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.     

Supersonic air flow past a sphere with account for vibrational relaxation

A numerical solution is obtained for the problem of air flow past a sphere under conditions when nonequilibrium excitation of the vibrational degrees of freedom of the molecular components takes place in the shock layer. The problem is solved using the method of [1]. In calculating the relaxation rates account was taken of two processes: 1) transition of the molecular translational energy into vibrational energy during collision; 2) exchange of vibrational energy between the air components. Expressions for the relaxation rates were computed in [2]. The solution indicates that in the state far from equilibrium a relaxation layer is formed near the sphere surface. A comparison is made of the calculated values of the shock standoff with the experimental data of [3].Notation uV_max, vV_max velocity components normal and tangential to the sphere surface - V_max maximal velocity - P V _max ² pressure - density - TT temperature - e_viRT vibrational energy of the i-th component per mole (i=–O₂, N₂) - =rb–1 shock wave shape - a _f the frozen speed of sound - HRT/m gas total enthalpy     

Propagation of a spherical wave in nonlinearly compressible and elastoplastic materials

The problem of spherical wave propagation in soil under the action of an intense uniformly decreasing load ₀(t) applied to the boundary of a cavity with radius r₀ is considered. Soil with a high stress level is modeled either by ideally nonlinearly compressible or elastoplastic material, taking account of linear irreversible unloading for the material. In contrast to [1–7], in order to describe material movement use is made of strain theory [8] with determining functions = (), _i=_i(_i), where , _i, , _i are the first and second invariants of strain and stress tensors. During material loading these functions are presented in the form of polynomials ()=(_i+₂¦¦), _ii)=(_i-_2i)_i, in which constant coefficients _i, _i=1, 2) are determined by experiment, taking account of the triaxial stressed state of soil. Solution of the problem is constructed by an analytically reversible method, with prescribed shape for the shock-wave (SW) surface in the form of a second-degree polynomial relating to time t and a numerical method of characteristics for a prescribed arbitrarily decreasing load _i(t). On the basis of the analytical equations obtained, calculations are carried out for material parameters (including loading profile) in a computer and stresses and mass velocity of plastic and elastoplastic materials are compared.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 95–100, July–August, 1986.The authors express their sincere thanks to Kh. A. Rakhmatulin for discussing the results of this work.     

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     

Some remarks on differential equations of quadratic type

In this paper we study differential equations of the formx(t) + x(t)=f(x(t)), x(0)=x ₀ C HereC is a closed, bounded convex subset of a Banach spaceX,f(C) C, and it is often assumed thatf(x) is a quadratic map. We study the differential equation by using the general theory of nonexpansive maps and nonexpansive, non-linear semigroups, and we obtain sharp results in a number of cases of interest. We give a formula for the Lipschitz constant off: C C, and we derive a precise explicit formula for the Lipschitz constant whenf is quadratic,C is the unit simplex inR ⁿ, and thel ₁ norm is used. We give a new proof of a theorem about nonexpansive semigroups; and we show that if the Lipschitz constant off: CC is less than or equal to one, then lim_tf(x(t))–x(t)=0 and, if {x(t):t 0} is precompact, then lim_tx(t) exists. Iff¦C=L¦C, whereL is a bounded linear operator, we apply the nonlinear theory to prove that (under mild further conditions on C) lim_t f(x(t))–x(t)=0 and that lim_t x(t) exists if {x(t):t 0} is precompact. However, forn 3 we give examples of quadratic mapsf of the unit simplex ofR ⁿ into itself such that lim_t x(t) fails to exist for mostx ₀ C andx(t) may be periodic. Our theorems answer several questions recently raised by J. Herod in connection with so-called model Boltzmann equations.     

