Viscoelastic fluids in a thin domain
Authors:
G. Bayada, L. Chupin and S. Martin
Journal:
Quart. Appl. Math. 65 (2007), 625-651
MSC (2000):
Primary 76A10, 35B40
DOI:
https://doi.org/10.1090/S0033-569X-07-01062-X
Published electronically:
October 19, 2007
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Abstract: The present paper deals with viscoelastic flows in a thin domain. In particular, we derive and analyse the asymptotic equations of the Stokes-Oldroyd system in thin films (including shear effects). We present a numerical method which solves the corresponding problem and we present some related numerical tests which evidence the effects of the elastic contribution on the flow.
References
- Ahiko Assemien, Guy Bayada, and Michèle Chambat, Inertial effects in the asymptotic behavior of a thin film flow, Asymptotic Anal. 9 (1994), no. 3, 177–208. MR 1295293
- Guy Bayada and Michèle Chambat, The transition between the Stokes equations and the Reynolds equation: a mathematical proof, Appl. Math. Optim. 14 (1986), no. 1, 73–93. MR 826853, DOI https://doi.org/10.1007/BF01442229
- G. Bayada, M. Chambat, and S. R. Gamouana, About thin film micropolar asymptotic equations, Quart. Appl. Math. 59 (2001), no. 3, 413–439. MR 1848526, DOI https://doi.org/10.1090/qam/1848526
- Franck Boyer, Laurent Chupin, and Pierre Fabrie, Numerical study of viscoelastic mixtures through a Cahn-Hilliard flow model, Eur. J. Mech. B Fluids 23 (2004), no. 5, 759–780. MR 2077449, DOI https://doi.org/10.1016/j.euromechflu.2004.03.001
- Laurent Chupin, Some theoretical results concerning diphasic viscoelastic flows of the Oldroyd kind, Adv. Differential Equations 9 (2004), no. 9-10, 1039–1078. MR 2098065
- James P. Denier and Paul P. Dabrowski, On the boundary-layer equations for power-law fluids, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460 (2004), no. 2051, 3143–3158. MR 2098711, DOI https://doi.org/10.1098/rspa.2004.1349
- Yurun Fan, Huayong Yang, and Roger I. Tanner, Stress boundary layers in the viscoelastic flow past a cylinder in a channel: limiting solutions, Acta Mech. Sin. 21 (2005), no. 4, 311–321. MR 2202171, DOI https://doi.org/10.1007/s10409-005-0040-z
- C. Guillopé and J.-C. Saut, Existence results for the flow of viscoelastic fluids with a differential constitutive law, Nonlinear Anal. 15 (1990), no. 9, 849–869. MR 1077577, DOI https://doi.org/10.1016/0362-546X%2890%2990097-Z
- Daniel D. Joseph, Fluid dynamics of viscoelastic liquids, Applied Mathematical Sciences, vol. 84, Springer-Verlag, New York, 1990. MR 1051193
- Roger E. Khayat and Runling Pan, Transient free-surface flow of a viscoelastic fluid in a narrow channel, Internat. J. Numer. Methods Fluids 46 (2004), no. 6, 637–661. MR 2088859, DOI https://doi.org/10.1002/fld.771
- J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969 (French). MR 0259693
- J.-L. Lions and G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math. 20 (1967), 493–519. MR 216344, DOI https://doi.org/10.1002/cpa.3160200302
- Luc Molinet and Raafat Talhouk, On the global and periodic regular flows of viscoelastic fluids with a differential constitutive law, NoDEA Nonlinear Differential Equations Appl. 11 (2004), no. 3, 349–359. MR 2090278, DOI https://doi.org/10.1007/s00030-004-1073-x
- David O. Olagunju, Local similarity solutions for boundary layer flow of a FENE-P fluid, Appl. Math. Comput. 173 (2006), no. 1, 593–602. MR 2203412, DOI https://doi.org/10.1016/j.amc.2005.04.051
- J. G. Oldroyd, On the formulation of rheological equations of state, Proc. Roy. Soc. London Ser. A 200 (1950), 523–541. MR 35192, DOI https://doi.org/10.1098/rspa.1950.0035
- J.-M. Sac-Épée and K. Taous, On a wide class of nonlinear models for non-Newtonian fluids with mixed boundary conditions in thin domains, Asymptot. Anal. 44 (2005), no. 1-2, 151–171. MR 2196672
- F. Talay Akyildiz and H. Bellout, Viscoelastic lubrication with Phan-Thein-Tanner fluid (PTT), ASME J. Tribol. 126 (2004), 288–291.
- R. I. Tanner and K. Walters, Rheology: an historical perspective, Elsevier, 1998, Rheology series, vol. 7.
- J. Tichy, Non-Newtonian lubrication with the convective Maxwell model, ASME J. Tribol. 118 (1996), 344–349.
- Rong Zhang and Xin Kai Li, Non-Newtonian effects on lubricant thin film flows, J. Engrg. Math. 51 (2005), no. 1, 1–13. MR 2132429, DOI https://doi.org/10.1007/s10665-004-1342-z
- Y. L. Zhang, O. K. Matar, and R. V. Craster, Surfactant spreading on a thin weakly viscoelastic film, J. Non-Newtonian Fluid Mech. 105 (2002), no. 1, 53–78.
