Flow in porous media III: Deformable media

Stokes flow in a deformable medium is considered in terms of an isotropic, linearly elastic solid matrix. The analysis is restricted to steady forms of the momentum equations and small deformation of the solid phase. Darcy's law can be used to determine the motion of the fluid phase; however, the determination of the Darcy's law permeability tensor represents part of the closure problem in which the position of the fluid-solid interface must be determined.Roman Letters A interfacial area of the- interface 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 macroscopic system, m² - A ^* interfacial area of the- interface contained within a unit cell, m² - A _e ^* area of entrances and exits for the-phase contained within a unit cell, m² - E Young's modulus for the-phase, N/m² - e i unit base vectors (i = 1, 2, 3) - g gravity vector, m²/s - H height of elastic, porous bed, m - k unit base vector (=e ₃) - characteristic length scale for the-phase, m - L characteristic length scale for volume-averaged quantities, m - n unit normal vector pointing from the-phase toward the-phase (n = -n ) - p pressure in the-phase, N/m² - P p – g·r, N/m² - r ₀ radius of the averaging volume, m - r position vector, m - t time, s - T total stress tensor in the-phase, N/m² - T ⁰ hydrostatic stress tensor for the-phase, N/m² - u displacement vector for the-phase, m - V averaging volume, m₃ - V volume of the-phase contained within the averaging volume, m³ - v velocity vector for the-phase, m/s Greek Letters V /V, volume fraction of the-phase - mass density of the-phase, kg/m³ - shear coefficient of viscosity for the-phase, Nt/m² - first Lamé coefficient for the-phase, N/m² - second Lamé coefficient for the-phase, N/m² - bulk coefficient of viscosity for the-phase, Nt/m² - T –T ⁰ , a deviatoric stress tensor for the-phase, N/m²     

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

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

Supersonic flow past axisymmetric flared cones

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

Asymptotic Nusselt numbers for Newtonian flow through a parallel-plate channel with recycle

General expressions for evaluating the asymptotic Nusselt number for a Newtonian flow through a parallel-plate channel with recycle at the ends have been derived. Numerical results with the ratio of thicknesses as a parameter for various recycle ratios are obtained. A regression analysis shows that the results can be expressed by log Nur^0.83=0.3589 (log)² -0.2925 (log) + 0.3348 forR 3, 0.1 0.9; logNu=0.5982(log)² +0.3755 × 10^–2 (log) +0.8342 forR 10^–2, 0.1 0.9.

---

Asymptotische Nusselt-Zahlen für die Newtonsche Strömung durch einen Kanal aus parallelen Platten mit Rückführung

Zusammenfassung In dieser Untersuchung wurden allgemeine Ausdrücke hergeleitet um die asymptotische Nusselt-Zahl für eine Newtonsche Strömung durch einen Kanal aus parallelen Platten mit Rückführung an den Enden berechnen zu können. Es wurden numerische Ergebnisse mit den Dicken-Verhältnissen, als Parameter für verschiedene Rückführungs-verhältnisse, erhalten. Eine Regressionsanalyse zeigt, daß die Ergebnisse wie folgt ausgedrückt werden können: log Nur^0,83=0,3589 (log)² -0,2925 (log) + 0,3348 fürR 3, 0,1 0,9; logNu=0,5982(log)² +0,3755 × 10^–2 (log) + 0,8342 fürR 10^–2, 0,1 0,9.

---

Nomenclature A₁ shooting value,d(0)/d - A₂ shooting value,d(1)/d - B channel width - Gz Graetz number, U_bW²/L - h _m logarithmic average convective heat transfer coefficient - h _x average local convective heat transfer coefficient - k thermal conductivity - L channel length - Nu average local Nusselt number, 2 h_xW/k - Nu _m logarithmic average Nusselt number, 2h_mW/k - R recycle ratio, reverse volume flow rate divided by input volume flow rate - T temperature of fluid - T _m bulk temperature, Eq. (8) - T ₀ temperature of feed stream - T _s wall temperature - U velocity distribution - U _b reference velocity,V/BW - V input volume flow rate - v dimensionless velocity distribution, U/U_b - W channel thickness - x longitudinal coordinate - y transversal coordinate - Z₁-z₆ functions defined in Eq. (A1) - thermal diffusivity - least squares error, Eq. (A7) - weight, Eqs. (A8), (A9) - dimensionless coordinate,y/W - dimensionless coordinate,x/GzL - function, Eq. (7)     

