Creep in macromolecular solutions according to rigid dumbbell kinetic theory

The creep experiment is analyzed using the rigid-dumbbell suspension model. It is found that the equilibrium shear compliance J _e is given by $$J_e = \frac{{\theta _0 }}{{2\eta _0^2 }} + O(\kappa _\infty ^2 )$$ where η₀ and θ₀ are the viscosity and primary normal stress functions at zero-shear rate, and κ _∞ is the velocity gradient for large time. It is found that, to the lowest level of approximation, τ _yy ?τ _zz and τ _xx ?τ _yy have the same sign during the creep experiment.     

Diffusion of a velocity discontinuity according to kinetic theory

Summary The problem of describing the smoothing out and diffusion of an initial velocity discontinuity is solved analytically by means of the linearized Bhatnager, Gross and Krook model of the Boltzmann equation. The solution is compared with previous approximate solutions and with the solution arising from a continuum treatment based on Navier-Stokes equations.

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Sommario Si risolve analiticamente per mezzo del modello linearizzato di Gross, Bhatnagar e Krook dell'equazione di Boltzmann, il problema dell'evoluzione temporale di una discontinuità iniziale di velocità.La soluzione che così si ottiene viene confrontata con precedenti soluzioni approssimate e con la soluzione ottenuta dalle equazioni di Navier-Stokes che descrivono macroscopicamente il fluido.

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Research developed in the frame of the activity of the CNR group for Fluid Dynamics at the Istituto di Scienze Fisiche dell'Università di Milano, Grant N. 02/100/5/2001.     

The langevin approach to the dumbbell kinetic problem in rheology

An alternative formalism to the dumbbell kinetic problem is proposed which is believed to be more fundamental than the classical Liouville one. The new formulation provides a logical approach to non-conservative systems and systems with varying frictional coefficients. A non-linear dumbbell with internal viscosity and varying frictional factor for the beads is examined. It is proved that the centre of gravity of the dumbbell must move affinely with the solvent continuum. A useful class of approximation is suggested to reduce the constitutive equation to an explicit form. The response of the model is computed for a number of flow fields. For shear flows, the introduction of the internal viscosity results in a shear-thinning phenomenon. The onset of non-Newtonian behaviour occurs at the correct order of magnitude of the dimensionless shear rate. Also, a negative second normal stress difference is found which varies with shear rate. In an oscillatory shear flow, the internal viscosity leads to a finite limiting value of the dynamic viscosity at high frequencies. In elongational flow the effects of the varying frictional coefficient dominate that of the internal viscosity. Interesting phenomena include the presence of a hysteresis loop and the ability of the dumbbell to maintain an extended configuration at moderate elongational rates. Clearly, these are relevant in turbulent drag reduction applications. The model has sufficient merits to deserve more investigation.     

Equilibration in momentum space in the kinetic theory of polymer solutions

In the kinetic theory of dilute polymer solutions it is customary to assume a Maxwellian velocity distribution for the “beads” in the bead-spring models. For the sake of simplicity this velocity distribution function is generally taken to be Maxwellian about the fluid velocity at the center of mass of the macromolecule. In Brownian dynamics simulations it is more natural to assume that the velocity distribution is Maxwellian about the fluid velocity at the “beads”. It is found that the differences between the two approaches are important only at very high velocity gradients. This could be a confusing point in high-shear-rate Brownian-dynamics simulations. The extra terms arising in the constitutive equation because of bead-centered Brownian motion are non-objective.     

An extended FENE dumbbell model theory for concentration dependent shear-induced anisotropy in dilute polymer solutions: addenda

Schneggenburger et al. [C. Schneggenburger, M. Kröger, S. Hess, An extended FENE dumbbell theory for concentration dependent shear-induced anisotropy in dilute polymer solutions, J. Non-Newtonian Fluid Mech. 62 (1996) 235] extended the original FENE dumbbell kinetic theory to describe concentration dependent shear-induced anisotropy in dilute polymer solutions by a mean-field approach. Besides providing an erratum to the above-mentioned paper and two revised figures we present related analytic results for steady shear and uniaxial elongational flow. Within the same framework we further consider a modified FENE potential and briefly discuss its implications.     

A possible alternative to the FENE dumbbell model of dilute polymer solutions

The dumbbell model of dilute polymer solutions is simple and successful, and its FENE version has progressively become a paradigm. In some transient extensional flows however, an increased dissipation was observed which could not be understood within the FENE model. This prompted us to look for a new dumbbell-like model in which the finite extensibility of the polymer is taken more thoroughly into account, i.e. not only through the Warner potential. The main lines of this alternative model are presented.     