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      

Representation of the rheological properties of polymer melts in terms of complex fluidity

In dynamic rheological experiments melt behavior is usually expressed in terms of complex viscosity ^* () or complex modulusG ^* (). In contrast, we attempted to use the complex fluidity ^* () = 1/µ ^* () to represent this behavior. The main interest is to simplify the complex-plane diagram and to simplify the determination of fundamental parameters such as the Newtonian viscosity or the parameter of relaxation-time distribution when a Cole-Cole type distribution can be applied. ^* () complex shear viscosity - () real part of the complex viscosity - () imaginary part of the complex viscosity - G ^* () complex shear modulus - G() storage modulus in shear - G() loss modulus in shear - J ^* () complex shear compliance - J() storage compliance in shear - J() loss compliance in shear - shear strain - rate of strain - angular frequency (rad/s) - shear stress - loss angle - ^* () complex shear fluidity - () real part of the complex fluidity - () imaginary part of the complex fluidity - ₀ zero-viscosity - ₀ average relaxation time - h parameter of relaxation-time distribution     

Global Behavior of a Nonlinear Quasiperiodic Mathieu Equation

In this paper, we investigate the interaction of subharmonicresonances in the nonlinear quasiperiodic Mathieu equation,x + [ + (cos ₁ t + cos ₂ t)] x + x³ = 0.We assume that 1 and that the coefficient of the nonlinearterm, , is positive but not necessarily small.We utilize Lie transform perturbation theory with elliptic functions –rather than the usual trigonometric functions – to study subharmonic resonances associated with orbits in 2m:1 resonance with a respective driver. In particular, we derive analytic expressions that place conditions on (, , ₁, ₂) at which subharmonic resonance bands in a Poincaré section of action space begin to overlap. These results are used in combination with Chirikov's overlap criterion to obtain an overview of the O() global behavior of equation (1) as a function of and ₂ with ₁, , and fixed.     

Deformationsbeständige,elastische und Relaxations-Eigenschaften der flüssigplastischen kolloiden Systeme

Zusammenfassung Es wurde gezeigt, daß es zweckmäßig ist, die strukturierten kolloiden Systeme in fest-plastische und flüssig-plastische Systeme einzuteilen, weil beide Systeme einen übereinstimmenden Eigenschaftskomplex aufweisen, sich jedoch quantitativ durch die RelaxationszeitenP >P _k im Schubspannungsbereich unterhalb der unteren FließgrenzeP < P _k unterscheiden. Es wurde darauf hingewiesen, daß für beide Systeme die Spannungsdeformationskurven sehr charakteristisch sind.Die Kurven werden unter der Bedingung der konstanten Deformationsgeschwindigkeit erhalten, wobei ihre Form von der gegenseitigen Beziehung von und abhängt und mit den Strukturelementetypen, die durch und _i charakterisiert sind, im Zusammenhang stehen.Die Methoden,die zur Messung der elastischen Deformation im breiten Bereich längs der KurveP () bei sowohl kleiner als auch größer _r entsprechend der kritischen SchubspannungP _r angewandt werden können, wurden entwickelt. Dabei wurde gezeigt, daß die Kurveine() durch das Maximum bei _m hindurchgeht.Der Einfluß von auf die kritische Deformation _r der Strukturzerstörung und auf die maximale Rückfederung _e max, die ihrerseits wiederum von der Gelkonzentration abhängen, wurde eingehend untersucht.Es wurden die Zahlenwerte der Grenzviskosität der Nachwirkung bestimmt und die Abhängigkeit der Geschwindigkeit (der Zeit) der Relaxation der elastischen Deformation von der gesamten und der elastischen Deformation ermittelt.Weiter wurde gezeigt, daß die größte elastische Deformation _e max des Systems größer als die kritische Deformation _r der Strukturzerstörung, die dem Maximum der kritischen Schubspannung der Struktur entspricht, sein kann.     

Modelling of heat transfer by conduction in a transition region between a porous medium and an external fluid