Effect of measuring volume length on two-component laser velocimeter measurements in a turbulent boundary layer

The effects of finite measuring volume length on laser velocimetry measurements of turbulent boundary layers were studied. Four different effective measuring volume lengths, ranging in spanwise extent from 7 to 44 viscous units, were used in a low Reynolds number (Re=1440) turbulent boundary layer with high data density. Reynolds shear stress profiles in the near-wall region show that u v strongly depends on the measuring volume length; at a given y-position, u v decreases with increasing measuring volume length. This dependence was attributed to simultaneous validations on the U and V channels of Doppler bursts coming from different particles within the measuring volume. Moments of the streamwise velocity showed a slight dependence on measuring volume length, indicating that spatial averaging effects well known for hot-films and hot-wires can occur in laser velocimetry measurements when the data density is high.List of symbols time-averaged quantity - u wall friction velocity, ( _w /)^1/2 - v kinematic viscosity - d _p pinhole diameter - l _eff spanwise extent of LDV measuring volume viewed by photomultiplier - l ⁺ non-dimensional length of measuring volume, l _eff u /v - y ⁺ non-dimensional coordinate in spanwise direction, y u /v - z ⁺ non-dimensional coordinate in spanwise direction, z u /v - U ⁺ non-dimensional mean velocity, /u - u instantaneous streamwise velocity fluctuation, U &#x2329;U - v instantaneous normal velocity fluctuation, V–V - u RMS streamwise velocity fluctuation, u ^21/2 - v RMS normal velocity fluctuation, v ^21/2 - Re Reynolds number based on momentum thickness, U 0/v - R _uv cross-correlation coefficient, u v/u v - R₁₂(0, 0, z) two point correlation between u and v with z-separation, <u(0, 0, 0) v (0, 0, z)>/<u(0, 0, 0) v (0, 0, 0)> - N rate at which bursts are validated by counter processor - T Taylor time microscale, u (dv/dt²)^–1/2     

On surface waves in generalized thermoelasticity

Knowles' representation theorem for harmonically time-dependent free surface waves on a homogeneous, isotropic elastic half-space is extended to include harmonically time-dependent free processes for thermoelastic surface waves in generalized thermoelasticity of Lord and Shulman and of Green and Lindsay.r , , r , , .This work was done when author was unemployed.     

Experimental analysis of scaling laws in low Re λgrid-generated turbulence

An experimental investigation of scaling laws in turbulence generated by a biplane grid for low Reynolds numbers (Re ) is presented. The present study covers a wide range of flow field conditions (from Re 2.5 to Re 36) that have not been analyzed from this point of view. It is shown that following the Kolmogorov theory a scaling range can not be observed since the separation between the energy production scales and the dissipative scales is too short. On the other hand, an extended form of scaling, the Extended Self-Similarity (ESS), permits the identification of a range of scales characterized by the same scaling exponent much wider than the one previously examined. Thus the scaling laws for the first six moments of the velocity structure function are accurately calculated by means of the ESS and an anomalous scaling with respect to the Kolmogorov theory is observed for Re down to the order of 10. As a matter of fact the scaling exponents are in good agreement with the ones that were determined at higher Re by previous experimental and numerical investigations. For Re 6 a regularization of the scaling exponents is observed as an effect of the dissipation. We also present an analysis of the universality properties of the ESS form of scaling by means of the form function and an analysis of the sensitivity of the scaling range to the Re .     

