Error estimates for pressure-driven Hele-Shaw flow
Authors:
John Fabricius, Salvador Manjate and Peter Wall
Journal:
Quart. Appl. Math. 80 (2022), 575-595
MSC (2020):
Primary 76A20, 76D27, 76D07, 76D08
DOI:
https://doi.org/10.1090/qam/1619
Published electronically:
March 30, 2022
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Abstract: We consider Stokes flow past cylindrical obstacles in a generalized Hele-Shaw cell, i.e. a thin three-dimensional domain confined between two surfaces. The flow is assumed to be driven by an external pressure gradient, which is modeled as a normal stress condition on the lateral boundary of the cell. On the remaining part of the boundary we assume that the velocity is zero. We derive a divergence-free (volume preserving) approximation of the flow by studying its asymptotic behavior as the thickness of the domain tends to zero. The approximation is verified by error estimates for both the velocity and pressure in $H^1$- and $L^2$-norms, respectively.
References
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- María Anguiano and Francisco Javier Suárez-Grau, Nonlinear Reynolds equations for non-Newtonian thin-film fluid flows over a rough boundary, IMA J. Appl. Math. 84 (2019), no. 1, 63–95. MR 3904247, DOI 10.1093/imamat/hxy052
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- William L. Barth and Graham F. Carey, On a boundary condition for pressure-driven laminar flow of incompressible fluids, Internat. J. Numer. Methods Fluids 54 (2007), no. 11, 1313–1325. MR 2341655, DOI 10.1002/fld.1427
- G. K. Batchelor, An introduction to fluid dynamics, Second paperback edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1999. MR 1744638
- Alain Bensoussan, Jacques-Louis Lions, and George Papanicolaou, Asymptotic analysis for periodic structures, Studies in Mathematics and its Applications, vol. 5, North-Holland Publishing Co., Amsterdam-New York, 1978. MR 503330
- M. E. Bogovskiĭ, Solution of the first boundary value problem for an equation of continuity of an incompressible medium, Dokl. Akad. Nauk SSSR 248 (1979), no. 5, 1037–1040 (Russian). MR 553920
- Doina Cioranescu and Patrizia Donato, An introduction to homogenization, Oxford Lecture Series in Mathematics and its Applications, vol. 17, The Clarendon Press, Oxford University Press, New York, 1999. MR 1765047
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- Antonija Duvnjak and Eduard Maru ić-Paloka, Derivation of the Reynolds equation for lubrication of a rotating shaft, Arch. Math. (Brno) 36 (2000), no. 4, 239–253. MR 1811168
- John Fabricius, Stokes flow with kinematic and dynamic boundary conditions, Quart. Appl. Math. 77 (2019), no. 3, 525–544. MR 3962580, DOI 10.1090/qam/1534
- J. Fabricius, S. Manjate, and P. Wall, On pressure-driven Hele-Shaw flow of power-law fluids, Appl. Anal. (2021), DOI: 10.1080/00036811.2021.1880570
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- Sanja Marušić and Eduard Maru ić-Paloka, Two-scale convergence for thin domains and its applications to some lower-dimensional models in fluid mechanics, Asymptot. Anal. 23 (2000), no. 1, 23–57. MR 1764338
- Eduard Maru ić-Paloka and Maja Starčević, High-order approximations for an incompressible viscous flow on a rough boundary, Appl. Anal. 94 (2015), no. 7, 1305–1333. MR 3345458, DOI 10.1080/00036811.2014.930823
- Andro Mikelić and Roland Tapiéro, Mathematical derivation of the power law describing polymer flow through a thin slab, RAIRO Modél. Math. Anal. Numér. 29 (1995), no. 1, 3–21 (English, with English and French summaries). MR 1326798, DOI 10.1051/m2an/1995290100031
- A. Mikelić, Remark on the result on homogenization in hydrodynamical lubrication by G. Bayada and M. Chambat, RAIRO Modél. Math. Anal. Numér. 25 (1991), no. 3, 363–370 (English, with French summary). MR 1103093, DOI 10.1051/m2an/1991250303631
- Sergueï A. Nazarov and Juha H. Videman, A modified nonlinear Reynolds equation for thin viscous flows in lubrication, Asymptot. Anal. 52 (2007), no. 1-2, 1–36. MR 2337025
- S. A. Nazarov, Asymptotic solution of the Navier-Stokes problem on the flow of a thin layer of fluid, Sibirsk. Mat. Zh. 31 (1990), no. 2, 131–144 (Russian); English transl., Siberian Math. J. 31 (1990), no. 2, 296–307. MR 1065588, DOI 10.1007/BF00970660
- Jindřich Nečas, Direct methods in the theory of elliptic equations, Springer Monographs in Mathematics, Springer, Heidelberg, 2012. Translated from the 1967 French original by Gerard Tronel and Alois Kufner; Editorial coordination and preface by Šárka Nečasová and a contribution by Christian G. Simader. MR 3014461, DOI 10.1007/978-3-642-10455-8
- G. Panasenko, Multi-scale modelling for structures and composites, Springer, Dordrecht, 2005. MR 2133084
- Ronald L. Panton, Incompressible flow, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1984. MR 802628
- O. Reynolds, On the theory of lubrication and its application to Mr. Beauchamp Tower’s experiments, Philos. Trans. Roy. Soc. Lond. 177 (1886), 157–234.