We study the modelling of purely conductive heat transfer between a porous medium and an external fluid within the framework of the volume averaging method. When the temperature field for such a system is classically determined by coupling the macroscopic heat conduction equation in the porous medium domain to the heat conduction equation in the external fluid domain, it is shown that the phase average temperature cannot be predicted without a generally negligible error due to the fact that the boundary conditions at the interface between the two media are specified at the macroscopic level.Afterwards, it is presented an alternative modelling by means of a single equation involving an effective thermal conductivity which is a function of point inside the interfacial region.The theoretical results are illustrated by means of some numerical simulations for a model porous medium. In particular, temperature fields at the microscopic level are presented.Roman Letters _sf interfacial area of thes-f interface contained within the macroscopic system m² - A _sf interfacial area of thes-f interface contained within the averaging volume m² - C _p mass fraction weighted heat capacity, kcal/kg/K - g vector that maps to _s , m - h vector that maps to _f , m - K _eff effective thermal conductivity tensor, kcal/m s K - l _s,l _f microscopic characteristic length m - L macroscopic characteristic length, m - n _fs outwardly directed unit normal vector for thef-phase at thef-s interface - n outwardly directed unit normal vector at the dividing surface. - R ₀ REV characteristic length, m - T _i macroscopic temperature at the interface, K - error on the external fluid temperature due to the macroscopic boundary condition, K - T ^* macroscopic temperature field obtained by solving the macroscopic Equation (3), K - V averaging volume, m³ - V _s,V _f volume of the considered phase within the averaging volume, m³. - _mp volume of the porous medium domain, m³ - _ex volume of the external fluid domain, m³ - _s , _f volume of the considered phase within the volume of the macroscopic system, m³ - dividing surface, m² - x, z spatial coordinates Greek Letters _s, _f volume fraction - ratio of the effective thermal conductivity to the external fluid thermal conductivity - ^* macroscopic thermal conductivity (single equation model) kcal/m s K - _s, _f microscopic thermal conductivities, kcal/m s K - spatial average density, kg/m³ - microscopic temperature, K - ^* microscopic temperature corresponding toT ^*, K - spatial deviation temperature K - error in the temperature due to the macroscopic boundary conditions, K - ^* _i macroscopic temperature at the interface given by the single equation model, K - spatial average - ^s , ^f intrinsic phase average.     

The extension of Beltrami's ellipsoid to anisotropic bodies

Summary The spectral decomposition of the compliance, stiffness, and failure tensors for transversely isotropic materials was studied and their characteristic values were calculated using the components of these fourth-rank tensors in a Cartesian frame defining the principal material directions. The spectrally decomposed compliance and stiffness or failure tensors for a transversely isotropic body (fiber-reinforced composite), and the eigenvalues derived from them define in a simple and efficient way the respective elastic eigenstates of the loading of the material. It has been shown that, for the general orthotropic or transversely isotropic body, these eigenstates consist of two double components, ₁ and ₂ which are shears (₂ being a simple shear and ₁, a superposition of simple and pure shears), and that they are associated with distortional components of energy. The remaining two eigenstates, with stress components ₃, and ₄, are the orthogonal supplements to the shear subspace of ₁ and ₂ and consist of an equilateral stress in the plane of isotropy, on which is superimposed a prescribed tension or compression along the symmetry axis of the material. The relationship between these superimposed loading modes is governed by another eigenquantity, the eigenangle .The spectral type of decomposition of the elastic stiffness or compliance tensors in elementary fourth-rank tensors thus serves as a means for the energy-orthogonal decomposition of the energy function. The advantage of this type of decomposition is that the elementary idempotent tensors to which the fourth-rank tensors are decomposed have the interesting property of defining energy-orthogonal stress states. That is, the stress-idempotent tensors are mutually orthogonal and at the same time collinear with their respective strain tensors, and therefore correspond to energy-orthogonal stress states, which are therefore independent of each other. Since the failure tensor is the limiting case for the respective _x, which are eigenstates of the compliance tensor S, this tensor also possesses the same remarkable property.An interesting geometric interpretation arises for the energy-orthogonal stress states if we consider the projections of _x in the principal₃D stress space. Then, the characteristic state ₂ vanishes, whereas stress states ₁, ₃ and ₄ are represented by three mutually orthogonal vectors, oriented as follows: The ₃ and ₄ lie on the principal diagonal plane (₃₁₂) with subtending angles equaling (–/2) and (-), respectively. On the positive principal ₃-axis, is the eigenangle of the orthotropic material, whereas the ₁-vector is normal to the (₃₁₂)-plane and lies on the deviatoric -plane. Vector ₂ is equal to zero.It was additionally conclusively proved that the four eigenvalues of the compliance, stiffness, and failure tensors for a transversely isotropic body, together with value of the eigenangle , constitute the five necessary and simplest parameters with which invariantly to describe either the elastic or the failure behavior of the body. The expressions for the _x-vector thus established represent an ellipsoid centered at the origin of the Cartesian frame, whose principal axes are the directions of the ₁-, ₃- and ₄-vectors. This ellipsoid is a generalization of the Beltrami ellipsoid for isotropic materials.Furthermore, in combination with extensive experimental evidence, this theory indicates that the eigenangle alone monoparametrically characterizes the degree of anisotropy for each transversely isotropic material. Thus, while the angle for isotropic materials is always equal to _i = 125.26° and constitutes a minimum, the angle || progressively increases within the interval 90–180° as the anisotropy of the material is increased. The anisotropy of the various materials, exemplified by their ratiosE _L/2G_L of the longitudinal elastic modulus to the double of the longitudinal shear modulus, increases rapidly tending asymptotically to very high values as the angle approaches its limits of 90 or 180°.     