Flow in porous media I: A theoretical derivation of Darcy's law

Stokes flow through a rigid porous medium is analyzed in terms of the method of volume averaging. The traditional averaging procedure leads to an equation of motion and a continuity equation expressed in terms of the volume-averaged pressure and velocity. The equation of motion contains integrals involving spatial deviations of the pressure and velocity, the Brinkman correction, and other lower-order terms. The analysis clearly indicates why the Brinkman correction should not be used to accommodate ano slip condition at an interface between a porous medium and a bounding solid surface.The presence of spatial deviations of the pressure and velocity in the volume-averaged equations of motion gives rise to aclosure problem, and representations for the spatial deviations are derived that lead to Darcy's law. The theoretical development is not restricted to either homogeneous or spatially periodic porous media; however, the problem ofabrupt changes in the structure of a porous medium is not considered.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 ^* interfacial area of the - interface contained within a unit cell, m² - Ae area of entrances and exits for the -phase contained within a unit cell, m² - B second order tensor used to represent the velocity deviation (see Equation (3.30)) - b vector used to represent the pressure deviation (see Equation (3.31)), m^–1 - d distance between two points at which the pressure is measured, m - g gravity vector, m/s² - K Darcy's law permeability tensor, m² - L characteristic length scale for volume averaged quantities, m - characteristic length scale for the -phase (see Figure 2), m - characteristic length scale for the -phase (see Figure 2), m - n unit normal vector pointing from the -phase toward the -phase (n =–n ) - n _e unit normal vector for the entrances and exits of the -phase contained within a unit cell - p pressure in the -phase, N/m² - p intrinsic phase average pressure for the -phase, N/m² - p –p , spatial deviation of the pressure in the -phase, N/m² - r ₀ radius of the averaging volume and radius of a capillary tube, m - v velocity vector for the -phase, m/s - v phase average velocity vector for the -phase, m/s - v intrinsic phase average velocity vector for the -phase, m/s - v –v , spatial deviation of the velocity vector for the -phase, m/s - V averaging volume, m³ - V volume of the -phase contained within the averaging volume, m³ Greek Letters V/V, volume fraction of the -phase - mass density of the -phase, kg/m³ - viscosity of the -phase, Nt/m² - arbitrary function used in the representation of the velocity deviation (see Equations (3.11) and (B1)), m/s - arbitrary function used in the representation of the pressure deviation (see Equations (3.12) and (B2)), s^–1     

Flow through a helically coiled annulus

In this paper the flow is studied of an incompressible viscous fluid through a helically coiled annulus, the torsion of its centre line taken into account. It has been shown that the torsion affects the secondary flow and contributes to the azimuthal component of velocity around the centre line. The symmetry of the secondary flow streamlines in the absence of torsion, is destroyed in its presence. Some stream lines penetrate from the upper half to the lower half, and if is further increased, a complete circulation around the centre line is obtained at low values of for all Reynolds numbers for which the analysis of this paper is valid, being the ratio of the torsion of the centre line to its curvature.Nomenclature A =constant - a outer radius of the annulus - b unit binormal vector to C - C helical centre line of the pipe - D rL - g 1000 - K Dean number=Re² - L 1+r sin - M (L ²+ ² r ²)^1/2 - n unit normal vector to C - P, P pressure and nondimensional pressure - p ₀, p pressures of O(1) and O() - Re Reynolds number=aW ₀/ - (r, , s), (r, , s) coordinates and nondimensional coordinates - nonorthogonal unit vectors along the coordinate directions - r ₀ radius of the projection of C - t unit tangent vector to C - V _r, V , V _s velocity components along the nonorthogonal directions - V_r, V, V _s nondimensional velocity components along - W ₀ average velocity in a straight annulus Greek symbols , curvature and nondimensional curvature of C - U, V, W lowest order terms for small in the velocity components along the orthogonal directions t - _r, , _s first approximations to V _r , V, V _s for small - =/=/ - kinematic viscosity - density of the fluid - , torsion and nondimensional torsion of C - , stream function and nondimensional stream function - nondimensional streamfunction for U, V - a inner radius of the annulus After this paper was accepted for publication, a paper entitled On the low-Reynolds number flow in a helical pipe, by C.Y. Wang, has appeared in J. Fluid. Mech., Vol 108, 1981, pp. 185–194. The results in Wangs paper are particular cases of this paper for =0, and are also contained in [9].     