Diffusion of macromolecular solutions in the boundary layer

Summary This paper concerns the study of a diffusion layer of macromolecular solutions annularly injected in a cylindric pipe in turbulent flow. The study includes firstly the analysis of the longitudinal and transversal projections of the concentration gradient, secondly the study of the state of turbulence in the diffusion layer.The concentrations are determined by the measurement of the concentration salt (salt was added to the injected solution). A mathematical model of the development of concentration is proposed.The turbulent state of diffusion layer has been studied with Laser-Doppler-Anemometry. An analysis of Eulerian autocorrelation and the power spectral density has been done.

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Zusammenfassung Es werden Ergebnisse über die turbulente Diffusionsschicht von nicht-newtonschen Flüssigkeiten (PÄO) mitgeteilt, die an der Wand eines kreiszylindrischen Rohrs injiziert werden. Diese Untersuchungen betreffen die Analyse der Längs- und Querkomponenten des Konzentrationsgradienten in der Diffusionsschicht und im Turbulenzfeld.Die Konzentrationen werden mit Hilfe von Salz (NaCl) gemessen, das der injizierten Polymerlösung beigefügt worden ist. Ein mathematisches Modell für die Konzentrationsverteilung wird vorgeschlagen.Der Turbulenzzustand der Diffusionsschicht wird mit Hilfe der Laser-Doppler-Anemometrie untersucht. Die Eulersche Autokorrelation und das Intensitätsspektrum werden analysiert.

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Nomenclature c polymer concentration (wppm) - c ₀ wall value ofc - c _i value ofc at the slot - D diameter of the pipe - e slot thickness - L ₀ distance from the slot at which the wall concentration drops toe ^–1of its original value - m distance in meter from the slot - U means freestream velocity - u _i injection velocity - x distance from the slot - y normal distance from the surface - boundary layer thickness - characteristic height of the diffusion, i. e. the value ofy at whichc/c ₀ = 0.5 With 4 figures     

The charged dumbbell model for dilute polyelectrolyte solutions in strong flows

A charged dumbbell model is used to investigate the behavior of dilute polyelectrolyte solutions in a general linear two-dimensional flow. The model studied has a nonlinear spring, conformation dependent friction and a Coulombic repulsive force due to an effective electrostatic charge on the two beads. The relative importance of the electrostatic charge is reflected by an effective charge density parameter,E. Equilibrium properties such as end-to-end distance and intrinsic viscosity are strongly dependent onE. In strong flows, which produce a dramatic increase in the dumbbell dimensions (a coil-stretch transition), the onset behavior is influenced byE. IncreasingE causes the onset velocity gradient to shift to much lower values. Large values ofE change the qualitative behavior to that of rigid (or slightly extensible) macromolecules or fibers. Results are presented for a charged dumbbell at equilibrium, in steady flows, and in transient flows.     

A study of conformation-dependent friction in a dumbbell model for dilute solutions

We consider predictions of rheological behavior in a variety of shear and extensional flows for an elastic dumbbell model with a nonlinear spring, and conformation-dependent hydrodynamic properties. The latter include a conformation-dependent anisotropic bead friction coefficient, and a related conformation-dependent degree of inefficiency for rotation in straining flows. With these features, the dumbbell exhibits hydrodynamic behavior consistent with a particle of finite axis ratio over the complete set of possible polymer conformations, from random-coil to a fully extended thread-like configuration. The predicted rheological behavior in shear flow is improved, relative to data, by the inclusion of anisotropy and strain-inefficiency in the frictional properties of the model, while other desirable features such as the sudden onset of fully extended states at a critical value of the velocity gradient, the presence of a hysteresis-loop in end-to-end dimension as a function of the velocity gradient, and the correlation of end-to-end distance (or birefringence) with the eigenvalue of the velocity gradient tensor for a wide variety of two-dimensional flows, are maintained.     

Propagation of infinitesimal disturbances in a gas according to a relativistic kinetic model

Summary In this paper a kinetic model describing a relativistic gas is considered. The propagation of infinitesimal disturbances is studied. Given the frequency of the disturbance, the complex wave number is shown to exhibit a continuous spectrum. A detailed analysis, however, indicates the possibility of isolating, at least in an asymptotic sense, a sort of discrete spectrum imbedded in the continuum. The spectrum obtained by such asymptotic analysis is in agreement with the corresponding results for the Boltzmann equation.

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Sommario In questo lavoro viene considerato un modello cinetico per descrivere un gas relativistico. Viene studiata in particolare la propagazione delle onde di piccola ampiezza e si mostra che, data la frequenza del disturbo, il numero d'onda possiede uno spettro continuo. Un'analisi dettagliata mostra però la possibilità di isolare, almeno in senso asintotico, una specie di spettro discreto immerso nel precedente. Questo spettro ricavato dall'analisi asintotica è in accordo coi corrispondenti risultati noti per l'equazione di Boltzmann.

