Continuum Mechanics and Thermodynamics

, Volume 30, Issue 2, pp 301–317

# Rigorous derivation of the effective model describing a non-isothermal fluid flow in a vertical pipe filled with porous medium

• Michal Beneš
• Igor Pažanin
Original Article

## Abstract

This paper reports an analytical investigation of non-isothermal fluid flow in a thin (or long) vertical pipe filled with porous medium via asymptotic analysis. We assume that the fluid inside the pipe is cooled (or heated) by the surrounding medium and that the flow is governed by the prescribed pressure drop between pipe’s ends. Starting from the dimensionless Darcy–Brinkman–Boussinesq system, we formally derive a macroscopic model describing the effective flow at small Brinkman–Darcy number. The asymptotic approximation is given by the explicit formulae for the velocity, pressure and temperature clearly acknowledging the effects of the cooling (heating) and porous structure. The theoretical error analysis is carried out to indicate the order of accuracy and to provide a rigorous justification of the effective model.

## Keywords

Darcy–Brinkman–Boussinesq model Thin pipe Newton cooling condition Asymptotic approximation Error estimates

## Notes

### Acknowledgements

The first author of this work has been supported by the project GAČR 16-20008S. The second author of this work has been supported by the Croatian Science Foundation (scientific project 3955: Mathematical modeling and numerical simulations of processes in thin or porous domains). The authors would like to thank the referee for his/her thorough review and highly appreciate the comments and suggestions, which significantly contributed to improving the quality of the publication.

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