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Fabricius, J., Miroshnikova, E., Tsandzana, A. & Wall, P. (2020). Pressure-driven flow in thin domains. Asymptotic Analysis, 116(1), 1-26
Open this publication in new window or tab >>Pressure-driven flow in thin domains
2020 (English)In: Asymptotic Analysis, ISSN 0921-7134, E-ISSN 1875-8576, Vol. 116, no 1, p. 1-26Article in journal (Refereed) Published
##### Abstract [en]

We study the asymptotic behavior of pressure-driven Stokes flow in a thin domain. By letting the thickness of the domain tend to zero we derive a generalized form of the classical Reynolds–Poiseuille law, i.e. the limit velocity field is a linear function of the pressure gradient. By prescribing the external pressure as a normal stress condition, we recover a Dirichlet condition for the limit pressure. In contrast, a Dirichlet condition for the velocity yields a Neumann condition for the limit pressure.

IOS Press, 2020
##### Keywords
Stokes equation, pressure boundary condition, two-scale convergence, thin domain, Bogovskii operator, Korn inequality
##### National Category
Mathematical Analysis
Mathematics
##### Identifiers
urn:nbn:se:ltu:diva-75853 (URN)10.3233/ASY-191535 (DOI)000501542500001 ()
##### Note

Available from: 2019-09-05 Created: 2019-09-05 Last updated: 2019-12-20Bibliographically approved
Fabricius, J. (2019). Stokes flow with kinematic and dynamic boundary conditions. Quarterly of Applied Mathematics, 77(3), 525-544
Open this publication in new window or tab >>Stokes flow with kinematic and dynamic boundary conditions
2019 (English)In: Quarterly of Applied Mathematics, ISSN 0033-569X, E-ISSN 1552-4485, Vol. 77, no 3, p. 525-544Article in journal (Refereed) Published
##### Abstract [en]

We review the first and second boundary value problems for the Stokes system posed in a bounded Lipschitz domain in . Particular attention is given to the mixed boundary condition: a Dirichlet condition is imposed for the velocity on one part of the boundary while a Neumann condition for the stress tensor is imposed on the remaining part. Some minor modifications to the standard theory are therefore required. The most noteworthy result is that both pressure and velocity are unique.

##### Place, publisher, year, edition, pages
American Mathematical Society (AMS), 2019
##### Keywords
Stokes equation, stress condition, traction condition, de Rham operator, pressure operator
##### National Category
Mathematical Analysis
Mathematics
##### Identifiers
urn:nbn:se:ltu:diva-74756 (URN)10.1090/qam/1534 (DOI)000469390700004 ()2-s2.0-85073716246 (Scopus ID)
##### Note

Available from: 2019-06-19 Created: 2019-06-19 Last updated: 2019-11-04Bibliographically approved
Fabricius, J., Tsandzana, A. F., Pérez-Ràfols, F. & Wall, P. (2017). A Comparison of the Roughness Regimes in Hydrodynamic Lubrication. Journal of tribology, 139(5), Article ID 051702.
Open this publication in new window or tab >>A Comparison of the Roughness Regimes in Hydrodynamic Lubrication
2017 (English)In: Journal of tribology, ISSN 0742-4787, E-ISSN 1528-8897, Vol. 139, no 5, article id 051702Article in journal (Refereed) Published
##### Abstract [en]

This work relates to previous studies concerning the asymptotic behavior of Stokes flow in a narrow gap between two surfaces in relative motion. It is assumed that one of the surfaces is rough, with small roughness wavelength l, so that the film thickness h becomes rapidly oscillating. Depending on the limit of the ratio h/l, denoted as k, three different lubrication regimes exist: Reynolds roughness (k-0), Stokes roughness (0<γ<1), and high-frequency roughness (γ = ∞). In each regime, the pressure field is governed by a generalized Reynolds equation, whose coefficients (so-called flow factors) depend on k. To investigate the accuracy and applicability of the limit regimes, we compute the Stokes flow factors for various roughness patterns by varying the parameter k. The results show that there are realistic surface textures for which the Reynolds roughness is not accurate and the Stokes roughness must be used instead.

