34567896 of 11
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Mathematical modelling of pressure-driven flow in thin domains
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science. Eduardo Mondlane University.ORCID iD: 0000-0002-6378-3781
2024 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The work presented in this thesis focuses on the mathematical modeling of pressure-driven flows in thin domains by analyzing the asymptotic behavior of solutions as a small parameter tends to zero.

The problem of describing asymptotic flows in thin domains arises in many scientific fields, where various physical phenomena are modeled, such as lubrication, liquid molding of fiber-reinforced polymer composites, fluid conduction in thin tubes, and blood circulation in capillaries. In such cases, the flow exhibits different characteristic lengths in different directions, particularly when the domain takes the form of a thin film or a slender tube.  Mathematically, the flow is described by a set of partial differential equations defined in a thin domain, depending on a small parameter ε related to the geometry, such as the ratio of two characteristic lengths. Lower-dimensional models, which retain the essential features of the original problem, are derived by letting ε approach zero. In this limiting process, all variables (e.g., velocity and pressure) depend on ε, and the resulting limit problem yields a simplified model of the flow. To address these problems, several mathematical approaches have been developed, including asymptotic expansions and two-scale convergence for thin domains.

The thesis summarizes the work presented in five papers, referred to as papers I through V, with complementary appendices. The results are discussed in a broader context in an introduction that also provides an overview of the subject. In all papers, the flow is assumed to be governed by the Stokes system posed in a three-dimensional thin domain, subject to a mixed boundary condition. The no-slip and no-penetration conditions require that the velocity vanishes on the solid surfaces of the domain. This is complemented by a normal stress condition on the remaining boundary, defined by an external pressure. Physically, this means that the fluid motion is driven by an external pressure gradient, acting parallel to the surfaces.

The thesis aims to provide a clearer explanation of the novel pressure-driven flow introduced in the papers, while also offering a deeper understanding of the properties of the solutions to such equations formulated in thin domains.

Two types of fluid configurations are considered in the thesis: In papers I and II, the fluid is confined within a generalized Hele-Shaw cell, a type of thin film domain, whereas in the remaining papers, the fluid flows through thin tubular domains. These tubular domains are further classified into two types: the thin straight tube, analyzed in papers III and IV, and the thin curved tube, examined in paper V.

In papers I, III, and V, a stationary incompressible Newtonian fluid is considered in thin domains, with results based on the formal asymptotic expansion method. The primary result is the construction of an approximate solution in an appropriate manner, which is rigorously justified by estimating the error, i.e., the difference between the exact solution and the approximation. In papers III and V, the approximate solution is refined by incorporating boundary layer corrections at the inlet and outlet boundaries.

In papers II and IV, the situation is similar that described in papers I and III, but the fluid follows a non-Newtonian law, modeled by a power-law. The results are obtained by developing functional analysis and calculus of variations techniques to justify theorems concerning the existence and uniqueness of weak solutions, along with a priori estimates for the corresponding Stokes problem. The limit problem is derived using compactness, the two-scale convergence procedure, and arguments such as monotonicity and variational inequality. Finally, the uniqueness and regularity of the solution to the limit problem are proved. In paper IV, strong two-scale convergence for the solution is also considered.

 

Place, publisher, year, edition, pages
Luleå: Luleå University of Technology, 2024.
Series
Doctoral thesis / Luleå University of Technology 1 jan 1997 → …, ISSN 1402-1544
Keywords [en]
fluid dynamics, thin domains, asymptotic analysis, power-law fluids, pressure boundary condition
National Category
Mathematical Analysis
Research subject
Applied Mathematics
Identifiers
URN: urn:nbn:se:ltu:diva-110350ISBN: 978-91-8048-672-9 (print)ISBN: 978-91-8048-673-6 (electronic)OAI: oai:DiVA.org:ltu-110350DiVA, id: diva2:1905200
Public defence
2024-12-18, E632, Luleå University of Technology, Luleå, 10:00 (English)
Opponent
Supervisors
Available from: 2024-10-14 Created: 2024-10-11 Last updated: 2024-11-27Bibliographically approved
List of papers
1. Error estimates for pressure-driven Hele-Shaw flow
Open this publication in new window or tab >>Error estimates for pressure-driven Hele-Shaw flow
2022 (English)In: Quarterly of Applied Mathematics, ISSN 0033-569X, E-ISSN 1552-4485, Vol. 80, no 3, p. 575-595Article in journal (Refereed) Published
Abstract [en]

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 H1- and L2-norms, respectively.

