Thermodynamically admissible form for discrete hydrodynamics

We construct a discrete model of fluid particles according to the GENERIC formalism. The model has the form of Smoothed Particle Hydrodynamics including correct thermal fluctuations. A slight variation of the model reproduces the Dissipative Particle Dynamics model with any desired thermodynamic behavior. The resulting algorithm has the following properties: mass, momentum and energy are conserved, entropy is a non-decreasing function of time and the thermal fluctuations produce the correct Einstein distribution function at equilibrium.Comment: 4 page

Foundations of Dissipative Particle Dynamics

We derive a mesoscopic modeling and simulation technique that is very close to the technique known as dissipative particle dynamics. The model is derived from molecular dynamics by means of a systematic coarse-graining procedure. Thus the rules governing our new form of dissipative particle dynamics reflect the underlying molecular dynamics; in particular all the underlying conservation laws carry over from the microscopic to the mesoscopic descriptions. Whereas previously the dissipative particles were spheres of fixed size and mass, now they are defined as cells on a Voronoi lattice with variable masses and sizes. This Voronoi lattice arises naturally from the coarse-graining procedure which may be applied iteratively and thus represents a form of renormalisation-group mapping. It enables us to select any desired local scale for the mesoscopic description of a given problem. Indeed, the method may be used to deal with situations in which several different length scales are simultaneously present. Simulations carried out with the present scheme show good agreement with theoretical predictions for the equilibrium behavior.Comment: 18 pages, 7 figure

On the microscopic foundation of dissipative particle dynamics

Mesoscopic particle based fluid models, such as dissipative particle dynamics, are usually assumed to be coarse-grained representations of an underlying microscopic fluid. A fundamental question is whether there exists a map from microscopic particles in these systems to the corresponding coarse-grained particles, such that the coarse-grained system has the same bulk and transport properties as the underlying system. In this letter, we investigate the coarse-graining of microscopic fluids using a Voronoi type projection that has been suggested in several studies. The simulations show that the projection fails in defining coarse-grained particles that have a physically meaningful connection to the microscopic fluid. In particular, the Voronoi projection produces identical coarse-grained equilibrium properties when applied to systems with different microscopic interactions and different bulk properties.Comment: First revisio

Asymptotic behaviour of Sobolev-type orthogonal polynomials on a rectifiable Jordan arc

22 pages, no figures.-- MSC2000 codes: Primary 42C05.MR#: MR1890494 (2002m:42023)Zbl#: Zbl 0991.42018Our object of study is the asymptotic behavior of the sequence of polynomials orthogonal with respect to the discrete Sobolev inner product $$\langle f, g \rangle = \int_{E} f(\xi) \overline{g(\xi)} \rho (\xi) \xi \xi f(Z) A g(Z)^H,$$ where $E$ is a rectifiable Jordan curve or arc in the complex plane $$f(Z) = (f(z_1), \ldots, f^{(l_1)}(z_1) , \ldots , f(z_m) , \ldots ,f^{(l_m)}(z_m)),$$ $A$ is an $M \times M$ Hermitian matrix, $M=l_{1} + \cdots + l_{m} + m$, $denotes the arc length measure,$\rho$is a nonnegative function on$E$, and$z_{i} \in \Omega$,$i=1,2,\ldots,m$, where$\Omega$is the exterior region to$E$.The work of the first author was supported by the Portuguese Ministry of Science and Technology, Fundação para a Ciência e Tecnología of Portugal under grant FMRH-BSAB-109-99 and by the Centro de Matemática da Universidade de Coimbra. The second author would also like to thank the Unidade de Investigação (Matemática e Aplicações) of the University of Aveiro for their support. The work of the second and third authors was supported by the Dirección General de Enseñanza Superior (DGES) of Spain under grant PB 96-0120-C03-01.Publicad

A scalar Riemann boundary value problem approach to orthogonal polynomials on the circle

8 pages, no figures.-- MSC2000 codes: 33C47; 42C05.MR#: MR2252097 (2007k:33010)Zbl#: Zbl 1130.42025A scalar Riemann boundary value problem defining orthogonal polynomials on the unit circle and the corresponding functions of the second kind is obtained. The Riemann problem is used for the asymptotic analysis of the polynomials orthogonal with respect to an analytical real-valued weight on the circle.The research was supported by INTAS Research Network NeCCA 03-51-6637. The first author was also supported by the Grants RFBR 05-01-00522, NSh-1551.2003.1 and by the program N1 DMS, RAS. The second authorwas supported by Ministerio de Ciencia y Tecnología under Grant number MTM2005-01320. The third author was supported by Ministerio de Ciencia y Tecnología under Grant number BFM2003-06335-C03-02.Publicad

