On the Ellipticity of the Middle Surface of a Shell and its Application to the Asymptotic Analysis of ‘Membrane’ Shells

We consider a surface S = (), where ² is a bounded, connected, open set with a smooth boundary and : ³ is a smooth map; let () denote the components of the two-dimensional linearized strain tensor of S and let 0 with length 0 > 0. We assume the the norm ,|| ()||0, in the space V0() = { H¹() × H¹() × L²(); = 0 on 0 } is equivalent to the usual product norm on this space. We then establish that this assumption implies that the surface S is uniformly elliptic and that we necessarily have 0 = .     

Evaluation of the dynamic-mechanical properties of solids from a cooperative model for stress relaxation

For many solid materials the stress relaxation process obeys the universal relationF = – (d/d lnt)_max = (0.1 ± 0.01) ( ₀ — _i ), regardless of the structure of the material. Here denotes the stress,t the time, ₀ the initial stress of the experiment and _i the internal stress. A cooperative model accounting for the similarity in relaxation behaviour between different materials was developed earlier. Since this model has a spectral character, the concepts of linear viscoelasticity are used here to evaluate the corresponding prediction of the dynamic mechanical properties, i.e. the frequency dependence of the storageE () and lossE () moduli. Useful numerical approximations ofE () andE () are also evaluated. It is noted that the universal relation in stress relaxation had a counterpart in the frequency dependence ofE (). The theoretical prediction of the loss factor for high-density polyethylene is compared with experimental results. The agreement is good.     

A Direct Method for the Identification of the Permeability Field Based on Flux Assimilation by a Discrete Analog of Darcy's Law

Inverse models to determine the permeability are generally based on existing forward models for the pressure. The permeabilities are adapted in such a way that the calculated pressures match the specified pressures in a number of points. To assimilate a priori knowledge about the flux, we introduce the flux assimilation method, which is based on the vector potential–pressure formulation of Darcy's law. Thanks to an unconventional discretization technique – the edge-based face element method – not only the specified pressures, but also specified information about the flux density can easily be assimilated. A relatively simple, but insightful analytical example illustrates the potential of this method.     

Heteroclinic orbits in singular systems: A unifying approach

We consider singularly perturbed systems , such that=f(, _o, 0). _o ^m , has a heteroclinic orbitu(t). We construct a bifurcation functionG(, ) such that the singular system has a heteroclinic orbit if and only ifG(, )=0 has a solution=(). We also apply this result to recover some theorems that have been proved using different approaches.     

Transport in ordered and disordered porous media II: Generalized volume averaging

In this paper we develop the averaged form of the Stokes equations in terms of weighting functions. The analysis clearly indicates at what point one must choose a media-specific weighting function in order to achieve spatially smoothed transport equations. The form of the weighting function that produces the cellular average is derived, and some important geometrical theorems are presented.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 _p surface area of a particle, m² - d _p 6V _p/A_p, effective particle diameter, m - g gravity vector, m/s² - I unit tensor - K _m permeability tensor for the weighted average form of Darcy's law, m² - L general characteristic length for volume averaged quantities, m - L _p general characteristic length for volume averaged pressure, m - L characteristic length for the porosity, m - L _v characteristic length for the volume averaged velocity, m - l characteristic length (pore scale) for the-phase - l _i i=1, 2, 3 lattice vectors, m - (y) weighting function - m(–y) (y), convolution product weighting function - _v special weighting function associated with the traditional averaging volume - m _v special convolution product weighting function associated with the traditional averaging volume - m _g general convolution product weighting function - m _V unit cell convolution product weighting function - m _C special convolution product weighting function for ordered media which produces the cellular average - m _D special convolution product weighting function for disordered media - m _M master convolution product weighting function for ordered and disordered media - n unit normal vector pointing from the-phase toward the-phase - p pressure in the-phase, N/m² - p_m superficial weighted average pressure, N/m² - p _m intrinsic weighted average pressure, N/m² - p traditional intrinsic volume averaged pressure, N/m² - p p p _m , spatial deviation pressure, N/m² - r ₀ radius of a spherical averaging volume, m - r _m support of the convolution product weighting function, m - r position vector, m - r position vector locating points in the-phase, m - V 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 - vm superficial weighted average velocity, m/s - v _m intrinsic weighted average velocity, m/s - V volume of the-phase contained in the averaging volume, m³ - V _p volume of a particle, m³ - v traditional superficial volume averaged velocity, m/s - v v p _m spatial deviation 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 - _m m * . weighted average porosity - mass density of the-phase, kg/m³ - viscosity of the-phase, Ns/m² - V /V, volume fraction of the-phase     

