652 research outputs found
The Generalized Graetz Problem in Finite Domains
No full textWe consider the generalized Graetz problem associated with stationary convection-diffusion inside a domain having any regular three-dimensional translationally invariant section and finite or semi-infinite extent. Our framework encompasses any previous “extended” and “conjugated” Graetz generalizations and provides theoretical bases for computing the orthogonal set of generalized two-dimensional Graetz modes. The theoretical framework includes both heterogeneous and possibly anisotropic diffusion tensors. In the case of semi-infinite domains, the existence of a bounded solution is shown from the analysis of two-dimensional operator eigenvectors which form a basis of L2 . In the case of finite domains a similar basis can be exhibited, and the mode’s amplitudes can be obtained from the inversion of newly defined finite domain operator. Our analysis includes both the theoretical and practical issues associated with this finite domain operator inversion as well as its interpretation as a multireflection image method. Error estimates are provided when numerically truncating the spectrum to a finite number of modes. Numerical examples are validated for reference configurations and provided in nontrivial cases. Our methodology shows how to map the solution of stationary convection-diffusion problems in finite three-dimensional domains into a two-dimensional operator spectrum, which leads to a drastic reduction in computational cost
Mathematical analysis of parallel convective exchangers with general lateral boundary conditions using generalized graetz modes
No full textWe propose a mathematical analysis of parallel convective exchangers for any general but longitudinally invariant domains. We analyze general Dirichlet or Neumann prescribed boundary conditions at the outer solid domain. Our study provides general mathematical expressions for the solution of convection/diffusion problems. Explicit form of generalized solutions along longitudinal coordinate are found from convoluting elementary base Graetz mode with the applied sources at the boundary. In the case of adiabatic zero flux counter-current configuration, we recover the longitudinally linearly varying solution associated with the zeroth eigenmode which can be considered as the fully developed behavior for heat-exchangers. We also provide general expression for the infinite asymptotic behavior of the solutions which depends on simple parameters such as total convective flux, outer domain perimeter and the applied boundary conditions. Practical considerations associated with the numerical precision of truncated mode decomposition is also analyzed in various configurations for illustrating the versatility of the formalism. Numerical quantities of interest are investigated, such as fluid/solid internal and external fluxes
Un point de repère dans l’histoire du roi David
No full textGraetz H. Un point de repère dans l’histoire du roi David. In: Revue des études juives, tome 21, n°42, octobre-décembre 1890. pp. 241-245
Numerical Analysis of a New Mixed Formulation for Eigenvalue Convection-Diffusion Problems
No full textA mixed formulation is proposed and analyzed mathematically for coupled convection-diffusion in heterogeneous medias. Transfer in solid parts driven by pure diffusion is coupled with convection-diffusion transfer in fluid parts. This study is carried out for translation-invariant geometries (general infinite cylinders) and unidirectional flows. This formulation brings to the fore a new convection-diffusion operator, the properties of which are mathematically studied: its symmetry is first shown using a suitable scalar product. It is proved to be self-adjoint with compact resolvent on a simple Hilbert space. Its spectrum is characterized as being composed of a double set of eigenvalues: one converging towards −∞ and the other towards +∞, thus resulting in a nonsectorial operator. The decomposition of the convection-diffusion problem into a generalized eigenvalue problem permits the reduction of the original three-dimensional problem into a two-dimensional one. Despite the operator being nonsectorial, a complete solution on the infinite cylinder, associated to a step change of the wall temperature at the origin, is exhibited with the help of the operator’s two sets of eigenvalues/eigenfunctions. On the computational point of view, a mixed variational formulation is naturally associated to the eigenvalue problem. Numerical illustrations are provided for axisymmetrical situations, the convergence of which is found to be consistent with the numerical discretization
Une initiative saint-simonienne pour l'émancipation des Juifs. Lettres de Gustave d'Eichthal sur son voyage en Autriche
No full textGraetz Michael. Une initiative saint-simonienne pour l'émancipation des Juifs. Lettres de Gustave d'Eichthal sur son voyage en Autriche. In: Revue des études juives, tome 129, n°1, janvier-mars 1970. pp. 67-84
Additions et rectifications à l'Histoire des Juifs, de Graetz
No full textHarkavy Albert. Additions et rectifications à l'Histoire des Juifs, de Graetz. In: Revue des études juives, tome 5, n°10, octobre-décembre 1882. pp. 199-218
Graetz (Michael) L'Alliance israélite universelle et l'héritage de la Révolution française
No full textGraetz (Michael) L'Alliance israélite universelle et l'héritage de la Révolution française. In: Archives de sciences sociales des religions, n°68/2, 1989. p. 184
Graetz (Michael) L'Alliance israélite universelle et l'héritage de la Révolution française
No full textGraetz (Michael) L'Alliance israélite universelle et l'héritage de la Révolution française. In: Archives de sciences sociales des religions, n°68/2, 1989. p. 184
Geschichte der Juden, νοn Dr Η. Graetz ; achter Band ; dritte verbesserte und stark vermehrte Auflage, 189
No full textGeschichte der Juden, νοn Dr Η. Graetz ; achter Band ; dritte verbesserte und stark vermehrte Auflage, 189. In: Revue des études juives, tome 21, n°41, juillet-septembre 1890. pp. 146-160
Michaël Graetz, Les Juifs en France au XIXe siècle : de la Révolution française à l'Alliance Israélite universelle
No full textGreen Nancy L. Michaël Graetz, Les Juifs en France au XIXe siècle : de la Révolution française à l'Alliance Israélite universelle. In: Annales. Économies, Sociétés, Civilisations. 45ᵉ année, N. 3, 1990. pp. 678-679
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