| 1. |
- Almqvist, Andreas, et al.
(författare)
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A new approach for studying cavitation in lubrication
- 2014
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Ingår i: Journal of tribology. - : ASME International. - 0742-4787 .- 1528-8897. ; 136:1
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Tidskriftsartikel (refereegranskat)abstract
- 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
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| 2. |
- Almqvist, Andreas, et al.
(författare)
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Flow in thin domains with a microstructure : Lubrication and thin porous media
- 2017
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Ingår i: AIP Conference Proceedings. - : AIP Publishing. - 0094-243X .- 1551-7616. ; 1798
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 3. |
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| 4. |
- Almqvist, Andreas, et al.
(författare)
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Homogenization of a Reynolds equation describing compressible flow
- 2012
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Ingår i: Journal of Mathematical Analysis and Applications. - : Elsevier BV. - 0022-247X .- 1096-0813. ; 390:2, s. 456-471
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Tidskriftsartikel (refereegranskat)abstract
- We homogenize a Reynolds equation with rapidly oscillating film thickness function hε, assuming a constant compressiblity factor in the pressure-density relation. The oscillations are due to roughness on the bounding surfaces of the fluid film. As shown by previous studies, homogenization is an effective approach for analyzing the effects of surface roughness in hydrodynamic lubrication. By two-scale convergence theory we obtain the limit problem (homogenized equation) and strong convergence in L2 for the unknown density ρε. By adding a small corrector term we also obtain strong convergence in the Sobolev norm.
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| 5. |
- Almqvist, Andreas, et al.
(författare)
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Multiscale homogenization of a class of nonlinear equations with applications in lubrication theory and applications
- 2011
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Ingår i: Journal of Function Spaces and Applications. - : Hindawi Limited. - 0972-6802 .- 1758-4965. ; 9:1, s. 17-40
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Tidskriftsartikel (refereegranskat)abstract
- We prove a homogenization result for monotone operators by using the method of multiscale convergence. More precisely, we study the asymptotic behavior as epsilon -> 0 of the solutions u(epsilon) of the nonlinear equation div a(epsilon)(x, del u(epsilon)) = div b(epsilon), where both a(epsilon) and b(epsilon) oscillate rapidly on several microscopic scales and a(epsilon) satisfies certain continuity, monotonicity and boundedness conditions. This kind of problem has applications in hydrodynamic thin film lubrication where the bounding surfaces have roughness on several length scales. The homogenization result is obtained by extending the multiscale convergence method to the setting of Sobolev spaces W-0(1,p)(Omega), where 1 < p < infinity. In particular we give new proofs of some fundamental theorems concerning this convergence that were first obtained by Allaire and Briane for the case p = 2.
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| 6. |
- Almqvist, Andreas, et al.
(författare)
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Reiterated homogenization applied in hydrodynamic lubrication
- 2008
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Ingår i: Proceedings of the Institution of mechanical engineers. Part J, journal of engineering tribology. - 1350-6501 .- 2041-305X. ; 222:7, s. 827-841
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Tidskriftsartikel (refereegranskat)abstract
- This work is devoted to studying the combined effect that arises due to surface texture and surface roughness in hydrodynamic lubrication. An effective approach in tackling this problem is by using the theory of reiterated homogenization with three scales. In the numerical analysis of such problems, a very fine mesh is needed, suggesting some type of averaging. To this end, a general class of problems is studied that, e.g. includes the incompressible Reynolds problem in both artesian and cylindrical coordinate forms. To demonstrate the effectiveness of the method several numerical results are presented that clearly show the convergence of the deterministic solutions towards the homogenized solution.Moreover, the convergence of the friction force and the load carrying capacity of the lubricant film is also addressed in this paper. In conclusion, reiterated homogenization is a feasible mathematical tool that facilitates the analysis of this type of problem.
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| 7. |
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| 8. |
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| 9. |
- Almqvist, Andreas, et al.
(författare)
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Similarities and differences between the flow factor method by Patir and Cheng and homogenization
- 2011
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Ingår i: Journal of tribology. - : ASME International. - 0742-4787 .- 1528-8897. ; 133:3, s. 031702-1
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Tidskriftsartikel (refereegranskat)abstract
- Different averaging techniques have proved to be useful for analyzing the effects of surface roughness in hydrodynamic lubrication. This paper compares two of these averaging techniques, namely the flow factor method by Patir and Cheng (P&C) and homogenization. It has been rigorously proved by many authors that the homogenization method provides a correct solution for arbitrary roughness. In this work it is shown that the two methods coincide if and only if the roughness exhibits certain symmetries. Hence, homogenization is always the preferred method.
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| 10. |
- Almqvist, Andreas, et al.