Symmetry groups of attractors

For maps equivariant under the action of a finite group on ⁿ, the possible symmetries of fixed points are known and correspond to the isotropy subgroups. This paper investigates the possible symmetries of arbitrary, possibly chaotic, attractors and finds that the necessary conditions of Melbourne, Dellnitz & Golubitsky [15] are sufficient, at least for continuous maps.This result shows that the reflection hyperplanes are important in determining those groups which are admissible; more precisely, a subgroup of is admissible as the symmetry group of an attractor if there exists a with / cyclic such that fixes a connected component of the complement of the set of reflection hyperplanes of reflections in but not in . For finite reflection groups this condition on reduces to the condition that is an isotropy subgroup. Our results are illustrated for finite subgroups of O(3).     

Effect of Streamwise Streaky Structures on Turbulization of a Circular Jet

Experimental investigations of the influence of streamwise streaky structures on turbulization of a circular laminar jet are described. The qualitative characteristics of jet evolution are studied by smoke visualization of the flow pattern in the jet and by filming the transverse and longitudinal sections of the jet illuminated by the laser sheet with image stroboscopy. It is shown that the streaky structures can be generated directly at the nozzle exit, and their interaction with the Kelvin–Helmholtz ring vortices leads to emergence of azimuthal beams ( structures) by a mechanism similar to threedimensional distortion of the twodimensional Tollmien–Schlichting wave at the nonlinear stage of the classical transition in nearwall flows. The effect of the jetexhaustion velocity and acoustic action on jet turbulization is considered.     

Normal forms for random diffeomorphisms

Given a dynamical system (,, ,) and a random diffeomorphism (): ^{d d} with fixed point at x=0. The normal form problem is to construct a smooth near-identity nonlinear random coordinate transformation h() to make the random diffeomorphism ()=h()^–1() h() as simple as possible, preferably linear. The linearization D(, 0)=:A() generates a matrix cocycle for which the multiplicative ergodic theorem holds, providing us with stochastic analogues of eigenvalues (Lyapunov exponents) and eigenspaces. Now the development runs pretty much parallel to the deterministic one, the difference being that the appearance of turns all problems into infinite-dimensional ones. In particular, the range of the homological operator is in general not closed, making the conceptof-normal form necessary. The stochastic versions of resonance and averaging are developed. The case of simple Lyapunov spectrum is treated in detail.     

Effect of suspended particles on the stability of plane poiseuille flow

The behavior of the neutral stability curves is investigated for various values of the particle relaxation time and mass concentration 0 100 and 0 f 0.1. It is shown that as increases from zero the flow is at first destabilized and then at >6 becomes stable, while at >40 the stabilizing effect of the dispersed phase grows weaker. It is found that there is a certain interval 10< <40 on which the flow is most stable.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 46–53, January–February, 1986.     

Unique Periodic Orbits for Delayed Positive Feedback and the Global Attractor

The delay differential equation, (t)=–x(t)+f(x(t–1)), with >0 and a real function f satisfying f(0)=0 and f>0 models a system governed by delayed positive feedback and instantaneous damping. Recently the geometric, topological, and dynamical properties of a three-dimensional compact invariant set were described in the phase space C=C([–1, 0], ) of initial data for solutions of the equation. In this paper, for a set of and f which include examples from neural network theory, we show that this three-dimensional set is the global attractor, i.e., the compact invariant set which attracts all bounded subsets of C. The proof involves, among others, results on uniqueness and absence of periodic orbits.     

