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1.
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.  相似文献   

2.
In this paper the fundamental solution of the singular governing equation of plate static bending is taken as the Green's function, which can satisfy the governing equation precisely in the plate region. Based on the principle of superposition, let the function values on the plate boundary, induced by a set of the Green's function sources (including the known sources in the plate region and the unknown sources in the fictitious region), satisfy the prescribed conditions on specially chosen boundary matching points, and the corresponding semi-analytical and semi-numerical solution can be obtained, which is free from the restraint of boundary forms and boundary conditions. The more matching points there are on the boundary, the better the accuracy of results is. Finally, in static bending problems a set of linear algebraic equations has to be computed; in buckling problems the minimum value of buckling eigenvalue equation has to be found; in natural vibration problems the eigenvalues of the frequency equation have to be calculated. Numerical examples are given and the results are compared with those by the analytical method and other methods. It can be seen that they are very close to each other.  相似文献   

3.
Experiments are conducted to test extant theory on the effect of uniform rotation on the angle of conical beam wave propagation excited by a sphere vertically oscillating at frequency in a density stratified fluid. The near-constant Brunt–Väisälä frequency stratification N produced in situ in a rotating cylindrical tank exhibits no effect of residual motion for the range of Froude numbers investigated. Good agreement between experiment and theory is found over the range of angles 15°<<65° using the synthetic schlieren visualization technique. In particular, the cut-off for wave propagation at =2, below which waves do not propagate, is clearly observed.  相似文献   

4.
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   相似文献   

5.
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 = .  相似文献   

6.
The dynamics of the structure of a liquid layer structure (with microbubbles of a free gas) behind a rarefaction wave front is studied numerically using the two-phase Iordansky–Kogarko–van Wijngaarden model and the frozen mass-velocity field model. An analysis of the initial stage of cavitation by the Iordansky–Kogarko–van Wijngaarden model showed that tensile stresses behind the rarefaction wave front relax quickly and the mass-velocity field in the cavitation zone turns out to be frozen. This effect is used to describe the late stage of the development of the cavitation zone. These models were combined to study the formation of cavitating spalls in a free-surface liquid under shock-wave loading.  相似文献   

7.
In this paper we continue previous studies of the closure problem for two-phase flow in homogeneous porous media, and we show how the closure problem can be transformed to a pair of Stokes-like boundary-value problems in terms of pressures that have units of length and velocities that have units of length squared. These are essentially geometrical boundary value problems that are used to calculate the four permeability tensors that appear in the volume averaged Stokes' equations. To determine the geometry associated with the closure problem, one needs to solve the physical problem; however, the closure problem can be solved using the same algorithm used to solve the physical problem, thus the entire procedure can be accomplished with a single numerical code.Nomenclature a a vector that maps V onto , m-1. - A a tensor that maps V onto . - A area of the - interface contained within the macroscopic region, m2. - A area of the -phase entrances and exits contained within the macroscopic region, m2. - A area of the - interface contained within the averaging volume, m2. - A area of the -phase entrances and exits contained within the averaging volume, m2. - Bo Bond number (= (=(–)g2/). - Ca capillary number (= v/). - g gravitational acceleration, m/s2. - H mean curvature, m-1. - I unit tensor. - permeability tensor for the -phase, m2. - viscous drag tensor that maps V onto V. - * dominant permeability tensor that maps onto v , m2. - * coupling permeability tensor that maps onto v , m2. - characteristic length scale for the -phase, m. - l characteristic length scale representing both and , m. - L characteristic length scale for volume averaged quantities, m. - n unit normal vector directed from the -phase toward the -phase. - n unit normal vector representing both n and n . - n unit normal vector representing both n and n . - P pressure in the -phase, N/m2. - p superficial average pressure in the -phase, N/m2. - p intrinsic average pressure in the -phase, N/m2. - p p , spatial deviation pressure for the -phase, N/m2. - r 0 radius of the averaging volume, m. - r position vector, m. - t time, s. - v fluid velocity in the -phase, m/s. - v superficial average velocity in the -phase, m/s. - v intrinsic average velocity in the -phase, m/s. - v v , spatial deviation velocity in the -phase, m/s. - V volume of the -phase contained within the averaging volmue, m3. - averaging volume, m3. Greek Symbols V /, volume fraction of the -phase. - viscosity of the -phase, Ns/m2. - density of the -phase, kg/m3. - surface tension, N/m. - (v +v T ), viscous stress tensor for the -phase, N/m2.  相似文献   