Analysis of a method and a system to determine heat transfer and properties of fluids in the absence of temperature measurements and a modified Fourier Equation

The technique to determine by capacitance measurements heat transfer, thermal transport and dielectric properties of fluids introduced recently is now analyzed for a simple system of spherical geometry. The temperature distribution under programmed heat input to a fluid annulus between solid walls is computed by finite difference method for the determination of the capacitance time function of the arrangement. A system of heavy wall structure and heated long enough will produce a capacitance-time curve which is a function of thermal conductivity only. Thermal diffusivity is of influence in thin wall systems. The capacitance change of a heavy wall arrangement is related to the thermal conductivity of the test fluid by a modified Fourier equation. This equation describes the heat flow through the fluid layer but includes the thermal expansion of the solid walls. The change of geometry with T is therefore accounted for. For other multicomposite structures the Fourier equation must be further modified by including the thermal expansion of all materials of the structure and possibly also their compressibilities.

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Zusammenfassung Die kürzlich eingeführte Methode der Bestimmung von Wärmeübergang, thermischen Transport und dielektrischen Größen mittels Kapazitäts-Zeit-Messung wird analysiert für ein einfaches kugeliges System. Die Temperaturverteilung in der Flüssigkeit im Kugelspalt zwischen zwei festen Körpern wird für konstante Wärmezufuhr von außen mittels der Differenzmethode bestimmt und daraus die Kapazitäts-Zeit-Funktion ermittelt. Es wird gezeigt, daß die Kapazitäts-Zeit-Kurve nur eine Funktion der Wärmeleitzahl ist für den Fall dickwandiger Anordnungen. Für dünnwandige Systeme wird sie auch abhängig von der Temperaturleitzahl. Es wird eine modifizierte Fourier-Gleichung eingeführt, die den Wärmetransport durch die Flüssigkeit beschreibt, dabei aber die Änderung der Geometrie der Schicht berücksichtigt, die sich wegen der thermischen Ausdehnung der festen Wände bei der Einstellung der Temperaturdifferenz ergibt. Für andere mehrschichtige Körper muß die Fourier-Gleichung weiterhin modifiziert werden durch Berücksichtigung der thermischen Ausdehnungskoeffizienten aller beteiligten Materialien und möglicherweise auch ihrer Kompressibilitäten.

