The stability of a differentially rotating thin disk
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On discute de l'etat d'equilibre d'un disque fin en rotation isotherme et en auto-gravitation. Pour cela on resout l'equation integrodifferentielle des perturbations dont on discute les solutions. On considere l'influence de l'exposant polytrope et l'effet de la sphere entourante sur la stabiliteKeywords:
Polytrope
Computational astrophysics
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In this note we discuss the influence of a large angular momentum on nonlinear axisymmetric motions of centrally condensed bodies. In order to illustrate the problem we consider finite-amplitude pulsations of rapidly rotating columns. An analysis involving the virial theorem gives an approximate description of adiabatic radial motions. It is shown that nonlinearity lessens the stabilizing influence of angular momentum predicted in a linear stability analysis.
Momentum (technical analysis)
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Des distributions fluides physiquement significatives, sous la forme d'un disque mince autour d'un objet compact en rotation, peuvent exister en equilibre ou la vitesse angulaire du fluide est due a l'entrainement induit par l'espace-temps entourant l'objet compact. L'equilibre de la pression au bord interieur est principalement du a l'egalite de la pression de radiation et de la pression hydrostatique du gaz
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La validite de l'approximation des 2 corps est etudiee dans le probleme collisionnel entre 2 particules kepleriennes en revolution dans le champ gravitationnel solaire au moyen d'une methode de perturbation basee sur l'equation de Hill. Apres une description breve de l'equation de Hill, la difference dans la plus proche distance entre les 2 particules est examinee dans le formalisme des 2 et 3 corps. La condition de validite de l'approximation des 2 corps est ensuite etablie comme fonction du degre de deviation par rapport au probleme des 3 corps, de la distance heliocentrique de la protoplanete et de l'excentricite du mouvement relatif des 2 particules
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We present models for evolving globular clusters with realistic stellar mass spectra, generated by direct numerical integration of the Fokker–Planck equation. The goal of this study is to compute surface brightness and projected velocity dispersion profiles that can be directly compared with observations. We use an evolved power-law mass function containing non-luminous objects of up to 1.2 Mʘ, red giants and horizontal branch stars of 0. 7 Mʘ, and main-sequence stars extending down to 0.1 Mʘ. The heavy non-luminous objects settle to the centre of the cluster and dominate the core within a half-mass relaxation time which corresponds to ~3 Gyr for our standard model resembling a Salpeter mass function. We follow the simulation until core collapse is well established at 10 Gyr. The surface density profile of the heavy non-luminous objects in the core at this time is |$\sigma\propto{r}^{-1.2}$|. The other less massive components have significantly flatter power-law surface density profiles in the core, with the power-law index depending on the individual stellar mass of each mass group. Since red giants and horizontal branch stars dominate the luminosity of the cluster, the resulting composite surface brightness profiles are substantially flatter than for single component models. This is consistent with the recent finding by Lugger et al. that the central power-law slopes of several candidate post-core-collapse clusters are flatter than the value of –1 expected for binary post-collapse expansion in a one-component cluster. Our models should place significant constraints on the stellar population in globular clusters, particularly the non-luminous remnant component.
Mass segregation
Initial mass function
Velocity dispersion
Stellar mass
Stellar density
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Les collisions decentrees entre une etoile de la sequence principale de 0.5 M ○. , modelisee par un polytrope d'indice 1.5, et une naine blanche de 0.5 M ○. , modelisee par un potentiel de masse ponctuelle, sont etudiees. La dynamique totale de la collision, les orbites, les energies globales et les temperatures maximum atteintes sont determinees pour differents parametres initiaux.
Polytrope
Star (game theory)
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Abstract An analytic expression has been deduced for the ultimate velocity of a free particle in a vertically oscillating fluid by evaluating asymptotic forms of Floquet's solution to the Mathieu equation arising from the nonlinear Langevin equation describing the particle‐fluid interactions. Using general Mathieu theory it has been rigorously demonstrated that directional particle motion in a gravitational field can be retarded by sinusoidal fluid oscillations when the particle‐fluid drag law is of square form. The earlier Rayleigh‐Ritz approximation (Houghton 1966) for the retarded particle velocity is shown to be in good agreement with the results of Mathieu theory, as summarized in graphical form.
Mathieu function
Particle (ecology)
Particle velocity
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Abstract A new model has been formulated and analyzed which accounts for the deformation of a drop in response to the pressure distribution in the draining film between it and a rigid plane. The rate of approach is related to the pressure distribution and drop shape. When the nonlinear partial differential equations are written in dimensionless form, one dimensionless group, β = Δρ gb /σ 2 , arises which represents the ratio of gravity forces to surface forces. Only β need be specified to solve these equations; viscosity enters only as a scaling factor on time. The equations were simplified to ordinary differential equations by assuming that the rate of change of shape was small relative to the change in position. Numerical soliitions were obtained from β = .15 to 10 −3 using an iterative Runge‐Kutta procedure at each separation.