Place, publisher, year, edition, pages
American Mathematical Society (AMS), 2022
Keywords
Hele-Shaw flow, asymptotic expansions, pressure boundary condition, thin film flow, error estimates
National Category
Probability Theory and Statistics Computer Sciences
Research subject
Applied Mathematics
Identifiers
urn:nbn:se:ltu:diva-91626 (URN)10.1090/qam/1619 (DOI)000807138600001 ()2-s2.0-85131407179 (Scopus ID)
Note

Validerad;2022;Nivå 2;2022-06-20 (joosat);

Available from: 2022-06-20 Created: 2022-06-20 Last updated: 2024-10-11Bibliographically approved
2. On pressure-driven Hele–Shaw flow of power-law fluids
Open this publication in new window or tab >>On pressure-driven Hele–Shaw flow of power-law fluids
2022 (English)In: Applicable Analysis, ISSN 0003-6811, E-ISSN 1563-504X, Vol. 101, no 14, p. 5107-5137Article in journal (Refereed) Published
Abstract [en]

We analyze the asymptotic behavior of a non-Newtonian Stokes system, posed in a Hele–Shaw cell, i.e. a thin three-dimensional domain which is confined between two curved surfaces and contains a cylindrical obstacle. The fluid is assumed to be of power-law type defined by the exponent 1< p<∞. By letting the thickness of the domain tend to zero we obtain a generalized form of the Poiseuille law, i.e. the limit velocity is a nonlinear function of the limit pressure gradient. The flow is assumed to be driven by an external pressure which is applied as a normal stress along the lateral part of the boundary. On the remaining part of the boundary we impose a no-slip condition. The two-dimensional limit problem for the pressure is a generalized form of the p′-Laplace equation, 1/p+1/p'=1, with a coefficient called ‘flow factor’, which depends on the geometry as well as the power-law exponent. The boundary conditions are preserved in the limit as a Dirichlet condition for the pressure on the lateral boundary and as a Neumann condition for the pressure on the solid obstacle.

Place, publisher, year, edition, pages
Taylor & Francis, 2022
Keywords
stress boundary condition, Hele-Shaw cell, power-law fluid, p-Laplace equation, thin film flow
National Category
Mathematical Analysis
Research subject
Applied Mathematics
Identifiers
urn:nbn:se:ltu:diva-82624 (URN)10.1080/00036811.2021.1880570 (DOI)000614510000001 ()2-s2.0-85100661967 (Scopus ID)
Note

Validerad;2022;Nivå 2;2022-09-26 (hanlid)

Available from: 2021-01-24 Created: 2021-01-24 Last updated: 2024-10-11Bibliographically approved
3. Pressure-driven flow in a thin straight tube with variable cross-section
Open this publication in new window or tab >>Pressure-driven flow in a thin straight tube with variable cross-section
(English)Manuscript (preprint) (Other academic)
National Category
Natural Sciences
Research subject
Applied Mathematics
Identifiers
urn:nbn:se:ltu:diva-110341 (URN)
Available from: 2024-10-11 Created: 2024-10-11 Last updated: 2024-10-11
4. Flow of power-law fluids in thin straight tubes with variable cross-section
Open this publication in new window or tab >>Flow of power-law fluids in thin straight tubes with variable cross-section
(English)Manuscript (preprint) (Other academic)
National Category
Natural Sciences Mathematics
Research subject
Applied Mathematics
Identifiers
urn:nbn:se:ltu:diva-110342 (URN)
Available from: 2024-10-11 Created: 2024-10-11 Last updated: 2024-10-11
5. Torsion-free frames and pressure-driven flow in thin curved tubes
Open this publication in new window or tab >>Torsion-free frames and pressure-driven flow in thin curved tubes
(English)Manuscript (preprint) (Other academic)
National Category
Mathematics
Research subject
Applied Mathematics
Identifiers
urn:nbn:se:ltu:diva-110344 (URN)
Available from: 2024-10-11 Created: 2024-10-11 Last updated: 2024-10-11

Open Access in DiVA

fulltext(2211 kB)11 downloads
File information
File name FULLTEXT01.pdfFile size 2211 kBChecksum SHA-512
66017ca87eabf7927f3118cbfcba315f87a0e9e06bf694d8c77bd7eecf9853124f9c2647eef3f2c10766dbacb6bdb8c4a0f1998a11fb4be3b5220580d0c29199
Type fulltextMimetype application/pdf

Authority records

Manjate, Salvador Rafael

Search in DiVA

By author/editor
Manjate, Salvador Rafael
By organisation
Mathematical Science
Mathematical Analysis

Search outside of DiVA

GoogleGoogle Scholar
Total: 11 downloads
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

isbn
urn-nbn

Altmetric score

isbn
urn-nbn
Total: 343 hits
34567896 of 11
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf