no. 2
Mathematical model of heat transfer through a conductive pipe
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021) no. 2, pp. 627-658

The standard engineer’s model for heat transfer between the fluid flowing through the pipe and the exterior medium neglects the effects of the pipe’s wall. The goal of this paper is to prove that they are not always negligible. Comparing the ratio between diffusivities of the fluid and the wall with the wall’s thickness, using rigorous asymptotic analysis, we find five different models for effective description of the heat exchange process.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1051/m2an/2021008
Classification : 35B40, 35Q79, 35J25
Keywords: Asymptotic analysis, two-scale convergence, convection-diffusion equation, mathematical modeling, heat transfer 
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     title = {Mathematical model of heat transfer through a conductive pipe},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {627--658},
     year = {2021},
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Ljulj, Matko; Marušić-Paloka, Eduard; Pažanin, Igor; Tambača, Josip. Mathematical model of heat transfer through a conductive pipe. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021) no. 2, pp. 627-658. doi: 10.1051/m2an/2021008

[1] N. S. Bakhvalov and G. Panasenko, Homogenisation: Averaging Processes in Periodic Media. Kluwer (1989). | MR | Zbl

[2] D. Caillerie, The effect of a thin inclusion of high rigidity in an elastic body. Math. Methods Appl. Sci. 2 (1980) 251–270. | MR | Zbl | DOI

[3] S. W. Churchill and H. Chu, Correlating equations for laminar and turbulent free convection from a vertical plate. Int. J. Heat Mass Trans. 18 (1975) 1323–1329. | DOI

[4] J. Droniou, Solving convection-diffusion equations with mixed, Neumann and Fourier boundary conditions and measures as data, by duality method. Adv. Differ. Equ. 5 (2000) 1341–1396. | MR | Zbl

[5] J. Fernandez-Seara, F. J. Uhia and J. A. Dopazo, Experimental transient natural convection heat transfer from a vertical cylindrical tank. Appl. Thermal Eng. 31 (2011) 1915–1922. | DOI

[6] F. Hecht, New development in FreeFem++. J. Numer. Math. 20 (2012) 251–266. | MR | Zbl | DOI

[7] J. P. Holman, Heat Transfer, 9th edition. Mc Graw Hill (2002).

[8] M. Ljulj and J. Tambača, 3D structure–2D plate interaction problem. Math. Mech. Solids 24 (2019) 3354–3377. | MR | Zbl | DOI

[9] S. Marušić and E. Marušić-Paloka, Two-scale convergence for thin domains and its applications to some lower-dimensional models in fluid mechanics. Asymptotic Anal. 23 (2000) 23–57. | MR | Zbl

[10] S. Marušić and E. Marušić-Paloka, Reduction of dimension for parabolic equations via two-scale convergence. In: Proceedings of ApplMat 99, Dubrovnik, 1999, Department of Mathematics, University of Zagreb (2001) 155–164. | MR | Zbl

[11] S. Marušić, E. Marušić-Paloka and I. Pažanin, Effects of strong convection on the cooling process for a long or thin pipe. C.R. Mec. 336 (2008) 493–499. | Zbl | DOI

[12] E. Marušić-Paloka, Mathematical modeling of junctions in fluid mechanics via two-scale convergence. J. Math. Anal. App. 480 (2019) 123399. | MR | Zbl | DOI

[13] E. Marušić-Paloka and I. Pažanin, Non-isothermal fluid flow through a thin pipe with cooling. Appl. Anal. 88 (2009) 495–515. | MR | Zbl | DOI

[14] E. Marušić-Paloka and I. Pažanin, Modelling of heat transfer in a laminar flow through a helical pipe. Math. Comput. Model. 50 (2009) 1571–1582. | MR | Zbl | DOI

[15] E. Marušić-Paloka and I. Pažanin, On the effects of curved geometry on heat conduction through a distorted pipe. Nonlinear Anal.: Real World App. 11 (2010) 4554–4564. | MR | Zbl | DOI

[16] E. Marušić-Paloka, I. Pažanin and M. Prša, Heat conduction problem in a dilated pipe: existence and uniqueness result. Mediterranean J. Math. 14 (2017) 97. | MR | Zbl | DOI

[17] E. Marušić-Paloka, I. Pažanin and M. Prša, Asymptotic analysis of the heat conduction problem in a dilated pipe. Appl. Math. Comput. 355 (2019) 135–150. | MR | Zbl

[18] J. Nečas, Les méthodes directes en théorie des équations elliptiques. Masson, Paris (1967). | MR | Zbl

[19] R. Nittka, Regularity of solutions of linear second order elliptic and parabolic boundary value problems on Lipschitz domains. J. Differ. Equ. 251 (2011) 860–880. | MR | Zbl | DOI

[20] G. Panasenko, Multiscale Modelling of Thin Structures and Composites. Springer (2005). | MR

[21] G. Stampacchia, Equations elliptiques du second ordre à coefficients discontinus. Séminaire Jean Leray 3 (1963–1964) 1–77. | Numdam

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This paper is devoted to the memory of our friend Andro Mikelić.