References
- A. Assemien, G. Bayada, and M. Chambat, Inertial effects in the asymptotic behavior of a thin film flow, Asymptotic Anal. 9 (1994), no. 3, 177–208. MR 1295293 (95m:76027)
- G. Bayada and M. Chambat, The transition between the Stokes equations and the Reynolds equation: a mathematical proof, Appl. Math. Optim. 14 (1986), no. 1, 73–93. MR 826853 (87g:76044)
- G. Bayada, M. Chambat, and S. R. Gamouana, About thin film micropolar asymptotic equations, Quart. Appl. Math. 59 (2001), no. 3, 413–439. MR 1848526 (2003c:76004)
- F. Boyer, L. Chupin, and P. Fabrie, Numerical study of viscoelastic mixtures through a Cahn-Hilliard flow model, Eur. J. Mech. B Fluids 23 (2004), no. 5, 759–780. MR 2077449 (2005c:76011)
- L. Chupin, Some theoretical results concerning diphasic viscoelastic flows of the Oldroyd kind, Adv. Differential Equations 9 (2004), no. 9-10, 1039–1078. MR 2098065 (2005i:76006)
- J. P. Denier and P. P. Dabrowski, On the boundary-layer equations for power-law fluids, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460 (2004), 3143–3158. MR 2098711 (2005f:76005)
- Y. Fan, H. Yang, and R. I. Tanner, Stress boundary layers in the viscoelastic flow past a cylinder in a channel: limiting solutions, Acta Mech. Sin. 21 (2005), no. 4, 311–321. MR 2202171
- C. Guillopé and J.-C. Saut, Existence results for the flow of viscoelastic fluids with a differential constitutive law, Nonlinear Anal. 15 (1990), no. 9, 849–869. MR 1077577 (91h:76007)
- D. D. Joseph, Fluid dynamics of viscoelastic liquids, Springer, New York, 1990. MR 1051193 (91d:76003)
- R. E. Khayat and R. Pan, Transient free-surface flow of a viscoelastic fluid in a narrow channel, Int. J. Numer. Meth. Fluids 46 (2004), no. 6, 637–661. MR 2088859 (2005e:76006)
- J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, 1969. MR 0259693 (41:4326)
- J.-L. Lions and G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math. 20 (1967), 493–519. MR 0216344 (35:7178)
- L. Molinet and R. Talhouk, On the global and periodic regular flows of viscoelastic fluids with a differential constitutive law, NoDEA Nonlinear Differential Equations Appl. 11 (2004), no. 3, 349–359. MR 2090278 (2005d:76002)
- D. O. Olagunju, Local similarity solutions for boundary layer flow of a FENE-P fluid, Appl.Math. Comput. 173 (2006), no. 1, 593–602. MR 2203412 (2006i:76029)
- J. G. Oldroyd, On the formulation of rheological equations of state, Proc. Roy. Soc. London. Ser. A. 200 (1950), 523–541. MR 0035192 (11:703a)
- J.-M. Sac-Épée and K. Taous, On a wide class of non linear models for non-newtonian fluids with mixed boundary conditions in thin domains, Asymptot. Anal. 44 (2005), no. 1-2, 151–171. MR 2196672 (2006j:76008)
- F. Talay Akyildiz and H. Bellout, Viscoelastic lubrication with Phan-Thein-Tanner fluid (PTT), ASME J. Tribol. 126 (2004), 288–291.
- R. I. Tanner and K. Walters, Rheology: an historical perspective, Elsevier, 1998, Rheology series, vol. 7.
- J. Tichy, Non-Newtonian lubrication with the convective Maxwell model, ASME J. Tribol. 118 (1996), 344–349.
- R. Zhang and X. K. Li, Non-Newtonian effects on lubricant thin film flows, J. Engrg. Math. 51 (2005), no. 1, 1–13. MR 2132429
- Y. L. Zhang, O. K. Matar, and R. V. Craster, Surfactant spreading on a thin weakly viscoelastic film, J. Non-Newtonian Fluid Mech. 105 (2002), no. 1, 53–78.
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Additional Information
G. Bayada
Affiliation:
INSA-Lyon, CNRS UMR 5208 (Institut Camille Jordan) & CNRS UMR 5514 (LAMCOS), Bât. Léonard de Vinci, 21 avenue Jean Capelle, F-69621 Villeurbanne Cedex, France
Email:
guy.bayada@insa-lyon.fr
L. Chupin
Affiliation:
INSA-Lyon, CNRS UMR 5208 (Institut Camille Jordan), Bât. Léonard de Vinci, 21 avenue Jean Capelle, F-69621 Villeurbanne Cedex, France
Email:
laurent.chupin@insa-lyon.fr
S. Martin
Affiliation:
INSA-Lyon, CNRS UMR 5208 (Institut Camille Jordan), Bât. Léonard de Vinci, 21 avenue Jean Capelle, F-69621 Villeurbanne Cedex, France
Email:
sebastien.martin@insa-lyon.fr
Received by editor(s):
February 9, 2006
Published electronically:
October 19, 2007
Article copyright:
© Copyright 2007
Brown University
The copyright for this article reverts to public domain 28 years after publication.