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The Generalized Graetz Problem in Finite Domains

2012
*
SIAM Journal on Applied Mathematics
*

We consider

doi:10.1137/11082542x
fatcat:iuwkosnczna7xh3zzzysdlcmwm
*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 ...##
###
Numerical computation of 3D heat transfer in complex parallel heat exchangers using generalized Graetz modes

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

doi:10.1016/j.jcp.2014.02.037
fatcat:74ifvskp6jab3feihzdtdy5uki
*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] . ...##
###
Numerical modeling of heat and mass transfer in cylindrical ducts

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. ...

##
###
Conjugated Graetz problems—I

1981
*
Chemical Engineering Science
*

These solid-fluid

doi:10.1016/0009-2509(81)80172-8
fatcat:wo26a53ekvex5mrta4qmj67ci4
*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. ...##
###
Numerical Solution of Extended Graetz Problem by Orthogonal Collocation and Crank-Nicholson Method

2015
*
American Journal of Heat and Mass Transfer
*

A numerical method was developed

doi:10.7726/ajhmt.2015.1011
fatcat:m6mv7sbhbvdzbp3q6nfrwpmihi
*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*. ...##
###
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

2015
*
Case Studies in Thermal Engineering
*

boundary condition" is commonly called

doi:10.1016/j.csite.2015.08.003
fatcat:mmd5uo2z5featij7kz3mfkz5x4
*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*. ...##
###
An unconditionally stable finite-difference method for the solution of multi-dimensional transport equation

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. ...

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

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. ...

##
###
Shape optimization for the generalized Graetz problem

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) . ...

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

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*. ...

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

2018
*
SIAM Journal on Applied Mathematics
*

This analysis is

doi:10.1137/17m1157507
fatcat:hxkigpks7jdnhhuuilw6zfblpm
*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*. ...##
###
An assessment of the accuracy of numerical solutions to the graetz problem

1985
*
International Communications in Heat and Mass Transfer
*

It has been well documented

doi:10.1016/0735-1933(85)90069-7
fatcat:4yk4bmf3hreotjodxg6xoyehwq
*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 ...##
###
Exact Analytical Solution of Boundary Value Problem in a Form of an Infinite Hypergeometric Series

2017
*
International Journal of Mathematical Sciences and Computing
*

This paper proposes an exact solution of

doi:10.5815/ijmsc.2017.01.03
fatcat:qv7x5vxrkrdsrksuct2u3cnpci
*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. ...##
###
Exact Graetz problem solution by using hypergeometric function

2017
*
International Journal of Heat and Technology
*

This paper proposes an exact solution of

doi:10.18280/ijht.350216
fatcat:5lg5xtrdjjfr3a6sohstfwm7xu
*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. ...
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