8.
Expressions are obtained for the pressure distribution in an externally pressurised thrust bearing for the condition when one bearing surface is rotated. The influence of centripetal acceleration and the combined effect of rotational and radial inertia terms are included in the analysis. Rotation of the bearing causes the lubricant to have a velocity component in an axial direction towards the rotating surface as it spirals radially outwards between the bearing surfaces. This results in an increase in the pumping losses and a decrease in the load capacity of the bearing. A further loss in the performance of the bearing is found when the radial inertia term, in addition to the rotational inertia term is included in the analysis.Nomenclature r, z, cylindrical co-ordinates - V r, V , V z velocity components in the r, and z directions respectively - U, X, W representative velocities - coefficient of viscosity - p static pressure at radius r - p mean static pressure at radius r - Q volume flow per unit time - 2h lubricant film thickness - density of the lubricant - r 2 outside radius of bearing = D/2 - angular velocity of bearing - R dimensionless radius = r/h - P dimensionless pressure = h 3 p/Q - Re channel Reynolds number = Q/h  相似文献   

9.
The detailed analysis of the dynamical process of coin tossing is made. Through calculations, it is illustrated how and why the result is extremely sensitive to the initial conditions. It is also shown that, as the initial height of the mass center of the coin increases, the final configuration, i.e. head or tail, becomes more and more sensitive to the initial parameters (the initial velocity angular velocity, and the initial orientation), the coefficient of the air drag, and the energy absorption factor of the surface on which the coin bounces. If we keep the head upward initially but allow a small range for the change of some other initial parameters, the frequency that the final configuration is head, would be 1 if the initial height h of the mass center is sufficiently small, and would be clo to 1/2 if h is sufficiently large. An interesting question is how this frequency changes continuously from 1 to 1/2 as h increases. Detailed calculations show that such a transition is very similar to the transition from laminar to turbulent flows. A basic difference between the transition stage and the completely random stage is indicated: In the completely random stage, the deterministic process of the individual case is extremely sensitive to the initial conditions and the dynamical parameters, out the statistical properties of the ensemble are insensitive to the small changes of the initial conditions and the dynamical parameters. On the contrary, in the transition stage, both the deterministic process of the individual case and the statistical properties of the ensemble are sensitive to the initial conditions and the dynamical parameters. The mechanism for this feature of the transition stage is the existence of the long-train structure in the parameter space. The illuminations of this analysis on some other random phenomena are discussed.  相似文献   

10.
The paper proposes a heuristic approach to constructing exact solutions of the hydrodynamic equations based on the specificity of these equations. A number of systems of hydrodynamic equations possess the following structure: they contain a reduced system of n equations and an additional equation for an extra function w. In this case, the reduced system, in which w = 0, admits a Lie group G. Taking a certain partially invariant solution of the reduced system with respect to this group as a seed:rdquo; solution, we can find a solution of the entire system, in which the functional dependence of the invariant part of the seed solution on the invariants of the group G has the previous form. Implementation of the algorithm proposed is exemplified by constructing new exact solutions of the equations of rotationally symmetric motion of an ideal incompressible liquid and the equations of concentrational convection in a plane boundary layer and thermal convection in a rotating layer of a viscous liquid.  相似文献   

11.
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.
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.
  相似文献   

12.
An effective numerical procedure, based on the Galerkin method, for finding solutions of the stationary traveling wave type in the complete formulation is proposed for the case of viscous liquid films. Examples of a viscous film flowing freely down a vertical surface have been calculated. The calculations have been made for various values of the dimensionless surface tension , including =0. The method makes it possible to predict a number of bifurcations that occur as decreases. The existence of numerous families of stationary traveling waves when 1 was demonstrated in [6]. The present study shows that as 1 all but one of these families of wave solutions disappear. The shape of the periodic and solitary waves and the pressure distribution in the film are found for various . When =0 and the wave number is fairly small, the periodic solution has a singularity, as predicted in [14]: at the crest of the wave a corner point appears; the angle between the tangents at this point =140–150. The method proposed can be used to calculate other wavy film flows.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 94–100, May–June, 1990.  相似文献   

13.
Barbera  Elvira  Müller  Ingo  Sugiyama  Masaru 《Meccanica》1999,34(2):103-113
This paper addresses the problem of the proper definition of temperature of a gas in nonequilibrium. It shows that the mean kinetic energy of the atoms of a rarefied gas is not a good measure for thethermodynamic temperature, because in general it jumps at a wall, and because it is nonmonotone in a onedimensional process of stationary heat conduction. The jump of the kinetic temperature is calculated and found to be about 5K in a rarefied gas. The basis for the calculations is provided by the arguments of extended thermodynamics of 14 moments. An essential tool is the minimax principle of entropy production recently postulated by Struchtrup Weiss [1], because it furnishes one important boundary condition.Sommario. Il lavoro riguarda la corretta definizione della temperatura di un gas in condizioni di nonequilibrio. Si mostra come lenergia cinetica media degli atomi di un gas rarefatto non sia una buona misura della temperatura termodinamica poiché in generale, essa risulta discontinua su una parete e nonmonotona in un processo unidimensionale di conduzione stazionaria del calore. Viene calcolato il salto della temperatura cinetica che risulta pari a circa 5K in un gas rarefatto. La base per il calcolo è fornita dal contesto della termodinamica estesa di 14 momenti. Uno strumento essenziale è rappresentato dal principio di minimax di produzione di entropia recentemente postulato da Struchtrup and Weiss [1], che fornisce unimportante condizione alcontorno.  相似文献   