- G. Stokes, Mathematical proof of the identity of the stream lines obtained by means of a viscous film with those of a perfect fluid moving in two dimensions, Mathematical and Physical Papers, Cambridge Library Collection—Mathematics, pp. 278–282, Cambridge University Press, Cambridge. DOI 10.1017/CBO9780511702297.032
- Luc Tartar, The general theory of homogenization, Lecture Notes of the Unione Matematica Italiana, vol. 7, Springer-Verlag, Berlin; UMI, Bologna, 2009. A personalized introduction. MR 2582099, DOI 10.1007/978-3-642-05195-1
- Luc Tartar, Topics in nonlinear analysis, Publications Mathématiques d’Orsay 78, vol. 13, Université de Paris-Sud, Département de Mathématiques, Orsay, 1978. MR 532371
- Jon Wilkening, Practical error estimates for Reynolds’ lubrication approximation and its higher order corrections, SIAM J. Math. Anal. 41 (2009), no. 2, 588–630. MR 2507463, DOI 10.1137/070695447
- Lailai Zhu and François Gallaire, A pancake droplet translating in a Hele-Shaw cell: lubrication film and flow field, J. Fluid Mech. 798 (2016), 955–969. MR 3512329, DOI 10.1017/jfm.2016.357
References
- D. J. Acheson, Elementary fluid dynamics, Oxford Applied Mathematics and Computing Science Series, The Clarendon Press, Oxford University Press, New York, 1990. MR 1069557
- María Anguiano and Francisco Javier Suárez-Grau, Nonlinear Reynolds equations for non-Newtonian thin-film fluid flows over a rough boundary, IMA J. Appl. Math. 84 (2019), no. 1, 63–95. MR 3904247, DOI 10.1093/imamat/hxy052
- G. Bayada, M. Boukrouche, and M. El-A. Talibi, The transient lubrication problem as a generalized Hele-Shaw type problem, Z. Anal. Anwendungen 14 (1995), no. 1, 59–87. MR 1327492, DOI 10.4171/ZAA/663
- 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 10.1007/BF01442229
- William L. Barth and Graham F. Carey, On a boundary condition for pressure-driven laminar flow of incompressible fluids, Internat. J. Numer. Methods Fluids 54 (2007), no. 11, 1313–1325. MR 2341655, DOI 10.1002/fld.1427
- G. K. Batchelor, An introduction to fluid dynamics, Second paperback edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1999. MR 1744638
- Alain Bensoussan, Jacques-Louis Lions, and George Papanicolaou, Asymptotic analysis for periodic structures, Studies in Mathematics and its Applications, vol. 5, North-Holland Publishing Co., Amsterdam-New York, 1978. MR 503330
- M. E. Bogovskiĭ, Solution of the first boundary value problem for an equation of continuity of an incompressible medium, Dokl. Akad. Nauk SSSR 248 (1979), no. 5, 1037–1040 (Russian). MR 553920
- Doina Cioranescu and Patrizia Donato, An introduction to homogenization, Oxford Lecture Series in Mathematics and its Applications, vol. 17, The Clarendon Press, Oxford University Press, New York, 1999. MR 1765047
- A. M. J. Davis, Rotational effects in Stokes flow; pressure-driven extrusion through an annular hole or concentric holes in parallel walls, J. Engrg. Math. 46 (2003), no. 3-4, 227–240. L. N. G. Filon and biharmonic problems in mechanics. MR 2015358, DOI 10.1023/A:1025072120091
- Antonija Duvnjak and Eduard Maru ić-Paloka, Derivation of the Reynolds equation for lubrication of a rotating shaft, Arch. Math. (Brno) 36 (2000), no. 4, 239–253. MR 1811168
