Non-isothermal fluid flow through a thin pipe with cooling

Abstract

The stationary flow of a Boussinesquian fluid with temperature-dependent viscosity through a thin straight pipe is considered. The fluid in the pipe is cooled by the exterior medium. The asymptotic approximation of the solution is built and rigorously justified by proving the error estimate in terms of domain thickness. The boundary layers for the temperature at the ends of the pipe are studied.

Keywords:

• Boussinesq approximation
• thin pipe
• asymptotic expansion
• boundary layers

AMS Subject Classifications::

• 35C20
• 76D05
• 76D10

Acknowledgements

This research was supported by the Ministry of Science, Education and Sports, Republic of Croatia, Grant 037-0372787-2797. The authors would like to thank the referees for their helpful comments and suggestions.

Notes

Notes

1. We need these technical conditions to prove the error estimate and rigorously justify obtained approximation. For instance, the hypothesis (39)₃ is used in the proof of a priori estimates, for estimating the term involving gravitational force see (59). Since, in standard Boussinesq approximation, f has the form -coefficient of thermic expansion), such an assumption is physically reasonable.

2. See the discussion at the end of the Section 2.2.