##### Place, publisher, year, edition, pages
The American Society of Mechanical Engineers (ASME), 2017
##### National Category
Mathematical Analysis Tribology (Interacting Surfaces including Friction, Lubrication and Wear)
##### Research subject
Mathematics; Machine Elements
##### Identifiers
urn:nbn:se:ltu:diva-64734 (URN)10.1115/1.4035868 (DOI)000406397500016 ()2-s2.0-85020933899 (Scopus ID)
##### Note

Available from: 2017-07-03 Created: 2017-07-03 Last updated: 2018-11-27Bibliographically approved
Almqvist, A., Fabricius, J., Lundström, S. & Wall, P. (2017). Flow in thin domains with a microstructure: Lubrication and thin porous media. Paper presented at 11th International Conference on Mathematical Problems in Engineering, Aerospace and Sciences, ICNPAA 2016, La Rochelle, France, 4-8 July 2016. AIP Conference Proceedings, 1798, Article ID 020172.
Open this publication in new window or tab >>Flow in thin domains with a microstructure: Lubrication and thin porous media
2017 (English)In: AIP Conference Proceedings, ISSN 0094-243X, E-ISSN 1551-7616, Vol. 1798, article id 020172Article in journal (Refereed) Published
##### Abstract [en]

This paper is devoted to homogenization of different models of flow in thin domains with a microstructure. The focus is on applications connected to the effect of surface roughness in full film lubrication, but a parallel to flow in thin porous media is also discussed. Mathematical models of such flows naturally include two small parameters. One is connected to the fluid film thickness and the other to the microstructure. The corresponding asymptotic analysis is a delicate problem, since the result depends on how fast the two small parameters tend to zero relative to each other. We give a review of the current status in this area and point out some future challenges.

##### Place, publisher, year, edition, pages
AIP Publishing, 2017
##### National Category
Mathematical Analysis Tribology (Interacting Surfaces including Friction, Lubrication and Wear) Fluid Mechanics and Acoustics
##### Research subject
Machine Elements; Mathematics; Fluid Mechanics
##### Identifiers
urn:nbn:se:ltu:diva-62224 (URN)10.1063/1.4972764 (DOI)000399203000171 ()2-s2.0-85013665597 (Scopus ID)
##### Conference
11th International Conference on Mathematical Problems in Engineering, Aerospace and Sciences, ICNPAA 2016, La Rochelle, France, 4-8 July 2016
##### Note

2017-03-03 (andbra);Konferensartikel i tidskrift

Available from: 2017-03-01 Created: 2017-03-01 Last updated: 2018-11-20Bibliographically approved
Fabricius, J., Miroshnikova, E. & Wall, P. (2017). Homogenization of the Stokes equation with mixed boundary condition in a porous medium. Cogent Mathamatics, 4(1), Article ID 1327502.
Open this publication in new window or tab >>Homogenization of the Stokes equation with mixed boundary condition in a porous medium
2017 (English)In: Cogent Mathamatics, E-ISSN 2331-1835, Vol. 4, no 1, article id 1327502Article in journal (Refereed) Published
##### Abstract [en]

We homogenize stationary incompressible Stokes flow in a periodic porous medium. The fluid is assumed to satisfy a no-slip condition on the boundary of solid inclusions and a normal stress (traction) condition on the global boundary. Under these assumptions, the homogenized equation becomes the classical Darcy law with a Dirichlet condition for the pressure.

##### Place, publisher, year, edition, pages
Taylor & Francis, 2017
##### National Category
Mathematical Analysis
Mathematics
##### Identifiers
urn:nbn:se:ltu:diva-64001 (URN)10.1080/23311835.2017.1327502 (DOI)000403291100001 ()
##### Note

Available from: 2017-06-14 Created: 2017-06-14 Last updated: 2018-07-10Bibliographically approved
Fabricius, J., Hellström, G., Lundström, S., Miroshnikova, E. & Wall, P. (2016). Darcy's law for flow in a periodic thin porous medium confined between two parallel plates (ed.). Paper presented at InterPore Industry Workshop on Thin Porous Mediat as part of the 5th International Conference of the Interpore-Society, Prague, May 13 2014.. Transport in Porous Media, 115(3), 473-493
Open this publication in new window or tab >>Darcy's law for flow in a periodic thin porous medium confined between two parallel plates
2016 (English)In: Transport in Porous Media, ISSN 0169-3913, E-ISSN 1573-1634, Vol. 115, no 3, p. 473-493Article in journal (Refereed) Published
##### Abstract [en]