Transport in ordered and disordered porous media I: The cellular average and the use of weighting functions

In this work we consider transport in ordered and disordered porous media using singlephase flow in rigid porous mediaas an example. We defineorder anddisorder in terms of geometrical integrals that arise naturally in the method of volume averaging, and we show that dependent variables for ordered media must generally be defined in terms of thecellular average. The cellular average can be constructed by means of a weighting function, thus transport processes in both ordered and disordered media can be treated with a single theory based on weighted averages. Part I provides some basic ideas associated with ordered and disordered media, weighted averages, and the theory of distributions. In Part II a generalized averaging procedure is presented and in Part III the closure problem is developed and the theory is compared with experiment. Parts IV and V provide some geometrical results for computer generated porous media.Roman Letters A interfacial area of the- interface contained within the macroscopic region, m² - A_e area of entrances and exits for the-phase contained within the macroscopic system, m² - g gravity vector, m/s² - I unit tensor - K traditional Darcy's law permeability tensor, m² - L general characteristic length for volume averaged quantities, m - characteristic length (pore scale) for the-phase - (y) weighting function - m(–y) (y), convolution product weighting function - _v special weighting function associated with the traditional averaging volume - N unit normal vector pointing from the-phase toward the-phase - p pressure in the-phase, N/m² - p0 reference pressure in the-phase, N/m² - p traditional intrinsic volume averaged pressure, N/m² - r0 radius of a spherical averaging volume, m - r position vector, m - r position vector locating points in the-phase, m - averaging volume, m³ - V volume of the-phase contained in the averaging volume, m³ - V _cell volume of a unit cell, m³ - v velocity vector in the-phase, m/s - v traditional superficial volume averaged velocity, m/s - x position vector locating the centroid of the averaging volume or the convolution product weighting function, m - y position vector 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 - mass density of the-phase, kg/m³ - viscosity of the-phase, Ns/m²     

LDA measurements in low Reynolds number turbulent boundary layer

LDA measurements of the mean velocity in a low Reynolds number turbulent boundary layer allow a direct estimate of the friction velocity U from the value of /y at the wall. The trend of the Reynolds number dependence of / is similar to the direct numerical simulations of Spalart (1988).     

Stability characteristics of condensate films

The linear stability theory is used to study stability characteristics of laminar condensate film flow down an arbitrarily inclined wall. A critical Reynolds number exists above which disturbances will be amplified. The magnitude of the critical Reynolds number is in all practical situations so small that a laminar gravity-induced condensate film can be expected to be unstable. Several stabilizing effects are acting on the film flow; at an inclined wall these effects are due to surface tension, gravity and condensation mass transfer.

---

Zusammenfassung Mit Hilfe der linearen Stabilitätstheorie werden die Stabilitätseigenschaften laminarer Kondensatfilme an einer geneigten Wand untersucht. Es zeigt sich, daß Kondensatfilme in jedem praktischen Fall ein unstabiles Verhalten aufweisen. Der stabilisierende Einfluß von Oberflächenspannung, Schwerkraft und Stoffübertragung durch Kondensation bewkkt jedoch, daß Störungen in bestimmten Wellenlängenbereichen gedämpft werden.

---

Nomenclature c=c^*/u₀ complex wave velocity, celerity, dimensionless - c^*=c _r ^* + i c _i ^* complex wave velocity, celerity, dimensional - c_p specific heat at constant pressure - g gravitational acceleration - h_fg latent heat - k thermal conductivity of liquid - p^* pressure - p=p^*/u₀ ² dimensionless pressure - Pe=Pr Re^* Peclet number - Pr Prandtl number - Re^*=u₀ / Reynolds number (defined with surface velocity) - S temperature perturbation amplitude - t^* time - t=t^* u₀/ dimensionless time - T temperature - T_s saturation temperature - T_w wall temperature - T=T_s-T_w temperature drop across liquid film - u^*, v^* velocity components - u=u^*/u₀ dimensionless velocity components - v=v^*/u₀ dimensionless velocity components - u₀ surface velocity of undisturbed film flow - v _g ^* vapor velocity - x^*, y^* coordinates - x=x^*/ dimensionless coordinates - y=y^*/ dimensionless coordinates Greek Symbols =^* wave number, dimensionless - ^*=2 /^* wave number dimensional - ^* wave length, dimensional - =^*/ wave length, dimensionless - local thickness of undisturbed condensate film - kinematic viscosity, liquid - density, liquid - _g density vapor - surface tension - = (1 +) film thickness of disturbed film, Fig. 1 - stream function perturbation amplitude - angle of inclination Base flow quantities are denoted by^–, disturbance quantities are denoted by.     