- John Fabricius, Stokes flow with kinematic and dynamic boundary conditions, Quart. Appl. Math. 77 (2019), no. 3, 525–544. MR 3962580, DOI 10.1090/qam/1534
- J. Fabricius, S. Manjate, and P. Wall, On pressure-driven Hele-Shaw flow of power-law fluids, Appl. Anal. (2021), DOI: 10.1080/00036811.2021.1880570
- John Fabricius, Elena Miroshnikova, Afonso Tsandzana, and Peter Wall, Pressure-driven flow in thin domains, Asymptot. Anal. 116 (2020), no. 1, 1–26. MR 4044383, DOI 10.3233/asy-191535
- John Fabricius, Elena Miroshnikova, and Peter Wall, Homogenization of the Stokes equation with mixed boundary condition in a porous medium, Cogent Math. 4 (2017), Art. ID 1327502, 13. MR 3772266, DOI 10.1080/23311835.2017.1327502
- John Fabricius, J. Gunnar I. Hellström, T. Staffan Lundström, Elena Miroshnikova, and Peter Wall, Darcy’s law for flow in a periodic thin porous medium confined between two parallel plates, Transp. Porous Media 115 (2016), no. 3, 473–493. MR 3575716, DOI 10.1007/s11242-016-0702-2
- Olivier Gipouloux and Eduard Marušić, Asymptotic behaviour of the incompressible Newtonian flow through thin constricted fracture, Multiscale problems in science and technology (Dubrovnik, 2000) Springer, Berlin, 2002, pp. 189–202. MR 1998797
- Roland Glowinski, Finite element methods for incompressible viscous flow, Handbook of numerical analysis, Vol. IX, Handb. Numer. Anal., IX, North-Holland, Amsterdam, 2003, pp. 3–1176. MR 2009826
- Pierre Grisvard, Behavior of the solutions of an elliptic boundary value problem in a polygonal or polyhedral domain, Numerical solution of partial differential equations, III (Proc. Third Sympos. (SYNSPADE), Univ. Maryland, College Park, Md., 1975) Academic Press, New York, 1976, pp. 207–274. MR 0466912
- P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683
- H. S. Hele-Shaw, The flow of water, Nature 58 (1898), 34–36
- Horace Lamb, Hydrodynamics, 6th ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1993. With a foreword by R. A. Caflisch [Russel E. Caflisch]. MR 1317348
- Eduard Marušić-Paloka, Mathematical modeling of junctions in fluid mechanics via two-scale convergence, J. Math. Anal. Appl. 480 (2019), no. 1, 123399, 25. MR 3994935, DOI 10.1016/j.jmaa.2019.123399
- Sanja Marušić and Eduard Marušić-Paloka, Two-scale convergence for thin domains and its applications to some lower-dimensional models in fluid mechanics, Asymptot. Anal. 23 (2000), no. 1, 23–57. MR 1764338
- Eduard Marušić-Paloka and Maja Starčević, High-order approximations for an incompressible viscous flow on a rough boundary, Appl. Anal. 94 (2015), no. 7, 1305–1333. MR 3345458, DOI 10.1080/00036811.2014.930823
- Andro Mikelić and Roland Tapiéro, Mathematical derivation of the power law describing polymer flow through a thin slab, RAIRO Modél. Math. Anal. Numér. 29 (1995), no. 1, 3–21 (English, with English and French summaries). MR 1326798, DOI 10.1051/m2an/1995290100031
- A. Mikelić, Remark on the result on homogenization in hydrodynamical lubrication by G. Bayada and M. Chambat, RAIRO Modél. Math. Anal. Numér. 25 (1991), no. 3, 363–370 (English, with French summary). MR 1103093, DOI 10.1051/m2an/1991250303631
- Sergueï A. Nazarov and Juha H. Videman, A modified nonlinear Reynolds equation for thin viscous flows in lubrication, Asymptot. Anal. 52 (2007), no. 1-2, 1–36. MR 2337025