Reservoir transport equations by compositional approach

The objective of this paper is to present an overview of the fundamental equations governing transport phenomena in compressible reservoirs. A general mathematical model is presented for important thermo-mechanical processes operative in a reservoir. Such a formulation includes equations governing multiphase fluid (gas-water-hydrocarbon) flow, energy transport, and reservoir skeleton deformation. The model allows phase changes due to gas solubility. Furthermore, Terzaghi's concept of effective stress and stress-strain relations are incorporated into the general model. The functional relations among various model parameters which cause the nonlinearity of the system of equations are explained within the context of reservoir engineering principles. Simplified equations and appropriate boundary conditions have also been presented for various cases. It has been demonstrated that various well-known equations such as Jacob, Terzaghi, Buckley-Leverett, Richards, solute transport, black-oil, and Biot equations are simplifications of the compositional model.Notation List B reservoir thickness - B formation volume factor of phase - Cⁱ mass of component i dissolved per total volume of solution - C ⁱ mass fraction of component i in phase - C heat capacity of phase at constant volume - C_p heat capacity of phase at constant pressure - D ⁱ hydrodynamic dispersion coefficient of component i in phase - D_MTf thermal liquid diffusivity for fluid f - F = F(x, y, z, t) defines the boundary surface - f_p fractional flow of phase - g gravitational acceleration - H_p enthalpy per unit mass of phase - J_p volumetric flux of phase - k_rf relative permeability to fluid f - k₀ absolute permeability of the medium - M_p ⁱ mass of component i in phase - n porosity - N rate of accretion - P_f pressure in fluid f - p_ca capillary pressure between phases and =p-p - Rⁱ rate of mass transfer of component i from phase to phase - Rⁱ _source source rate of component i within phase - S saturation of phase - s gas solubility - T temperature - t time - U displacement vector - u velocity in the x-direction - v velocity in the y-direction - V volume of phase - V_s velocity of soil solids - W_i body force in coordinate direction i - x horizontal coordinate - z vertical coordinate Greek Letters _p volumetric coefficient of compressibility - _T volumetric coefficient of thermal expansion - _ij Kronecker delta - volumetric strain - _m thermal conductivity of the whole matrix - internal energy per unit mass of phase - gf suction head - density of phase - _ij tensor of total stresses - _ij tensor of effective stresses - volumetric content of phase - f viscosity of fluid f     

On laminar flow through a uniformly porous pipe

Numerous investigations ([1] and [4–9]) have been made of laminar flow in a uniformly porous circular pipe with constant suction or injection applied at the wall. The object of this paper is to give a complete analysis of the numerical and theoretical solutions of this problem. It is shown that two solutions exist for all values of injection as well as the dual solutions for suction which had been noted by previous investigators. Analytical solutions are derived for large suction and injection; for large suction a viscous layer occurs at the wall while for large injection one solution has a viscous layer at the centre of the channel and the other has no viscous layer anywhere. Approximate analytic solutions are also given for small values of suction and injection.

Nomenclature

General r distance measured radially - z distance measured along axis of pipe - u velocity component in direction of z increasing - v velocity component in direction of r increasing - p pressure - density - coefficient of kinematic viscosity - a radius of pipe - V velocity of suction at the wall - r ²/a ² - R wall or suction Reynolds number, Va/ - f() similarity function defined in (6) - u ₀() eigensolution - U(0) a velocity at z=0 - K an arbitrary constant - B _K Bernoulli numbers Particular Section 5 perturbation parameter, –2/R - ² a constant, –K - x / - g(x) f()/ Section 6 perturbation parameter, –R/2 - ² a constant, –K - g() f() - g _c ()=g() near centre of pipe - * point where g()=0 Section 7 2/R - ² K - t (1–)/ - w(t, ) [1–f(t)]/ - ₀, ₁ constants - g() f()– ₀ - ₀/ - ₀ a constant - * point where f()=0     

An experimental study on the two-dimensional opposed wall jet in a turbulent boundary layer: Change in the flow pattern with velocity ratio