Dielectric properties of the composite system polyurethane — sodium chloride

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

---

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

     

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.     

The value of the critical exponent for reaction-diffusion equations in cones

Let D R ^N be a cone with vertex at the origin i.e., D = (0, )x where S ^N–1 and x D if and only if x = (r, ) with r=¦x¦, . We consider the initial boundary value problem: u _t = u+u ^p in D×(0, T), u=0 on Dx(0, T) with u(x, 0)=u ₀(x) 0. Let ₁ denote the smallest Dirichlet eigenvalue for the Laplace-Beltrami operator on and let ₊ denote the positive root of (+N–2) = ₁. Let p ^* = 1 + 2/(N + ₊). If 1 < p < p ^*, no positive global solution exists. If p>p ^*, positive global solutions do exist. Extensions are given to the same problem for u _t=+¦x¦ u ^p .This research was supported in part by the Air Force Office of Scientific Research under Grant # AFOSR 88-0031 and in part by NSF Grant DMS-8 822 788. The United States Government is authorized to reproduce and distribute reprints for governmental purposes not withstanding any copyright notation therein.     

On the localness of the spectral energy transfer in turbulence

By utilizing available experimental data for net energy transfer spectra for homogeneous turbulence, contributions P(, ) to the energy transfer at a wavenumber from various other wavenumbers are calculated. This is done by fitting a truncated power-exponential series in and to the experimental data for the net energy transfer T(), and using known properties of P(, ). Although the contributions P(, ) obtained by using this procedure are not unique, the results obtained by using various assumptions do not differ significantly. It seems clear from the results that for a region where the energy entering a wavenumber band dominates that leaving, much of the energy entering the band comes from wavenumbers which are about an order of magnitude smaller. That is, the energy transfer is rather nonlocal. This result is not significantly dependent on Reynolds number (for turbulence Reynolds numbers based on microscale from 3 to 800). For lower wavenumbers, where more energy leaves than enters a wavenumber band, the energy transfer into the band is more local, but much of the energy then leaves at distant wavenumbers.     

ON THE UNIFICATION OF THE HAMILTON PRINCIPLES IN NONHOLONOMIC SYSTEM AND IN HOLONOMIC SYSTEM

ＯＮＴＨＥＵＮＩＦＩＣＡＴＩＯＮＯＦＴＨＥＨＡＭＩＬＴＯＮＰＲＩＮＣＩＰＬＥＳＩＮＮＯＮＨＯＬＯＮＯＭＩＣＳＹＳＴＥＭＡＮＤＩＮＨＯＬＯＮＯＭＩＣＳＹＳＴＥＭ（梁立孚）（韦扬）ＯＮＴＨＥＵＮＩＦＩＣＡＴＩＯＮＯＦＴＨＥＨＡＭＩＬＴＯＮＰＲＩＮＣＩＰＬ...     

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.     

The problems of nonlinear bending for orthotropic rectangular plate with four clamped edges

（黄家寅）（秦圣立）ＴＨＥＰＲＯＢＬＥＭＳＯＦＮＯＮＬＩＮＥＡＲＢＥＮＤＩＮＧＦＯＲＯＲＴＨＯＴＲＯＰＩＣＲＥＣＴＡＮＧＵＬＡＲＰＬＡＴＥＷＩＴＨＦＯＵＲＣＬＡＭＰＥＤＥＤＧＥＳ￥ＨｕａｎｇＪｉａｙｉｎ;ＱｉｎＳｈｅｎｇｌｉ（ＱｕｆｕＮｏｒｍａｌＵｎ...     

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

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