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The Generalized Graetz Problem in Finite Domains

Jérôme Fehrenbach, Frédéric de Gournay, Charles Pierre, Franck Plouraboué
2012 SIAM Journal on Applied Mathematics  
We consider the generalized Graetz problem associated with stationary convectiondiffusion inside a domain having any regular three-dimensional translationally invariant section and finite or semi-infinite  ...  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.  ...  The aim of the present contribution is to provide the mathematical analysis and numerical methods for solving the generalized Graetz problem in semi-infinite and finite domains, as well as effective numerical  ... 
doi:10.1137/11082542x fatcat:iuwkosnczna7xh3zzzysdlcmwm

Numerical computation of 3D heat transfer in complex parallel heat exchangers using generalized Graetz modes

Charles Pierre, Julien Bouyssier, Frédéric de Gournay, Franck Plouraboué
2014 Journal of Computational Physics  
It is based on a spectral method that allows to re-cast three-dimensional heat exchangers into a two-dimensional eigenvalue problem, named the generalized Graetz problem.  ...  Several examples of heat exchangers are analyzed, their numerical convergence is tested and the numerical efficiency of the approach is illustrated in the case of Poiseuille flow in tubes.  ...  The mathematical properties of the generalized Graetz modes have been studied in [21] [22] [23] . Generalized Graetz modes have been first used to solve problem (1) on infinite domains in [21] .  ... 
doi:10.1016/j.jcp.2014.02.037 fatcat:74ifvskp6jab3feihzdtdy5uki

Numerical modeling of heat and mass transfer in cylindrical ducts

Piotr Duda, Grzegorz Mazurkiewicz
2010 Archives of Thermodynamics  
The problems under consideration are often denoted as extended Graetz problems.  ...  Calculations were carried out gradually decreasing the mesh size in order to examine the convergence of numerical method to analytical solution.  ...  analytically extended Graetz problem. 2 Cylindrical duct ANSYS ICEM CFD software was used to generate the geometry of a cylindrical duct and finite volume grid.  ... 
doi:10.2478/v10173-010-0002-0 fatcat:jh3rt7j7zfbwtkbsondz4sqe2u

Conjugated Graetz problems—I

Eleftherios Papoutsakis, Doraiswami Ramkrishna
1981 Chemical Engineering Science  
These solid-fluid problems may be viewed as extensions of the extended, one-phase, Graetz problem with prescribed wall flux [2] .  ...  The aforementioned operator is shown to be symmetric in its domain, possessing a denumerable set of eigenvalues and a complete set of eigenvectors.  ...  Acknowledgemen&-The computer facilities for this work were generously provided by Purdue University.  ... 
doi:10.1016/0009-2509(81)80172-8 fatcat:wo26a53ekvex5mrta4qmj67ci4

Numerical Solution of Extended Graetz Problem by Orthogonal Collocation and Crank-Nicholson Method

Belhocine Ali, Wan Zaidi Wan Omar
2015 American Journal of Heat and Mass Transfer  
A numerical method was developed in this work, for visualization of the temperature profile, whose strategy of calculation is based on the orthogonal collocation method followed by the finite difference  ...  The present set of themes related to the investigations of heat transfer by convection and the transport phenomenon in a cylindrical pipe in laminar flow, which is to explore the evolution of the temperature  ...  Braga et al. (2014) considered an extended version of a Graetz problem with axial diffusion in an infinite domain.  ... 
doi:10.7726/ajhmt.2015.1011 fatcat:m6mv7sbhbvdzbp3q6nfrwpmihi

Page 789 of Mathematical Reviews Vol. , Issue 84b [page]

1984 Mathematical Reviews  
Authors’ summary: “A finite domain D, is enveloped by a finite domain D,. The domain D,, in turn, is in contact with two thermal baths with time-dependent periodic temperatures 7,(¢) and 7,(1).  ...  In the two articles cited in the heading of the first paper under review, a “complicated Hilbert space approach” was used to solve the extended Graetz problem.  ... 

Numerical study of heat convective mass transfer in a fully developed laminar flow with constant wall temperature

A. Belhocine, W.Z. Wan Omar
2015 Case Studies in Thermal Engineering  
boundary condition" is commonly called the Graetz Problem that our goal is to get the steady temperature distribution in the fluid.  ...  The simplified energy equation is solved numerically by the orthogonal collocation method followed by the finite difference method (Crank-Nicholson method).The calculations were effected through a FORTRAN  ...  [2] considered an extended version of a Graetz problem with axial diffusion in an infinite domain.  ... 
doi:10.1016/j.csite.2015.08.003 fatcat:mmd5uo2z5featij7kz3mfkz5x4

An unconditionally stable finite-difference method for the solution of multi-dimensional transport equation

Hany Saad, Hamada G. Asker
2020 Ain Shams Engineering Journal  
The visibility of using this method for the solution of the Multi-dimensional Transport equations is demonstrated through the solution of each of the Burgers' equation and the Graetz-Nusselt problem for  ...  The first-order derivative of the Transport equation is represented by a one-sided multi-level finite-difference. The resulting scheme is an explicit marching type iterative solution.  ...  not exist in the solution in Graetz problem where it has not a rapid change.  ... 
doi:10.1016/j.asej.2020.05.008 fatcat:rsd35qrvrzg4vl2ahtbjipkltq

