity and yield stress in dense slurries: Coal and other irregular particles

Illinois coal was ground and wet-sieved to prepare three powder stocks whose particle-size distributions were characterized. Three suspending fluids were used (glycerin, bromonaphthalene, Aroclor), with viscosities _s that differed by a factor of 100 and with very different chemistries, but whose densities matched that of the coal. Suspensions were prepared under vacuum, with coal volume fractions that ranged up to 0.46. Viscosities were measured in a cone-and-plate over a shear rate range 10^–3–10² s^–1. Reduced viscosity _r = /_s is correlated in the high-shear limit ( ) with/ _M, where _M is the maximum packing fraction for the high-shear microstructure, to reveal the roles of size distribution and suspending fluid character. A new model that invokes the stress-dependence of _M is found to correlate _r well under non-Newtonian conditions with simultaneous prediction of yield stress at sufficiently high; a critical result is that stress and not governs the microstructure and rheology. Numerous experimental anomalies provide insight into suspension behavior.     

Transcendentally small transversality in the rapidly forced pendulum

The rapidly forced pendulum equation with forcing sin((t/), where =<₀^p,p = 5, for ₀, sufficiently small, is considered. We prove that stable and unstable manifolds split and that the splitting distanced(t) in the ( ,t) plane satisfiesd(t) = sin(t/) sech(/2) +O( ₀ exp(–/2)) (2.3a) and the angle of transversal intersection,, in thet = 0 section satisfies 2 tan/2 = 2S _s = (/2) sech(/2) +O(( ₀ /) exp(–/2)) (2.3b) It follows that the Melnikov term correctly predicts the exponentially small splitting and angle of transversality. Our method improves a previous result of Holmes, Marsden, and Scheuerle. Our proof is elementary and self-contained, includes a stable manifold theorem, and emphasizes the phase space geometry.