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Nomenclature A average cross-sectional area of fluid layer - A coefficient matrix - B matrix defined by Eq. (20) - B₀ geometric constant of fluid layer (A/L) at reference temperature - C capacitance of arrangement - C_i, C_r capacitance of layer of fluid i and reference fluid at temperature T - capacitances at reference temperature - C_H, c_l specific heats of outer and inner wall - FA...FE constants defined in Eqs. (13 ... 17) - L thickness of fluid layer - M_H, M_L mass of outer and inner wall - P power input to the system - R constant defined by Eq. (24) - T temperature - T_ref reference temperature - T (O, t), T (L, t) temperatures of outer and inner wall at time t - T _i ⁿ , T _i+0 ^n+m temperatures at location i and time n (m=number of t's; 0=number of x's) - T temperature difference across fluid layer - T apparent temperature difference - t_h, T_l temperature increases of outer and inner wall - T_max temperature change of system from one to another thermal equilibrium condition a thermal diffusivity - k, k_i, k_r thermal qonductivity of fluids and of fluid i and reference fluid - q heat flow through fluid layer - ^rh,^rl inner radius of outer wall and outer radius of inner wall - ^rOH,^rOL radii at reference temperature - t time - t time interval - x coordinate - ¯x vector of unknown T_i ⁿ⁺¹ - x length interval Greek symbols linear thermal expansion coefficient - _H, _L linear thermal expansion coefficient of materials of outer and inner wall - dielectric constant - _i, _ref dielectric constant of fluid i and reference fluid - ₀ permittivity of free space - multiplyer of conduction Eq. (7) in finite difference form - time needed to establish quasi-steady state conditions in the system heated by a constant power input In honor of Prof. Dr. E. Schmidt to his 80th Birthday.     

Scheme of Investigation of the Spectrum of a Family of Perturbed Operators and Its Application to Spectral Problems in Thick Junctions

We develop a scheme for the investigation of the asymptotic behavior of eigenvalues and eigenvectors of a family of self-adjoint compact operators {A: > 0} that act in different spaces and lose their compactness in the limit case 0. We prove the Hausdorff convergence of the spectrum of the operator A to the spectrum of the limit operator A₀, obtain asymptotic estimates for this convergence both to points of the discrete spectrum and to points of the essential spectrum of the operator A₀, and prove asymptotic estimates for eigenvectors of A. This scheme is applied to the investigation of the asymptotic behavior of eigenvalues and eigenfunctions of the Neumann problem in a thick singularly degenerate junction that consists of two domains connected by an -periodic system of thin rods of fixed length.     

Parameter determination for thermodynamical models of the pseudoelastic behaviour of shape memory alloy by resistivity measurements and infrared thermography

Two thermodynamical models of pseudoelastic behaviour of shape memory alloys have been formulated. The first corresponds to the ideal reversible case. The second takes into account the hysteresis loop characteristic of this shape memory alloys.Two totally independent techniques are used during a loading-unloading tensile test to determine the whole set of model parameters, namely resistivity and infrared thermography measurements. In the ideal case, there is no difficulty in identifying parameters.Infrared thermography measurements are well adapted for observing the phase transformation thermal effects.Notations 1 austenite 2 martensite - ₍₎ Macroscopic infinitesimal strain tensor of phase - ₍₂₎ ^f Traceless strain tensor associated with the formation of martensite phase - Macroscopic infiniesimal strain tensor - Macroscopic infinitesimal strain tensor deviator - _f Trace - Equivalent strain - ^pe Macroscopic pseudoelastic strain tensor - x Distortion due to parent (austenite =1)product (martensite =2) phase transformation (traceless symmetric second order tensor) - M Total mass of a system - M() Total mass of phase - V Total volume of a system - V() Total volume of phase - z=M(2)/M Weight fraction of martensite - 1-z=M(1)/M Weight fraction of austenite - u ₀ ^* () Specific internal energy of phase (=1,2) - s ₀ ^* () Specific internal entropy of phase - Specific configurational energy - Specific configurational entropy - ₀ ^f (T) Driving force for temperature-induced martensitic transformation at stress free state ( ₀ ^f T) = T ^*–Ts ^*) - Kirchhoff stress tensor - Kirchhoff stress tensor deviator - Equivalent stress - Cauchy stress tensor - Mass density - K Bulk moduli (K ₀=K) - L Elastic moduli tensor (order 4) - E Young modulus - Energetic shear (₀ = ) - Poisson coefficient - M _s ^o (M _F ^o ) Martensite start (finish) temperature at stress free state - A _s ^o (A _F ^o ) Austenite start (finish) temperature at stress free state - C _v Specific heat at constant volume - k Conductivity - Pseudoelastic strain obtained in tensile test after complete phase transformation (AM) (unidimensional test) - ₀ Thermal expansion tensor - r Resistivity - 1MPa 10⁶ N/m ² - ⁽⁾ Specific free energy of phase - _n Specific free energy at non equilibrium (R model) - _n ^eq Specific free energy at equilibrium (R model) - _n ^v Volumic part of ^eq - Specific free energy at non equilibrium (R _L model) - _conf Specific coherency energy (R _L model) - _c Specific free energy at constrained equilibria (R _L model) - _it (T) Coherency term (R _L model)     