Analytical properties of Graetz modes in parallel and concentric configurations [article]

Charles Pierre
2020 arXiv   pre-print
The generalized Graetz problem refers to stationary convection-diffusion in uni-directional flows.  ...  In this contribution we demonstrate the ana-lyticity of generalized Graetz solutions associated with layered domains: either cylindrical (possibly concentric) or parallel.  ...  effective algorithmic framework for the computation of generalized Graetz mode decomposition in cylindrical or parallel configurations.  ... 
arXiv:1812.05351v2 fatcat:uxkixlet7fdwpl3nmsxuseb7ey

Shape optimization for the generalized Graetz problem

Frédéric de Gournay, Jérôme Fehrenbach, Franck Plouraboué
2014 Structural And Multidisciplinary Optimization  
The problem boils down to the optimization of generalized eigenvalues on a two phases domain.  ...  This is an author-deposited version published in : http://oatao.univ-toulouse.fr/ Eprints ID : 10965 Abstract We apply shape optimization tools to the generalized Graetz problem which is a convection-diffusion  ...  The mathematical analysis and numerical methods designed to solve the generalized Graetz problem in semi-infinite and finite domains are presented in Fehrenbach et al. (2012) .  ... 
doi:10.1007/s00158-013-1032-4 fatcat:6pxk2rtnlzg2bc4g2dl7syuauq

A characteristic/finite element algorithm for time-dependent 3-D advection-dominated transport using unstructured grids

M.R. Kaazempur-Mofrad, P.D. Minev, C.R. Ethier
2003 Computer Methods in Applied Mechanics and Engineering  
The practical utility and effectiveness of the proposed numerical scheme was further demonstrated by solving the Graetz-Nusselt problem, i.e. high Peclet number mass/heat transport in a fully developed  ...  The algorithm showed the expected error behaviour and good performance in modeling advection-dominated transport problems.  ...  Fig. 10 . 10 A schematic of the mapping used to create the mesh in test case 3 (Graetz-Nusselt problem). Solid lines represent the box in which the structured mesh was generated.  ... 
doi:10.1016/s0045-7825(02)00627-8 fatcat:kpxmpfp235d2npa5w77ypncbhi

The Case of Neumann, Robin, and Periodic Lateral Conditions for the Semi-infinite Generalized Graetz Problem and Applications

Valentin Debarnot, Jérôme Fehrenbach, Frédéric de Gournay, Léo Martire
2018 SIAM Journal on Applied Mathematics  
This analysis is the cornerstone of numerical methods to solve Graetz problem 11 in finite pipes, semi infinite pipes and exchangers of arbitrary cross section.  ...  The authors of [15] coined this problem as the "generalized Graetz problem". 39 Numerical methods for the approximation of this operator and error estimates where 40 provided in [4]. 41 The objective of  ...  Resolution of the problem in a finite domain.  ... 
doi:10.1137/17m1157507 fatcat:hxkigpks7jdnhhuuilw6zfblpm

An assessment of the accuracy of numerical solutions to the graetz problem

Nancy Conley, Adeniyi Lawal, Arun S. Mujumdar
1985 International Communications in Heat and Mass Transfer  
It has been well documented in prior studies that numerical solutions to the classical Graetz problem in the entrance region of pipes produce estimates of Nusselt number that are greatly in error.  ...  In this report, we examine this problem and attempt to provide plausible explanation for the discrepancy. 209 210 N. Conley, A. Lawal and A.S. Mujumdar Vol. 12, No. 2 * -6 X M with X =I0 .  ...  Acknowledgements The authors are indebted to Professor W. Thorpe for providing extensive computer time on the Amdahl V7A of the McGill Computing Centre. Nomenclature  ... 
doi:10.1016/0735-1933(85)90069-7 fatcat:4yk4bmf3hreotjodxg6xoyehwq

Exact Analytical Solution of Boundary Value Problem in a Form of an Infinite Hypergeometric Series

Ali Belhocine
2017 International Journal of Mathematical Sciences and Computing  
This paper proposes an exact solution of the classical Graetz problem in terms of an infinite series represented by a nonlinear partial differential equation considering two space variables, two boundary  ...  The mathematical derivation is based on the method of separation of variables whose several stages were illustrated to reach the solution of the Graetz problem.  ...  [5] presented an exact solution for a Graetz problem with axial diffusion and flow heating effects in a semi-infinite domain with a given inlet condition.  ... 
doi:10.5815/ijmsc.2017.01.03 fatcat:qv7x5vxrkrdsrksuct2u3cnpci

Exact Graetz problem solution by using hypergeometric function

Ali Belhocine, Wan Omar
2017 International Journal of Heat and Technology  
This paper proposes an exact solution of the classical Graetz problem in terms of an infinite series represented by a nonlinear partial differential equation considering two space variables, two boundary  ...  In addition, the analytical solution was compared to the numerical values obtained previously by Shah and London.  ...  [6] presented an exact solution for a Graetz problem with axial diffusion and flow heating effects in a semi-infinite domain with a given inlet condition.  ... 
doi:10.18280/ijht.350216 fatcat:5lg5xtrdjjfr3a6sohstfwm7xu
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