14.
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.  相似文献   

15.
A method is proposed for calculating hypersonic ideal-gas flow past blunt-edged delta wings with aspect ratios = 100–200. Systematic wing flow calculations are carried out on the intervals 6 M 20, 0 20, 60 80; the results are analyzed in terms of hypersonic similarity parameters.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 175–179, September–October, 1990.  相似文献   

16.
The drag coefficient for bubbles with mobile or immobile interface rising in shear-thinning elastic fluids described by an Ellis or a Carreau model is discussed. Approximate solutions based on linearization of the equations of motion are presented for the highly elastic region of flow. These solutions are in reasonably good agreement with the theoretical predictions based on variational principles and with published experimental data. C D Drag coefficient - E * Differential operator [E * 2 = 2/2 + (sin/ 2)/(1/sin /)] - El Ellis number - F D Drag force - K Consistency index in the power-law model for non-Newtonian fluid - n Flow behaviour index in the Carreau and power-law models - P Dimensionless pressure [=(p – p 0)/0 (U /R)] - p Pressure - R Bubble radius - Re 0 Reynolds number [= 2R U /0] - Re Reynolds number defined for the power-law fluid [= (2R) n U 2–n /K] - r Spherical coordinate - t Time - U Terminal velocity of a bubble - u Velocity - Wi Weissenberg number - Ellis model parameter - Rate of deformation - Apparent viscosity - 0 Zero shear rate viscosity - Infinite shear rate viscosity - Spherical coordinate - Parameter in the Carreau model - * Dimensionless time [=/(U /R)] - Dimensionless length [=r/R] - Second invariant of rate of deformation tensors - * Dimensionless second invariant of rate of deformation tensors [=/(U /R)2] - Second invariant of stress tensors - * Dimensionless second invariant of second invariant of stress tensor [= / 0 2 (U /R)2] - Fluid density - Shear stress - * Dimensionless shear stress [=/ 0 (U /R)] - 1/2 Ellis model parameter - 1 2/* Dimensionless Ellis model parameter [= 1/2/ 0(U /R)] - Stream function - * Dimensionless stream function [=/U R 2]  相似文献   

17.
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. , , . , . . .  相似文献   

18.
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  相似文献   

19.
This paper presents a theoretical and numerical investigation of the natural convection boundary-layer along a vertical surface, which is embedded in a porous medium, when the surface heat flux varies as (1 +x 2)), where is a constant andx is the distance along the surface. It is shown that for > -1/2 the solution develops from a similarity solution which is valid for small values ofx to one which is valid for large values ofx. However, when -1/2 no similarity solutions exist for large values ofx and it is found that there are two cases to consider, namely < -1/2 and = -1/2. The wall temperature and the velocity at large distances along the plate are determined for a range of values of .Notation g Gravitational acceleration - k Thermal conductivity of the saturated porous medium - K Permeability of the porous medium - l Typical streamwise length - q w Uniform heat flux on the wall - Ra Rayleigh number, =gK(q w /k)l/(v) - T Temperature - Too Temperature far from the plate - u, v Components of seepage velocity in the x and y directions - x, y Cartesian coordinates - Thermal diffusivity of the fluid saturated porous medium - The coefficient of thermal expansion - An undetermined constant - Porosity of the porous medium - Similarity variable, =y(1+x ) /3/x 1/3 - A preassigned constant - Kinematic viscosity - Nondimensional temperature, =(T – T )Ra1/3 k/qw - Similarity variable, = =y(loge x)1/3/x 2/3 - Similarity variable, =y/x 2/3 - Stream function  相似文献   

20.
Suddenly started laminar flow in the entrance region of a circular tube, with constant inlet velocity, is investigated analytically by using integral momentum approach. A closed form solution to the integral momentum equation is obtained by the method of characteristics to determine boundary layer thickness, entrance length, velocity profile, and pressure gradient.Nomenclature M(, , ) a function - N(, , ) a function - p pressure - p* p/1/2U 2, dimensionless pressure - Q(, , ) a function - R radius of the tube - r radial distance - Re 2RU/, Reynolds number - t time - U inlet velocity, constant for all time, uniform over the cross section - u velocity in the boundary layer - u* u/U, dimensionless velocity - u 1 velocity in the inviscid core - x axial distance - y distance perpendicular to the axis of the tube - y* y/R, dimensionless distance perpendicular to the axis - boundary layer thickness - * displacement thickness - /R, dimensionless boundary layer thickness - momentum thickness - absolute viscosity of the fluid - /, kinematic viscosity of the fluid - x/(R Re), dimensionless axial distance - density of the fluid - tU/(R Re), dimensionless time - w wall shear stress  相似文献   

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