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Partially supported by G.N.F.M. of C.N.R. and M.P.I. contract number (40%) 20120201/81 - (A) - 218142055.     

Remarks on relativistic kinetic theory to first order in the gradients

In this paper we emphasize some conceptual points related to the kinetic foundations of relativistic hydrodynamics. We summarize previous work and focus on the construction of the heat flux from a kinetic theory point of view. A thorough discussion addressing aspects concerning stability and causality under linear perturbations, and the construction of an appropriate stress-energy tensor, in the case of a high temperature (special relativistic) single component dilute fluid is included.     

New solutions of classical problems in rigid body dynamics

The equations of motion of a rigid body acted upon by general conservative potential and gyroscopic forces were reduced by Yehia to a single second-order differential equation. The reduced equation was used successfully in the study of stability of certain simple motions of the body. In the present work we use the reduced equation to construct a new particular solution of the dynamics of a rigid body about a fixed point in the approximate field of a far Newtonian centre of attraction. Using a transformation to a rotating frame we also construct a new solution of the problem of motion of a multiconnected rigid body in an ideal incompressible fluid. It turns out that the solutions obtained generalize a known solution of the simplest problem of motion of a heavy rigid body about a fixed point due to Dokshevich.     

Stability in kinetic theory of vehicular traffic

Summary We prove existence and uniqueness of the solution of a linearized version of the kinetic equation of vehicular traffic on a highway by using the theory of semigroups of linear bounded operators.We assume that a perturbation is introduced at the entrance of the highway, i. e., at x=0 and that it continues for a given time interval [0, ].We then study the behaviour of such a perturvation at values of x close to zero by using perturbation theory for linear operators.Finally, we prove that the perturbation attenuates and approaches zero as x+, i. e. at a large distance from the entrance of the highway.

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Sommario Si prova l'esistenza e l'unicità della soluzione dell'equazione linearizzata del traffico automobilistico su un'autostrada con un'entrata in corrispondenza di x=0.Si studia poi il comportamento di una perturbazione, dovuta per esempio all'uscita da una partita di calcio, e si mostra come tale perturbazione si attenui al crescere della distanza dall'ingresso dell'autostrada.

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Work performed under the auspices of the National Research Council (C.N.R., Gruppo Nazionale per la Fisica Matematica).     

A study of the interacting FENE dumbbell model for semi-dilute polymer solutions in extensional flows

The effect of polymer concentration on the conformation of semidilute polymer solutions in extensional flows is studied via the interacting elastic dumbbell model proposed by Hess (1984), here modified to include a nonlinear Warner spring (FENE dumbbell) instead of the linear Hookean spring of the original model. The length of flow-induced conformation changes for the polymer is predicted to be a decreasing function of concentration. In particular, increasing concentration tends to inhibit large extension of the polymer due to polymer-polymer interaction. The specific birefringence is thus proportional to c ^–1 for semi-dilute solutions, in contrast to dilute solutions where it is known to be independent of concentration. However, the correlation between birefringence and the principle eigenvalue of the velocity gradient tensor, also found originally for dilute solutions, is predicted to occur in the semi-dilute regime. All of these predictions agree qualitatively with experimental observations.Some recent exceptions to the neglect of segmental stretch can be found in Marrucci and Grizzuti (1988), Pearson et al. (1991), Mead et al. (1992).     