(författare)
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Variational bounds applied to unstationary hydrodynamic lubrication
- 2008
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Ingår i: International Journal of Engineering Science. - : Elsevier BV. - 0020-7225 .- 1879-2197. ; 46:9, s. 891-906
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Tidskriftsartikel (refereegranskat)abstract
- This paper is devoted to the effects of surface roughness in hydrodynamic lubrication. The numerical analysis of such problems requires a very fine mesh to resolve the surface roughness, hence it is often necessary to do some type of averaging. Previously, homogenization (a rigorous form of averaging) has been successfully applied to Reynolds type differential equations. More recently, the idea of finding upper and lower bounds on the effective behavior, obtained by homogenization, was applied for the first time in tribology. In these pioneering works, it has been assumed that only one surface is rough. In this paper we develop these results to include the unstationary case where both surfaces may be rough. More precisely, we first use multiple-scale expansion to obtain a homogenization result for a class of variational problems including the variational formulation associated with the unstationary Reynolds equation. Thereafter, we derive lower and upper bounds corresponding to the homogenized (averaged) variational problem. The bounds reduce the numerical analysis, in that one only needs to solve two smooth problems, i.e. no local scale has to be considered. Finally, we present several examples, where it is shown that the bounds can be used to estimate the effects of surface roughness with very high accuracy.
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| 11. |
- Cumming, Graeme S., et al.
(författare)
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Research priorities for the sustainability of coral-rich western Pacific seascapes
- 2023
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Ingår i: Regional Environmental Change. - 1436-3798 .- 1436-378X. ; 23:2
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Tidskriftsartikel (refereegranskat)abstract
- Nearly a billion people depend on tropical seascapes. The need to ensure sustainable use of these vital areas is recognised, as one of 17 policy commitments made by world leaders, in Sustainable Development Goal (SDG) 14 (‘Life below Water’) of the United Nations. SDG 14 seeks to secure marine sustainability by 2030. In a time of increasing social-ecological unpredictability and risk, scientists and policymakers working towards SDG 14 in the Asia–Pacific region need to know: (1) How are seascapes changing? (2) What can global society do about these changes? and (3) How can science and society together achieve sustainable seascape futures? Through a horizon scan, we identified nine emerging research priorities that clarify potential research contributions to marine sustainability in locations with high coral reef abundance. They include research on seascape geological and biological evolution and adaptation; elucidating drivers and mechanisms of change; understanding how seascape functions and services are produced, and how people depend on them; costs, benefits, and trade-offs to people in changing seascapes; improving seascape technologies and practices; learning to govern and manage seascapes for all; sustainable use, justice, and human well-being; bridging communities and epistemologies for innovative, equitable, and scale-crossing solutions; and informing resilient seascape futures through modelling and synthesis. Researchers can contribute to the sustainability of tropical seascapes by co-developing transdisciplinary understandings of people and ecosystems, emphasising the importance of equity and justice, and improving knowledge of key cross-scale and cross-level processes, feedbacks, and thresholds.
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| 12. |
- Erwin, Peter, et al.
(författare)
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Composite bulges : the coexistence of classical bulges and discy pseudo-bulges in S0 and spiral galaxies
- 2015
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Ingår i: Monthly notices of the Royal Astronomical Society. - : Oxford University Press (OUP). - 0035-8711 .- 1365-2966. ; 446:4, s. 4039-4077
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Tidskriftsartikel (refereegranskat)abstract
- We present an analysis of nine S0-Sb galaxies which have (photometric) bulges consisting of two distinct components. The outer component is a flattened, kinematically cool, disc-like structure: a 'discy pseudo-bulge'. Embedded inside is a rounder, kinematically hot spheroidal structure: a 'classical bulge'. This indicates that pseudo-bulges and classical bulges are not mutually exclusive phenomena: some galaxies have both. The discy pseudo-bulges almost always consist of an exponential disc (scalelengths = 125-870 pc, mean size similar to 440 pc) with one or more disc-related subcomponents: nuclear rings, nuclear bars, and/or spiral arms. They constitute 11-59 per cent of the galaxy stellar mass (mean PB/T = 0.33), with stellar masses similar to 7 x 10(9)-9 x 10(10) M-circle dot. The classical-bulge components have Sersic indices of 0.9-2.2, effective radii of 25-430 pc and stellar masses of 5 x 10(8)-3 x 10(10) M-circle dot; they are usually <10 per cent of the galaxy's stellar mass (mean B/T = 0.06). The classical bulges do show rotation, but are clearly kinematically hotter than the discy pseudo-bulges. Dynamical modelling of three systems indicates that velocity dispersions are isotropic in the classical bulges and equatorially biased in the discy pseudo-bulges. In the mass-radius and mass-stellar mass density planes, classical-bulge components follow sequences defined by ellipticals and (larger) classical bulges. Discy pseudo-bulges also fall on this sequence; they are more compact than large-scale discs of similar mass. Although some classical bulges are quite compact, they are as a class clearly distinct from nuclear star clusters in both size and mass; in at least two galaxies they coexist with nuclear clusters. Since almost all the galaxies in this study are barred, they probably also host boxy/peanut-shaped bulges (vertically thickened inner parts of bars). NGC 3368 shows isophotal evidence for such a zone just outside its discy pseudo-bulge, making it a clear case of a galaxy with all three types of 'bulge'.