The virtual origin of riblets in an open channel flow

In this paper, a method using the mean velocity profiles for the buffer layer was developed for the estimation of the virtual origin over a riblets surface in an open channel flow. First, the standardized profiles of the mixing length were estimated from the velocity measurement in the inner layer, and the location of the edge of the viscous layer was obtained. Then, the virtual origins were estimated by the best match between the measured velocity profile and the equations of the velocity profile derived from the mixing length profiles. It was made clear that the virtual origin and the thickness of the viscous layer are the function of the roughness Reynolds number. The drag variation coincided well with other results.Nomenclature f _r skin friction coefficient - f _ro skin friction coefficient in smooth channel at the same flow quantity and the same energy slope - g gravity acceleration - H water depth from virtual origin to water surface - H ⁺ u*H/ - H false water depth from top of riblets to water surface - H ⁺ u*H/ - I _e streamwise energy slope - I _b bed slope - k riblet height - k ⁺ u*k/ - l mixing length - l _s standardized mixing length - Q flow quantity - Re Reynolds number volume flow/unit width/v - s riblet spacing - u mean velocity - u* friction velocity = - u* false friction velocity = - y distance from virtual origin - y distance from top of riblet - y ₀ distance from top of riblet to virtual origin - y _v distance from top of riblet to edge of viscous layer - y ⁺ u*y/ - y ⁺ u*y/ - y ₀ ⁺ u*y ₀/ - u ⁺ u*y/ - shifting coefficient for standardization - thickness of viscous layer=y ₀+y - ⁺ u*/ - ⁺ u*/ - eddy viscosity - ridge angle - v kinematic viscosity - density - shear stress     

On Perturbative Expansions to the Stochastic Flow Problem

When analyzing stochastic steady flow, the hydraulic conductivity naturally appears logarithmically. Often the log conductivity is represented as the sum of an average plus a stochastic fluctuation. To make the problem tractable, the log conductivity fluctuation, f, about the mean log conductivity, lnK _G, is assumed to have finite variance, _f ². Historically, perturbation schemes have involved the assumption that _f ²<1. Here it is shown that _f may not be the most judicious choice of perturbation parameters for steady flow. Instead, we posit that the variance of the gradient of the conductivity fluctuation, _f ², is more appropriate hoice. By solving the problem withthis parameter and studying the solution, this conjecture can be refined and an even more appropriate perturbation parameter, , defined. Since the processes f and f can often be considered independent, further assumptions on f are necessary. In particular, when the two point correlation function for the conductivity is assumed to be exponential or Gaussian, it is possible to estimate the magnitude of f in terms of _f and various length scales. The ratio of the integral scale in the main direction of flow ( _x ) to the total domain length (L^*), _x ²=_x/L^*, plays an important role in the convergence of the perturbation scheme. For _x smaller than a critical value _c, _x < _c, the scheme's perturbation parameter is =_f/_x for one- dimensional flow, and =_f/_x ² for two-dimensional flow with mean flow in the x direction. For _x > _c, the parameter =_f/_x ³ may be thought as the perturbation parameter for two-dimensional flow. The shape of the log conductivity fluctuation two point correlation function, and boundary conditions influence the convergence of the perturbation scheme.     

A method for the analysis of recirculating turbulent flows in a cavity

The complex fluid-dynamic aspects of a turbulent recirculating flow in a cavity with axial throughflow, and a rotating wall, were investigated by adopting a simple procedure for evaluating the turbulent stresses. The flow field was divided into two regions, a core and a wall region respectively. A wall function was adopted in the zones near to the solid boundaries, while a constant eddy diffusivity was assumed, in the core, following the indications of computed heat transfer coefficients in comparison with existing experimental data. The distributions of the stream function and of the tangential velocity are presented for a range of the rotational Reynolds number of the rotating wall and of the Reynolds number of the throughflow.