We study stationary incompressible fluid flow in a thin periodic porous medium. The medium under consideration is a bounded perforated 3D-domain confined between two parallel plates. The distance between the plates is $$\delta$$, and the perforation consists of $$\varepsilon$$-periodically distributed solid cylinders which connect the plates in perpendicular direction. Both parameters $$\varepsilon$$, $$\delta$$ are assumed to be small in comparison with the planar dimensions of the plates. By constructing asymptotic expansions, three cases are analysed: (1) $$\varepsilon \ll \delta$$, (2) $$\delta /\varepsilon \sim \text {constant}$$ and (3) $$\varepsilon \gg \delta$$. For each case, a permeability tensor is obtained by solving local problems. In the intermediate case, the cell problems are 3D, whereas they are 2D in the other cases, which is a considerable simplification. The dimensional reduction can be used for a wide range of $$\varepsilon$$ and $$\delta$$ with maintained accuracy. This is illustrated by some numerical examples.

##### Keywords
Thin porous media, Asymptotic analysis, Homogenization, Darcy’s law, Mixed boundary condition, Stress boundary condition, Permeability
##### National Category
Fluid Mechanics and Acoustics Mathematical Analysis
##### Research subject
Fluid Mechanics; Mathematics
##### Identifiers
urn:nbn:se:ltu:diva-8685 (URN)10.1007/s11242-016-0702-2 (DOI)000388983600005 ()2-s2.0-84966705369 (Scopus ID)7396f0ae-22ce-48c1-b5d8-eaeed32c8ba5 (Local ID)7396f0ae-22ce-48c1-b5d8-eaeed32c8ba5 (Archive number)7396f0ae-22ce-48c1-b5d8-eaeed32c8ba5 (OAI)
##### Conference
InterPore Industry Workshop on Thin Porous Mediat as part of the 5th International Conference of the Interpore-Society, Prague, May 13 2014.
##### Funder
Swedish Research Council
##### Note

Validerad; 2016; Nivå 1; 2016-11-25 (andbra); Konferensartikel i tidskrift

Available from: 2016-09-29 Created: 2016-09-29 Last updated: 2019-09-25Bibliographically approved
Fabricius, J., Tsandzana, A. F. & Wall, P. (2015). Homogenization of a compressible cavitation model (ed.). Paper presented at . European journal of applied mathematics (Print), 26(3), 383-399
Open this publication in new window or tab >>Homogenization of a compressible cavitation model
2015 (English)In: European journal of applied mathematics (Print), ISSN 0956-7925, E-ISSN 1469-4425, Vol. 26, no 3, p. 383-399Article in journal (Refereed) Published
##### Abstract [en]

We develop a mathematical model in hydrodynamic lubrication that takes into account three phenomena: cavitation, surface roughness and compressibility of the fluid. Like the classical Reynolds equation, the model is mass preserving. We compute the homogenized coefficients in the case of unidirectional roughness. A one-dimensional problem is also solved explicitly

##### National Category
Mathematical Analysis
Mathematics
##### Identifiers
urn:nbn:se:ltu:diva-15347 (URN)10.1017/S0956792515000078 (DOI)000353580600005 ()2-s2.0-84928640588 (Scopus ID)ed7d137c-6164-4914-abea-4210d97ee63f (Local ID)ed7d137c-6164-4914-abea-4210d97ee63f (Archive number)ed7d137c-6164-4914-abea-4210d97ee63f (OAI)
##### Note
Validerad; 2015; Nivå 2; 20150319 (andbra)Available from: 2016-09-29 Created: 2016-09-29 Last updated: 2018-07-10Bibliographically approved
Almqvist, A., Fabricius, J., Larsson, R. & Wall, P. (2014). A new approach for studying cavitation in lubrication (ed.). Paper presented at . Journal of tribology, 136(1), Article ID 11706.
Open this publication in new window or tab >>A new approach for studying cavitation in lubrication
2014 (English)In: Journal of tribology, ISSN 0742-4787, E-ISSN 1528-8897, Vol. 136, no 1, article id 11706Article in journal (Refereed) Published
##### Abstract [en]