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| 13. |
- Fabricius, John, et al.
(författare)
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A Comparison of the Roughness Regimes in Hydrodynamic Lubrication
- 2017
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Ingår i: Journal of tribology. - : The American Society of Mechanical Engineers (ASME). - 0742-4787 .- 1528-8897. ; 139:5
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 14. |
- Fabricius, John, et al.
(författare)
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A rigorous derivation of the time-dependent Reynolds equation
- 2013
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Ingår i: Asymptotic Analysis. - 0921-7134 .- 1875-8576. ; 84:1-2, s. 103-121
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Tidskriftsartikel (refereegranskat)abstract
- We study the asymptotic behavior of solutions of the evolution Stokes equation in a thin three-dimensional domain bounded by two moving surfaces in the limit as the distance between the surfaces approaches zero. Using only a priori estimates and compactness it is rigorously verified that the limit velocity field and pressure are governed by the time-dependent Reynolds equation.
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| 15. |
- Fabricius, John, et al.
(författare)
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Asymptotic behaviour of Stokes flow in a thin domain with amoving rough boundary
- 2014
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Ingår i: Proceedings of the Royal Society. Mathematical, Physical and Engineering Sciences. - : The Royal Society. - 1364-5021 .- 1471-2946. ; 470:2167
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 16. |
- Fabricius, John, et al.
(författare)
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Darcy's law for flow in a periodic thin porous medium confined between two parallel plates
- 2016
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Ingår i: Transport in Porous Media. - : Springer Science and Business Media LLC. - 0169-3913 .- 1573-1634. ; 115:3, s. 473-493
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 17. |
- Fabricius, John, et al.
(författare)
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Error estimates for pressure-driven Hele-Shaw flow
- 2022
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Ingår i: Quarterly of Applied Mathematics. - : American Mathematical Society (AMS). - 0033-569X .- 1552-4485. ; 80:3, s. 575-595
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Tidskriftsartikel (refereegranskat)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 H1- and L2-norms, respectively.
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| 18. |
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| 19. |
- Fabricius, John, et al.
(författare)
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Homogenization of a compressible cavitation model
- 2015
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Ingår i: European journal of applied mathematics (Print). - 0956-7925 .- 1469-4425. ; 26:3, s. 383-399
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Tidskriftsartikel (refereegranskat)abstract
- 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
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| 20. |
- Fabricius, John
(författare)
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Homogenization of some problems in hydrodynamic lubrication involving rough boundaries
- 2011
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis is devoted to the study of some homogenization problems with applications in lubrication theory. It consists of an introduction, five research papers (I–V) and a complementary appendix.Homogenization is a mathematical theory for studying differential equations with rapidly oscillating coefficients. Many important problems in physics with one or several microscopic scales give rise to this kind of equations, whence the need for methods that enable an efficient treatment of such problems. To this end several mathematical techniques have been devised. The main homogenization method used in this thesis is called multiscale convergence. It is a notion of weak convergence in Lp spaces which is designed to take oscillations into account. In paper II we extend some previously obtained results in multiscale convergence that enable us to homogenize a nonlinear problem with a finite number of microscopic scales. The main idea in the proof is closely related to a decomposition of vector fields due to Hermann Weyl. The Weyl decomposition is further explored in paper III.Lubrication theory is devoted to the study of fluid flows in thin domains. More generally, tribology is the science of bodies in relative motion interacting through a mechanical contact. An important aspect of tribology is to explain the principles of friction, lubrication and wear. The mathematical foundations of lubrication theory are given by the Navier–Stokes equation which describes the motion of a viscous fluid. In thin domains several simplifications are possible, as shown in the introduction of this thesis. The resulting equation is named after Osborne Reynolds and is much simpler to analyze than the Navier--Stokes equation.The Reynolds equation is widely used by engineers today. For extremely thin films, it is well-known that the surface micro-topography is an important factor in hydrodynamic performance. Hence it is important to understand the influence of surface roughness with small characteristic wavelengths upon the solution of the Reynolds equation. Since the 1980s such problems have been increasingly studied by homogenization theory. The idea is to replace the original equation with a homogenized equation where the roughness effects are “averaged out”. One problem consists of finding an algorithm for computing the solution of the homogenized equation. Another problem consists of showing, on introducing the appropriate mathematical definitions, that the homogenized equation is the correct method of averaging. Papers I, II, IV and V investigate the effects of surface roughness by homogenization techniques in various situations of hydrodynamic lubrication. To compare the homogenized solution with the solution of the deterministic Reynolds equation, some numerical examples are also included.
