Abstract
The paper presents the mathematical model of a two dimensional steady viscous incompressible flow through a rotating radial blade machine. The flow is described and studied in the rotating frame. The paper provides the classical and weak formulation of the corresponding boundary value problem. The boundary condition on the outflow is the so called “natural” boundary condition, with the additional nonlinear term proposed by Bruneau and Fabrie (Math Model Numer Anal 30(7):815–840, 1996), and a new term arising from the rotation of the machine. The existence of a weak solution is proved.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bruneau C. H., Fabrie P.: New efficient boundary conditions for incompressible Navier–Stokes equations: A well–posedness result. Mathematical Modelling and Numerical Analysis 30, No. 7, 815–840, 1996.
Hishida T.: An existence theorem for the Navier–Stokes flow in the exterior of a rotating obstacle. Arch. Rational Mech. Anal. 150, 307–348, 1999.
Farwig R., Neustupa J.: On the spectrum of a Stokes–type operator arising from flow around a rotating body. Manuscripta Mathematica 122, 419–437, 2007.
Glowinski R.: Numerical Methods for Nonlinear Variational Problems. Springer–Verlag, New York-Berlin-Heidelberg-Tokyo, 1984.
Kučera P., Skalák Z.: Local solutions to the Navier-Stokes equations with mixed boundary conditions. Acta Applicandae Mathematicae 54, No. 3, 275–288, 1998.
Feistauer M., Neustupa T.: On some aspects of analysis of incompressible flow through cascades of profiles. Operator Theory, Advances and Applications, Vol. 147, Birkhauser, Basel, 257–276, 2004.
Temam R.: Navier-Stokes Equations. North-Holland, Amsterdam-New York-Oxford, 1977.
Acknowledgements
The research was supported by the research plan of the Ministry of Education of the Czech Republic No. MSM 6840770010 and by the Grant Agency of the Czech Rep., grant No. 201/09/P413.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Neustupa, T. (2013). Weak Formulation of the Problem of Modelling the Steady Flow of a Viscous Incompressible Liquid Through a Rotating Radial Blade Machine. In: Cangiani, A., Davidchack, R., Georgoulis, E., Gorban, A., Levesley, J., Tretyakov, M. (eds) Numerical Mathematics and Advanced Applications 2011. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33134-3_58
Download citation
DOI: https://doi.org/10.1007/978-3-642-33134-3_58
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-33133-6
Online ISBN: 978-3-642-33134-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)