Existence of turning points for the response diagram of the Poiseuille flow with prescribed flow-rate
Author:
Giovanni Cimatti
Journal:
Quart. Appl. Math. 65 (2007), 523-528
MSC (2000):
Primary 76D03, 76D05
DOI:
https://doi.org/10.1090/S0033-569X-07-01058-1
Published electronically:
June 6, 2007
MathSciNet review:
2354885
Full-text PDF Free Access
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Abstract: We study the stationary Poiseuille flow in a cylindrical channel of arbitrary cross-section with temperature dependent viscosity and internal dissipation. We assume the flow-rate $\Phi$ given and the axial pressure gradient $\mu$ unknown. This leads to a non-local problem. We show the existence in the response diagram, the plane $(\Phi ,\mu )$, of two turning points.
References
- H. Beirão da Veiga, Time periodic solutions of the Navier-Stokes equations in unbounded cylindrical domains—Leray’s problem for periodic flows, Arch. Ration. Mech. Anal. 178 (2005), no. 3, 301–325. MR 2196495, DOI https://doi.org/10.1007/s00205-005-0376-3
- G. P. Galdi and A. M. Robertson, The relation between flow rate and axial pressure gradient for time-periodic Poiseuille flow in a pipe, J. Math. Fluid Mech. 7 (2005), no. suppl. 2, S215–S223. MR 2192849, DOI https://doi.org/10.1007/s00021-005-0154-x
- Giovanni P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations. Vol. I, Springer Tracts in Natural Philosophy, vol. 38, Springer-Verlag, New York, 1994. Linearized steady problems. MR 1284205
- G. Cimatti, The Poiseuille solution with temperature dependent viscosity, Appl. Anal. 84 (2005), no. 5, 451–461. MR 2151274, DOI https://doi.org/10.1080/00036810500047386
References
- H. Beirão Da Veiga, Time-Periodic Solutions of the Navier-Stokes Equations inUnbounded Cylindrical Domains - Leray’s Problem for Periodic Flow, Arch. Rational Mech. Anal. 178 (2005) 301-325. MR 2196495 (2006k:35209)
- G. P. Galdi and A. M. Robertson, The Relation Between Flow Rate and Axial Pressure Gradient for Time-Periodic Poiseuille Flow in a Pipe, J. Math. Fluid. Mech. 7 (2005) 215-223. MR 2192849 (2006j:76038)
- G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Linearized Steady Problems, Revised Edition, Springer Tracts in Natural Philosophy, Vol. 38, Springer-Verlag, New York, 1998. MR 1284205 (95i:35216a)
- G. Cimatti, The Poiseuille Solution with Temperature Dependent Viscosity, Applicable Analysis, Vol. 84 (2005) 451-461. MR 2151274 (2006a:35247)
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Additional Information
Giovanni Cimatti
Affiliation:
Dipartimento di Matematica, Largo Bruno Pontecorvo 5, Pisa, Italy
Email:
cimatti@dm.unipi.it
Keywords:
Poiseuille flow,
temperature dependent viscosity,
existence and uniqueness of solutions.
Received by editor(s):
September 28, 2006
Published electronically:
June 6, 2007
Article copyright:
© Copyright 2007
Brown University
The copyright for this article reverts to public domain 28 years after publication.