Results of the measurement of flow properties in a two-dimensional turbulent wall jet which is injected into the turbulent boundary layer in the direction opposite to that of the boundary layer flow are presented by varying the ratio of the jet issuing velocity to the mainstream velocity . This flow includes the flow separation and the recirculating flow, and so it requires the magnitude and direction of instantaneous velocity be measured. In the present work, a tandem hot-wire probe is manufactured and its characteristics are examined experimentally. With the use of this probe the change in the penetration length of the jet with the velocity ratio is investigated. It is clarified that two regimes of flow patterns exist: in the low velocity ratio the penetration length of the jet changes little with , and in the high velocity ratio it goes far from the nozzle with increasing . Streamlines, turbulence intensity contours and static pressure contours are presented in the two typical velocity ratios corresponding to each of two flow patterns, and they are compared.List of symbols b ₀ nozzle width (Fig. 1) - , e mean and fluctuating output voltages of hot-wire anemometer - p, p mean static pressure, p = p – p _o - p ₀ static pressure in undisturbed mainstream - p _w , p _w mean wall pressure, p _w = p _w – p _o - U ₀ mainstream velocity - U _j jet velocity at the nozzle exit - , u mean and fluctuating velocity components in x-direction - u _e effective cooling velocity - x distance along the wall from nozzle exit - x _pmax, x _pmin positions where the wall pressure has maximum and minimum values respectively - x _s penetration length of jet - y distance from the wall - forward flow fraction - ₁, ₂ yaw and pitch angles of flow direction (Fig. 4) - velocity ratio, = U _j /U _o - density of the fluid - nondimensional stream function The authors wish to express their appreciation to Prof. Toshio Tanaka of Gifu University for his advice in the course of the experiment. They also would like to thank the Research Foundation for the Electrotechnology of Chubu which partly supported this work.     

On the properties of compressible gas flow in a porous media

The flow of an adiabatic gas through a porous media is treated analytically for steady one- and two-dimensional flows. The effect on a compressible Darcy flow by inertia and Forchheimer terms is studied. Finally, wave solutions are found which exhibit a cut-off frequency and a phase shift between pressure and velocity of the gas, with the velocity lagging behind the pressure.Nomenclature A area of tube for one-dimensional flow - B drag coefficient associated with Forchheimer term - c speed of sound - M Mach number - p ^* gas pressure - p dimensionless gas pressure - s coordinate along the axis of tube - t ^* time variable - t dimensionless time variable - V^* gas velocity in the porous media - V dimensionless gas velocity Greek Letters ratio of specific heat capacities - phase angle between gas pressure and velocity for linear waves - parameter indicating the importance of the inertia term - viscosity - _p natural frequency of the porous media - ^* gas density - dimensionless gas density - parameter indicating the importance of the Forchheimer term - porosity of porous media - velocity potential - stream function     

Mass transfer with chemical reaction in a laminar falling film

Zusammenfassung Es wird eine analytische Lösung für die Absorption in einem laminaren Rieselfilm mit homogener und heterogener chemischer Reaktion 1. Ordnung vorgestellt, wobei der Stofftransportwiderstand auf der Gasseite liegt. Die Lösung ist eine Funktion von drei dimensionslosen ParameternBi, und, welche die BiotZahl und einen homogenen bzw. heterogenen Reaktionsparameter darstellen. Es wird gezeigt, daß für feste Werte vonBi und die Absorptionsrate (bezogen auf die Breite 1 des Rieselfilms) über eine gewisse Länge (dimensionslos) des Rieselfilms unabhängig von ist, wenn, < 0,6 ist. Die laufende Länge wird von der Stelle aus gemessen, an der die Absorption beginnt. Für b 0,6 nimmt der FlußQ mit zu, erreicht aber einen Sättigungswert bei=10, wonachQ nurmehr sehr langsam anwächst. Jedoch für ein gegebenes und ohne Übergangswiderstand im Film (Bi ) nimmtQ mit für alle 0 zu.

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Mass transfer with chemical reaction in a laminar falling film

An analytical solution is presented for gas absorption in a laminar falling film with first-order homogeneous and heterogeneous chemical reaction and external gas-phase mass transfer resistance. The solution depends on three dimensionless parametersBi, and, wich represent the Biot number, homogeneous and heterogeneous reaction parameters, respectively. It is shown that for fixed values ofBi and, the rate of gas absorption (per unit breadth) over a certain length; (dimensionless) along the falling film measured from the point where surface absorption begins is independent of if < 0.6. For 0.6, this fluxQ increases with but reaches a saturation value at=10 beyond whichQ increases very slowly. But for given and zero gas film resistance (Bi ),Q increases with for all 0.