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Turbulente Rezirkulationsströmung in einem Hohlraum

Zusammenfassung Die komplizierten fluiddynamischen Aspekte einer turbulenten Rezirkulationsströmung in einem Hohlraum mit axialem Durchfluß und einer rotierenden Wand werden unter Verwendung einer vereinfachten Methode zur Berechnung der turbulenten Spannungen betrachtet. Das Strömungsfeld wird in einen Kern und einen Wandbereich aufgeteilt. Für die wandnahen Zonen wird eine Wandfunktion angenommen, während im Kern mit konstanter Wirbeldiffusivität gerechnet wird, was durch den Vergleich berechneter mit gemessenen Wärmeübergangskoeffizienten gerechtfertigt erscheint. Verteilungen der Stromfunktion und der tangentialen Geschwindigkeit sind für einen bestimmten Bereich der Reynoldszahlen für die Wandrotation und der für den Durchfluß angegeben.

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Nomenclature L axial length of enclosure - P dimensionless pressure, p^*2 - p static pressure - R dimensionless radial coordinate, r/r^* - r radial coordinate - r^* reference length, equal to r_O for enclosure - r_i radii of inlet and exit apertures - Re Reynolds number, v^*r^*/ - Re_i pipe Reynolds number, ¯v_zi(2r_i)/ - Re_t turbulent Reynolds number, Re(/) - Re rotational Reynolds number, r ₀ ² / - t dimensionless time,t/(r^*/v^*) - t time - V_r, V, Vz dimensionless velocity components, V_r/v^*, v, v_z/v^* - v_i turbulent fluctuation of the i-component of velocity - v_r, v, v_z velocity components - v^* reference velocity, equal to ¯v_zi for enclosure - X coordinate along a wall, x/r^* - Y coordinate normal to a wall, y/r^* - Z dimensionless axial coordinate, z/r^* - z axial coordinate - eddy diffusivity for momentum - dynamic viscosity - kinematic viscosity - density - shear stress - dimensionless shear stress, /v^*2 - dimensionless stream function, /r*²v*² - stream function - angular velocity - tangential vorticity component - ()_eff effective - ()_l laminar - ()_t turbulent - mean over the time     

Existence and regularity for solutions of the Korteweg-de Vries equation

The construction suggested by an inverse-scattering analysis establishes the existence of solutions u(x, t) of the Korteweg-de Vries equation subject to an initial condition u(x, 0)=U(x), where U has certain regularity and decay properties. It is assumed that UC³(), that U is piecewise of class C ⁴, and that U ^(j) decays at an algebraic rate for j4. The faster the decay of U ^(j) the smoother the solution will be for t0. If U and its first four derivatives decay faster than ¦x¦^–n for all n, then the solution will be infinitely differentiable for t0. For t>0, the decay rate of u(x, t) as x + increases with the decay rate of U; but the decay rate as x - depends on the regularity of U. A solution u ₁ of the Korteweg-de Vries equation such that u ₁(·, 0)C() may fail to remain in class C for all time if u ₁(x, 0) does not decay fast enough as ¦x¦.This research was performed in part as a Visiting Member of the Courant Institute of Mathematical Science.     

On the Jacobian conjecture for global asymptotic stability

An old conjecture says that, for the two-dimensional system of ordinary differential equationsx=f(x), wheref: ^{2 2},f C ¹, andf(0)=0 the originx=0 should beglobally asymptotically stable (i.e., a stable equilibrium and all trajectoriesx(t) converge to it ast +) whenever the following conditions on the Jacobian matrixJ(x) off hold: trJ(x) < 0, detJ(x) > 0, x ² It is known that if such anf is globallyone-to-one as a mapping of the plane into itself, then the origin is a globally asymptotically stable equilibrium point for the systemx =f(x). In this paper we outline a new strategy to tackle the injectivity off, based on anauxiliary boundary value problem. The strategy is shown to be successful if the norm of the matrixJ(x) ^T J(x)t/det J(x) is bounded or, at least, grows slowly (for instance, linearly) as ¦x¦ t.