The Graetz problem in tubes of arbitrary cross section

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Author list: Barrera C, Letelier M, Siginer D, Stockle J

Publisher: Springer

Place: WIEN

Publication year: 2016

Journal: Acta Mechanica (0001-5970)

Journal acronym: ACTA MECH

Volume number: 227

Issue number: 11

Start page: 3239

End page: 3246

Number of pages: 8

ISSN: 0001-5970

eISSN: 1619-6937

Languages: English-Great Britain (EN-GB)

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Abstract

The Graetz problem is investigated analytically for the steady laminar flow of Newtonian fluids in tubes of arbitrary cross section. The one-to-one mapping introduced satisfies the no-slip condition and allows the determination of the velocity field in a wide spectrum of arbitrary tube contours. The energy equation is solved for the temperature field in the corresponding tube contours, and the temperature distribution for the triangular, square and circular cross-sectional tubes is presented as particular case. Furthermore, in order to illustrate its relevance for a moderate P,clet number (Pe) regime, the solution applied to the square cross section is compared to numerical simulations for two scenarios, Case I with Pe = 100 and Case II with Pe = 500. It is found that for Case I the relative error does not exceed 2.9 %, being maximum at the center of the tube, while for Case II both analytical and numerical solutions match rather precisely, with less than 1 % of difference near the edges.

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