The underlying theory, in this paper, is based on clear physical arguments related to conservation of mass flow and considers both incompressible and compressible fluids. The result of the mathematical modeling is a system of equations with two unknowns, which are related to the hydrodynamic pressure and the degree of saturation of the fluid. Discretization of the system leads to a linear complementarity problem (LCP), which easily can be solved numerically with readily available standard methods and an implementation of a model problem in matlab code is made available for the reader of the paper. The model and the associated numerical solution method have significant advantages over today's most frequently used cavitation algorithms, which are based on Elrod-Adams pioneering work

##### National Category
Tribology (Interacting Surfaces including Friction, Lubrication and Wear) Mathematical Analysis
##### Research subject
Machine Elements; Mathematics
##### Identifiers
urn:nbn:se:ltu:diva-13843 (URN)10.1115/1.4025875 (DOI)000327760400018 ()2-s2.0-84888398410 (Scopus ID)d21f407a-71f4-4b91-9fc4-890b771ddabe (Local ID)d21f407a-71f4-4b91-9fc4-890b771ddabe (Archive number)d21f407a-71f4-4b91-9fc4-890b771ddabe (OAI)
##### Note
Validerad; 2014; 20131209 (andbra)Available from: 2016-09-29 Created: 2016-09-29 Last updated: 2018-07-10Bibliographically approved
Fabricius, J., Koroleva, Y., Tsandzana, A. F. & Wall, P. (2014). Asymptotic behaviour of Stokes flow in a thin domain with amoving rough boundary (ed.). Paper presented at . Proceedings of the Royal Society. Mathematical, Physical and Engineering Sciences, 470(2167), Article ID 20130735.
Open this publication in new window or tab >>Asymptotic behaviour of Stokes flow in a thin domain with amoving rough boundary
2014 (English)In: Proceedings of the Royal Society. Mathematical, Physical and Engineering Sciences, ISSN 1364-5021, E-ISSN 1471-2946, Vol. 470, no 2167, article id 20130735Article in journal (Refereed) Published
##### Abstract [en]

We consider a problem that models fluid flow in a thin domain bounded by two surfaces. One of the surfaces is rough and moving, whereas the other is flat and stationary. The problem involves two small parameters ε and μ that describe film thickness and roughness wavelength, respectively. Depending on the ratio λ = ε/μ, three different flow regimes are obtained in the limit as both of them tend to zero. Time-dependent equations of Reynolds type are obtained in all three cases (Stokes roughness, Reynolds roughness and high-frequency roughness regime). The derivations of the limiting equations are based on formal expansions in the parameters ε and μ.

##### National Category
Mathematical Analysis
Mathematics
##### Identifiers
urn:nbn:se:ltu:diva-4639 (URN)10.1098/rspa.2013.0735 (DOI)000336184600003 ()2-s2.0-84901270334 (Scopus ID)29cc4d7b-7c7c-4e3c-9b00-a9bce047ccd3 (Local ID)29cc4d7b-7c7c-4e3c-9b00-a9bce047ccd3 (Archive number)29cc4d7b-7c7c-4e3c-9b00-a9bce047ccd3 (OAI)
##### Note
Validerad; 2014; 20140610 (johsod)Available from: 2016-09-29 Created: 2016-09-29 Last updated: 2018-07-10Bibliographically approved
Wall, P., Koroleva, Y. U. U., Tsandzana, A. F. & Fabricius, J. (2014). On the effects of surface roughness in thin film flow governed by the time dependent Stokes equations (ed.). Paper presented at . Doklady. Mathematics, 90(1), 445-449
Open this publication in new window or tab >>On the effects of surface roughness in thin film flow governed by the time dependent Stokes equations
2014 (English)In: Doklady. Mathematics, ISSN 1064-5624, E-ISSN 1531-8362, Vol. 90, no 1, p. 445-449Article in journal (Refereed) Published
##### National Category
Mathematical Analysis
Mathematics
##### Identifiers
urn:nbn:se:ltu:diva-7771 (URN)10.1134/S106456241405010X (DOI)000341595000015 ()6309bf9f-c82b-41d2-8278-f1f5a9b4f551 (Local ID)6309bf9f-c82b-41d2-8278-f1f5a9b4f551 (Archive number)6309bf9f-c82b-41d2-8278-f1f5a9b4f551 (OAI)
##### Note
Validerad; 2014; 20140917 (andbra)Available from: 2016-09-29 Created: 2016-09-29 Last updated: 2018-07-10Bibliographically approved
##### Identifiers
ORCID iD: orcid.org/0000-0003-1993-8229

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