Approximate kinetic equations in rarefied gas theory

The fundamental kinetic equation of gas theory, the Boltzniann equation, is a complex integrodiffcrential equation. The difficulties associated with its solution are the result not only of the large number of independent variables, seven in the general case, but also of the very complicated structure of the collision integral. However, for the mechanics of rarefied gases the primary interest lies not in the distribution function itself, which satisfies the Boltzmann equation, but rather in its first few moments, i.e., the averaged characteristics. This circumstance suggests the possibility of obtaining the averaged quantities by a simpler way than the direct method of direct solution of the Boltzmann equation with subsequent calculation of the integrals.It is well known that if a distribution function satisfies the Boltzmann equation, then its moments satisfy an infinite system of moment equations. Consequently, if we wish to obtain with satisfactory accuracy some number of first moments, then we must require that these moments satisfy the exact system of moment equations. However, this does not mean that to determine the moments of interest to us we must solve this system, particularly since the system of moment equations is not closed. The closure of the system by specifying the form of the distribution function (method of moments) can be considered only as a rough approximate method of solving problems. First, in this case it is not possible to satisfy all the equations and we must limit ourselves to certain of the equations; second, generally speaking, we do not know which equation the selected distribution function satisfies, and, consequently, we do not know to what degree it has the properties of the distribution function which satisfies the Boltzmann equation.A more reliable technique for solving the problems of rarefied gasdynamics is that based on the approximation of the Boltzmann equation, more precisely, the approximation of the collision integral. The idea of replacing the collision integral by a simpler expression is not new [1–4]. The kinetic equations obtained as a result of this replacement are usually termed model equations, since their derivation is usually based on physical arguments and not on the direct use of the properties of the Boltzmann collision integral. In this connection we do not know to what degree the solutions of the Boltzmann equation and the model equations are close, particularly since the latter do not yield the possibility of refining the solution. Exceptions are the kinetic model for the linearized Boltzmann equation [5] and the sequence of model equations of [6], constructed by a method which is to some degree analogous with that of [5].In the present paper we suggest for the simplification of the solution of rarefied gas mechanics problems a technique for constructing a sequence of approximate kinetic equations which is based on an approximation of the collision integral. For each approximate equation (i.e., equation with an approximate collision operator) the first few moment equations coincide with the exact moment equations. It is assumed that the accuracy of the approximate equation increases with increase of the number of exact moment equations. Concretely, the approximation for the collision integral consists of a suitable approximation of the reverse collision integral and the collision frequency. The reverse collision integral is represented in the form of the product of the collision frequency and a function which characterizes the molecular velocity distribution resulting from the collisions, where the latter is selected in the form of a locally Maxwellian function multiplied by a polynomial in terms of the components of the molecular proper velocities. The collision frequency is approximated by a suitable expression which depends on the problem conditions. For the majority of problems it may obviously be taken equal to the collision frequency calculated from the locally Maxwellian distribution function; if necessary the error resulting from the inexact calculation of the collision frequency may be reduced by iterations.To illustrate the method, we solve the simplest problem of rarefied gas theory-the problem on the relaxation of an initially homogeneous and isotropic distribution in an unbounded space to an equilibrium distribution.The author wishes to thank A. A. Nikol'skii for discussions of the study and V. A. Rykov for the numerical results presented for the exact solution.     

On kinetic theory methods in vehicular flow

Similarly to the treatment of diluted gases, kinetic methods are formulated for the study of unidirectional freeway traffic. From these it is possible to construct fluid-dynamic equations which in comparison with heuristic fluid-dynamic models have the advantage of just some adjustable parameters although they have some other restrictions. In this work the comparison between two macroscopic models which are based on a kinetic traffic equation is shown. On the other hand, there will be presented some advances to attempt to generalize some restrictions of the kinetic formulation in order to study the synchronization phenomena, which is a very interesting transitory phase between free flow and traffic jams.     

Behavior of beams under transverse impact according to higher-order beam theory

Using Reissner’s principle, we formulated an equation of motion for a beam according to higher-order beam theory. We derived the Laplace transform of the equation and investigated wave-propagation behavior under transverse impact. In other words, we studied the effect of the nonlinear component of axial-warping, which cannot be determined by a conventional approach such as the Timoshenko beam theory. Specifically, we derived the transfer matrices for finite and semi-infinite beams. By choosing the appropriate state quantities, arrangement as a vector, and definition of sign convention, we were able to derive a perfect “reciprocal relation.” In spite of the complicated Laplace inverse transform, we obtained an accurate and rapid solution by investigating appropriate branch points and poles and setting branch cuts. The extent of the nonlinear warping effect and its region of influence were also clearly demonstrated.     

Simulation method in the kinetic theory of gases

We develop a method that is based on use of the simulation equation vf/v=vl ^–1(f ₀–f), where v is the modulus of the molecular velocity andl is the mean free-path length. A number of general properties of the model are clarified and the transition to the limit of a continuous medium is discussed in detail.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 110–115, November–December, 1973.     

A linear theory for Ekman's boundary layer according to a triple deck

The problem of the interaction between Ekman's classical boundary layer and that induced by a thermally non-homogeneous site shows that the latter is strongly linked to the order of magnitude of the horizontal scale of the site. Our purpose is the analysis of the local interaction equations (Boussinesq equations) starting from a triple-deck model. This analysis yields a system of quasi-linear equations for the viscous lower deck. The linear theory of this system shows that the thermal non-homogeneity has a significant influence on the Ekman boundary layer flow owing to the interactive nature of the triple-deck structure. The numerical solution of the quasi-linear system confirms to a large extent this influence. The numerical results are given in graph form. © 1997 John Wiley & Sons, Ltd.