     

Shear-strain contours from moiré interferometry

Whole-field contour maps of shear strains _xy are derived from displacement fields obtained by moiré interferometry with 2400 /mm (60,960 /in.). Cross-derivatives of displacements are obtained by mechanical differentiation. They are summed by graphical additive moiré. The high sensitivity of moiré interferometry permits quantitative analysis in the small-strain domain.Paper was presented at V International Congress on Experimental Mechanics held in Montreal, Quebec, Canada on June 10–15, 1984.     

Filament stretching equation in Lagrangian coordinates

The general integral of the very simple equation ^21/n/() was found to describe the cross sectional area of filaments of isothermal power law fluids while in transient stretching where is time and is the initial location of fluid molecules at time = 0 given as the distance from a reference point fixed in space. Any such stretching transient given as a solution of the above equation is physically realizable subject to the restrictions > 0 and/ < 0.     

Inhomogeneous ‘longitudinal’ plane waves in a deformed elastic material

By definition, a homogeneous isotropic compressible Hadamard material has the property that an infinitesimal longitudinal homogeneous plane wave may propagate in every direction when the material is maintained in a state of arbitrary finite static homogeneous deformation. Here, as regards the wave, homogeneous means that the direction of propagation of the wave is parallel to the direction of eventual attenuation; and longitudinal means that the wave is linearly polarized in a direction parallel to the direction of propagation. In other words, the displacement is of the form u = ncos k(n · x – ct), where n is a real vector. It is seen that the Hadamard material is the most general one for which a longitudinal inhomogeneous plane wave may also propagate in any direction of a predeformed body. Here, inhomogeneous means that the wave is attenuated, in a direction distinct from the direction of propagation; and longitudinal means that the wave is elliptically polarized in the plane containing these two directions, and that the ellipse of polarization is similar and similarly situated to the ellipse for which the real and imaginary parts of the complex wave vector are conjugate semi-diameters. In other words, the displacement is of the form u = {S exp i(S · x – ct)}, where S is a complex vector (or bivector). Then a Generalized Hadamard material is introduced. It is the most general homogeneous isotropic compressible material which allows the propagation of infinitesimal longitudinal inhomogeneous plane circularly polarized waves for all choices of the isotropic directional bivector. Finally, the most general forms of response functions are found for homogeneously deformed isotropic elastic materials in which longitudinal inhomogeneous plane waves may propagate with a circular polarization in each of the two planes of central circular section of the ⁿ -ellipsoid, where is the left Cauchy-Green strain tensor corresponding to the primary pure homogeneous deformation.     

Lösungen und Schmelzen von Hochpolymeren

Zusammenfassung Die Modelle der ideal verdünnten Lösung und des unendlichen Netzes (Bueche) werden besprochen; für praktische Fälle werden die Modelle verdünnte Lösung mit Gelteilchen und unendliches Netz mit Verhängungsbereichen vorgeschlagen und diskutiert. Es wird eine Methode angegeben, aus dem Übergangspunkt verdünnte Lösung unendliches Netz den Verknäuelungsparameter Am auszurechnen.Einleitungsvortrag anläßlich der Arbeitstagung der Sektion Rheologie des Vereins Österreichischer Chemiker am 28. September 1965 in Graz.     

Asymptotic behavior of solutions of nonlinear elliptic equations

We study and obtain formulas for the asymptotic behavior as ¦x¦ of C ² solutions of the semilinear equation u=f(x, u), x (*) where is the complement of some ball in ⁿ and f is continuous and nonlinear in u. If, for large x, f is nearly radially symmetric in x, we give conditions under which each positive solution of (*) is asymptotic, as ¦x¦, to some radially symmetric function. Our results can also be useful when f is only bounded above or below by a function which is radially symmetric in x or when the solution oscillates in sign. Examples when f has power-like growth or exponential growth in the variables x and u usefully illustrate our results.