Dimensionless quantity
Position (finance)
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This thesis deals with the dynamics of a thin liquid film falling down a heated plate. The heating yields surface tension gradients that induce thermocapillary stresses on the free surface, thus affecting the stability and the evolution of the film. Accounting for the coherence of the flow due to viscosity, two main approaches that reduce the dimensionality of the original problem are usually considered depending on the flow rate (as measured by the Reynolds number): the `long wave' asymptotic expansion for small Reynolds numbers and the `integral boundary layer' approximation for moderate Reynolds numbers. The former suffers from singularities and the latter from incorrectness of the instability threshold for the occurrence of hydrodynamic waves. Thus, the aim of this thesis is twofold: in a first part, we define quantitatively the validity of the `long wave' evolution equation (Benney equation) for the film thickness h including the thermocapillary effect; and in a second part, we improve the `integral boundary layer' approach by combining a gradient expansion to a weighted residual method. In the first part, we further investigate the Benney equation in its validity domain in the case of periodically inhomogeneous heating in the streamwise direction. It induces steady-state deformations of the free surface with increased transfer rate in regions where the film is thinner, and also in average. The inhomogeneities of the heating also modify the nature of travelling wave solutions at moderate temperature gradients and allows for suppressing wave motion at larger ones.Moreover, large temperature gradients (for instance positive ones) in the streamwise direction produce large local film thickening that may in turn become unstable with respect to transverse disturbances such that the flow may organize in rivulet-like structures. The mechanism of such instability is elucidated via an energy analysis. The main features of the rivulet pattern are described experimentally and recovered by direct numerical simulations.In the second part, various models are obtained, which are valid for larger Reynolds numbers than the Benney equation and account for second-order viscous and inertial effects. We then elaborate a strategy to select the optimal model in terms of linear stability properties and existence of nonlinear solutions (solitary waves), for the widest possible range of parameters. This model -- called reduced model -- is a system of three coupled evolution equations for the local film thickness h, the local flow rate q and the surface temperature Ts. Solutions of this model indicate that the interaction of the hydrodynamic and thermocapillary modes is non-trivial, especially in the region of large-amplitude solitary waves.Finally, the three-dimensional evolution of the solutions of the reduced model in the presence of periodic forcing and noise compares favourably with available experimental data in isothermal conditions and with direct numerical simulations in non-isothermal conditions.------------------------------------------------Cette these analyse la dynamique d'un film mince s'ecoulant le long d'une paroi chauffee. Le chauffage cree des gradients de tension superficielle qui induisent des tensions thermocapillaires a la surface libre, alterant ainsi la stabilite et l'evolution du film. Grâce a la coherence de l'ecoulement assuree par la viscosite, deux approches permettant de reduire la dimensionnalite du probleme original sont habituellement considerees suivant le debit (mesure par le nombre de Reynolds): l'approximation asymptotique dite `longues ondes' pour les faibles nombres de Reynolds et l'approximation `integrale couche limite' pour les nombres de Reynolds moderes. Cependant, la premiere approximation souffre de singularites et la derniere de predictions imprecises du seuil de stabilite des ondes hydrodynamiques a la surface du film. Le but de cette these est donc double: dans une premiere partie, il s'agit de determiner, de maniere quantitative, la validite de l'equation d'evolution `longues ondes' (ou equation de Benney) pour l'epaisseur du film h, en y incluant l'effet thermocapillaire; et dans une seconde partie, il s'agit d'ameliorer l'approche `integrale couche limite' en combinant un developpement en gradients avec une methode aux residus ponderes.Dans la premiere partie, nous etudions l'equation de Benney, dans son domaine de validite, dans le cas d'un chauffage inhomogene et periodique dans la direction de l'ecoulement. Cela induit des deformations permanentes de la surface libre avec un accroissement du transfert de chaleur dans les regions ou le film est plus mince, mais aussi en moyenne. Un chauffage inhomogene modifie egalement la nature des solutions d'ondes progressives pour des gradients de temperatures moderes et conduit meme a leur suppression pour des gradients de temperatures plus importants. De plus, ceux-ci, lorsqu'ils sont par exemple positifs le long de l'ecoulement, produisent des epaississements localises du film qui peuvent a leur tour devenir instables par rapport a des perturbations suivant la direction transverse a l'ecoulement. Ce dernier s'organise alors sous forme d'une structure en rivulets. Le mecanisme de cette instabilite est elucide via une analyse energetique des perturbations. Les principales caracteristiques des structures en rivulets sont decrites experimentalement et retrouvees par l'intermediaire de simulations numeriques. Dans la seconde partie, nous derivons une famille de modeles valables pour des nombres de Reynolds plus grands que l'equation de Benney, qui prennent en compte les effets visqueux et inertiels du second ordre. Nous elaborons ensuite une strategie pour selectionner le modele optimal en fonction de ses proprietes de stabilite lineaire et de l'existence de solutions non-lineaires (ondes solitaires), et ce pour la gamme de parametres la plus large possible. Ce modele -- appele modele reduit -- est un systeme de trois equations d'evolution couplees pour l'epaisseur locale de film h, le debit local q et la temperature de surface Ts. Les solutions de ce modele indiquent que l'interaction des modes hydrodynamiques et thermocapillaires n'est pas triviale, specialement dans le domaine des ondes solitaires de grande amplitude. Finalement, l'evolution tri-dimensionnelle des solutions du modele reduit en presence d'un forcage periodique ou d'un bruit se compare favorablement aux donnees experimentales disponibles en conditions isothermes, ainsi qu'aux simulations numeriques directes en conditions non-isothermes
Weber number
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En considerant que les vents de disque sont des gaz chauds qui emanent hydrodynamiquement de la surface d'un disque d'accretion dans le champ gravitationnel de l'objet central a travers le chauffage interne ou l'irradiation, ces vents sont analyses a partir d'une approximation simple dans laquelle le bilan des forces est decouple en composantes paralleles et perpendiculaires aux lignes de courant. Les proprietes globales a deux dimensions des vents de disque sont analysees.
Accretion disc
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