Static and Dynamic Properties of Dissipative Particle Dynamics

The algorithm for the DPD fluid, the dynamics of which is conceptually a combination of molecular dynamics, Brownian dynamics and lattice gas automata, is designed for simulating rheological properties of complex fluids on hydrodynamic time scales. This paper calculates the equilibrium and transport properties (viscosity, self-diffusion) of the thermostated DPD fluid explicitly in terms of the system parameters. It is demonstrated that temperature gradients cannot exist, and that there is therefore no heat conductivity. Starting from the N-particle Fokker-Planck, or Kramers' equation, we prove an H-theorem for the free energy, obtain hydrodynamic equations, and derive a non-linear kinetic equation (the Fokker-Planck-Boltzmann equation) for the single particle distribution function. This kinetic equation is solved by the Chapman-Enskog method. The analytic results are compared with numerical simulations.Comment: 22 pages, LaTeX, 3 Postscript figure

Shear-thickening of a non-colloidal suspension with a viscoelastic matrix

In this work we study the rheology of a non-colloidal suspension of rigid spherical particles interacting with a viscoelastic matrix. Three-dimensional numerical simulations under shear flow are performed using the smoothed particle hydrodynamics method and compared with experimental data available in the literature using different constant- viscosity elastic Boger fluids. The rheological properties of the Boger matrices are matched in simulation under viscometric flow conditions. Suspension rheology under dilute to semi-concentrated conditions (i.e. up to solid volume fraction φ = 0.3) is explored. It is found that at small Deborah numbers (based on the macroscopic imposed shear rate), relative suspension viscosities ηr exhibit a plateau at every concentration investigated. By increasing the Deborah number De shear-thickening is observed which is related to the extensional-thickening of the underlying viscoelastic matrix. Under dilute conditions (φ = 0.05) numerical results for ηr agree quantitatively with experimental data both in the De- and φ-dependencies. Even under dilute conditions, simulations of full many-particle systems with no ’a priori’ specification of their spatial distribution need to be considered to recover precisely experimental values. By increasing the solid volume fraction towards φ = 0.3, despite the fact that the trend is well captured, the agreement remains qualitative with discrepancies arising in the absolute values of ηr obtained from simulations and experiments but also with large deviations existing among different experiments. With regard to the specific mechanism of elastic thickening, the microstructural analysis shows that elastic thickening correlates well with the averaged viscoelastic dissipation function θ_elast, requiring a scaling as θ_elasti ∼De^α with α > 2 to take place. Locally, despite the fact that regions of large polymer stretching (and viscoelastic dissipation) can occur everywhere in the domain, flow regions uniquely responsible of the elastic thickening are well correlated to areas with significant extensional component

Orthogonal polynomials with respect to a differential operator. Existence and uniqueness

15 pages, no figures.-- MSC1991 codes: Primary 42C05.MR#: MR1934900 (2003i:42033)Zbl#: Zbl 1029.42018A new type of orthogonal polynomial connected withlin ear differential operators, intimately related withS obolev orthogonal polynomials and Hermite-Padé polynomials, is introduced. We study the question of uniqueness of the sequence of orthogonal polynomials arising from this construction. As we show, this problem is related to the analytic properties of the fundamental system of solutions of the operator. The notion of T-system of Tchebyshev plays a key role in the analysis. Some examples of general classes of operators which produce a unique system of polynomials are given.The research of the first author (A.I.A.) was supported by grant 93-01278 of the Russian Basic Research Foundation. The research of the second (G.T.L.L.) and third (F.M.) authors was supported by Dirección General de Investigación (Ministerio de Ciencia y Tecnología) of Spain under grant BFM2000-0206-C04-01, and INTAS 2000-272.Publicad

The role of thermal fluctuations in the motion of a free body

The motion of a rigid body is described in Classical Mechanics with the venerable Euler's equations which are based on the assumption that the relative distances among the constituent particles are fixed in time. Real bodies, however, cannot satisfy this property, as a consequence of thermal fluctuations. We generalize Euler's equations for a free body in order to describe dissipative and thermal fluctuation effects in a thermodynamically consistent way. The origin of these effects is internal, i.e. not due to an external thermal bath. The stochastic differential equations governing the orientation and central moments of the body are derived from first principles through the theory of coarse-graining. Within this theory, Euler's equations emerge as the reversible part of the dynamics. For the irreversible part, we identify two distinct dissipative mechanisms; one associated with diffusion of the orientation, whose origin lies in the difference between the spin velocity and the angular velocity, and one associated with the damping of dilations, i.e. inelasticity. We show that a deformable body with zero angular momentum will explore uniformly, through thermal fluctuations, all possible orientations. When the body spins, the equations describe the evolution towards the alignment of the body's major principal axis with the angular momentum vector. In this alignment process, the body increases its temperature. We demonstrate that the origin of the alignment process is not inelasticity but rather orientational diffusion. The theory also predicts the equilibrium shape of a spinning body.Comment: 24 pages, 1 figure with Supplemental Materia