- S. A. Nazarov, Asymptotic solution of the Navier-Stokes problem on the flow of a thin layer of fluid, Sibirsk. Mat. Zh. 31 (1990), no. 2, 131–144 (Russian); English transl., Siberian Math. J. 31 (1990), no. 2, 296–307. MR 1065588, DOI 10.1007/BF00970660
- Jindřich Nečas, Direct methods in the theory of elliptic equations, Springer Monographs in Mathematics, Springer, Heidelberg, 2012. Translated from the 1967 French original by Gerard Tronel and Alois Kufner; Editorial coordination and preface by Šárka Nečasová and a contribution by Christian G. Simader. MR 3014461, DOI 10.1007/978-3-642-10455-8
- G. Panasenko, Multi-scale modelling for structures and composites, Springer, Dordrecht, 2005. MR 2133084
- Ronald L. Panton, Incompressible flow, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1984. MR 802628
- O. Reynolds, On the theory of lubrication and its application to Mr. Beauchamp Tower’s experiments, Philos. Trans. Roy. Soc. Lond. 177 (1886), 157–234.
- G. Stokes, Mathematical proof of the identity of the stream lines obtained by means of a viscous film with those of a perfect fluid moving in two dimensions, Mathematical and Physical Papers, Cambridge Library Collection—Mathematics, pp. 278–282, Cambridge University Press, Cambridge. DOI 10.1017/CBO9780511702297.032
- Luc Tartar, The general theory of homogenization, Lecture Notes of the Unione Matematica Italiana, vol. 7, Springer-Verlag, Berlin; UMI, Bologna, 2009. A personalized introduction. MR 2582099, DOI 10.1007/978-3-642-05195-1
- Luc Tartar, Topics in nonlinear analysis, Publications Mathématiques d’Orsay 78, vol. 13, Université de Paris-Sud, Département de Mathématique, Orsay, 1978. MR 532371
- Jon Wilkening, Practical error estimates for Reynolds’ lubrication approximation and its higher order corrections, SIAM J. Math. Anal. 41 (2009), no. 2, 588–630. MR 2507463, DOI 10.1137/070695447
- Lailai Zhu and François Gallaire, A pancake droplet translating in a Hele-Shaw cell: lubrication film and flow field, J. Fluid Mech. 798 (2016), 955–969. MR 3512329, DOI 10.1017/jfm.2016.357
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Additional Information
John Fabricius
Affiliation:
Department of Engineering Sciences and Mathematics, Luleå University of Technology, SE-971 87 Luleå, Sweden
MR Author ID:
843760
ORCID:
0000-0003-1993-8229
Email:
john.fabricius@ltu.se
Salvador Manjate
Affiliation:
Department of Engineering Sciences and Mathematics, Luleå University of Technology, SE-971 87 Luleå, Sweden, and Department of Mathematics and Informatics, Eduardo Mondlane University, Av. Julius Nyerere, 3453 Maputo, Mozambique
ORCID:
0000-0002-6378-3781
Email:
salvador.manjate@ltu.se
Peter Wall
Affiliation:
Department of Engineering Sciences and Mathematics, Luleå University of Technology, SE-971 87 Luleå, Sweden
MR Author ID:
605963
ORCID:
0000-0001-8211-3671
Email:
peter.wall@ltu.se
Keywords:
Hele-Shaw flow,
asymptotic expansions,
pressure boundary condition,
thin film flow,
error estimates
Received by editor(s):
November 10, 2021
Received by editor(s) in revised form:
January 25, 2022
Published electronically:
March 30, 2022
Article copyright:
© Copyright 2022
Brown University