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| 21. |
- Fabricius, John, et al.
(författare)
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Homogenization of the Stokes equation with mixed boundary condition in a porous medium
- 2017
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Ingår i: Cogent Mathamatics. - : Taylor & Francis. - 2331-1835. ; 4:1
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 22. |
- Fabricius, John
(författare)
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Homogenization theory with applications in tribology
- 2008
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Homogenization is a mathematical theory for studying differential equations with rapidly oscillating coefficents. Many important problems in physics with one or several microscopic length scales give rise to this kind of equations. Hence there is a need for methods that enable an efficient treatment of such problems. To this end several homogenization techniques exist, ranging from the fairly abstract ones to those that are more oriented towards applications. This thesis is concerned with two such methods, namely the "asymptotic expansion method", also known as the "method of multiple scales", and multiscale convergence. The former method, sometimes referred to as the "engineering approach to homogenization" has, due to its versatility and intutive appeal, gained wide acceptance and popularity in the applied fields. However, it is not rigorous by mathematical standards. Multiscale convergence, introduced by Nguetseng in 1989, is a notion of weak convergence in Lp spaces that is designed to take oscillations into account. Although not the most general method around, multiscale convergence has become widely used by homogenizers because of its simplicity. In spite of its success, the multiscale theory is not yet sufficiently developed to be used in connection with certain nonlinear problems with several microscopic scales. In Paper A we extend some previously obtained results in multiscale convergence that enable us to homogenize a nonlinear problem with three scales. In Appendix to Paper A we present in more detail some results that were used in the proof of some of the main theorems in Paper A. Tribology is the science of bodies in relative motion interacting through a mechanical contact. An important aspect of tribology is to explain the principles of friction, lubrication and wear. Tribological phenomena are encountered everywhere in nature and technology and have a huge economical impact on society. An important example is that of two sliding solid surfaces interacting through a thin film of viscous fluid (lubricant). Hydrodynamic lubrication occurs when the pressure generated within the lubricant, through the viscosity of the fluid, is able to sustain an externally applied load. Many common bearings, e.g. journal bearings or slider bearings, operate according to this principle. As a branch of fluid dynamics, the mathematical foundations of lubrication theory are given by the Navier-Stokes equations, describing the motion of a viscous fluid. Because of the thin film assumption several simplifications are possible, leading to various reduced equations named after Osborne Reynolds, the founding father of lubrication theory. The Reynolds equation is used by engineers to compute the pressure distribution in various situations of thin film lubrication. For extremely thin films, it has been observed that the surface micro topography is an important factor in hydrodynamic performance. Hence it is important to understand the influence of surface roughness with small characteristic wavelength upon the pressure solution. Since the 1980s such problems have been increasingly studied by homogenization theory. The idea is to replace the original equation with a homogenized equation where the roughness effects are "averaged out". One problem consists of finding an algorithm that gives the homogenized equation. Another problem, consists of showing, by introducing the appropriate mathematical defintions, that the homogenized equation really is the correct one. Papers B and C investigate the effects of surface roughness by means of multiscale expansion of the pressure in various situations of hydrodynamic lubrication. Paper B, for which Paper A constitutes a rigorous basis, considers homogenization of the stationary Reynolds equation and roughness with two characteristic wavelengths. This leads to a multiscale problem and adds to the complexity of the homogenization process. To compare the homogenized solution to the solution of the unaveraged Reynolds equation, some numerical examples are also included. Paper C is devoted to homogenization of a variational principle which is a generalization of the unstationary Reynolds equation (both surfaces are rough). The advantage of adopting the calculus of variations viewpoint is that the recently introduced "variational bounds" can be computed. Bounds can be seen as a "cheap" alternative to computing the realtively costly homogenized solution. Several numerical examples are included to illustrate the utility of bounds.
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| 23. |
- Fabricius, John, et al.
(författare)
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On pressure-driven Hele–Shaw flow of power-law fluids
- 2022
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Ingår i: Applicable Analysis. - : Taylor & Francis. - 0003-6811 .- 1563-504X. ; 101:14, s. 5107-5137
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 24. |
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| 25. |
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| 26. |
- Fabricius, John, et al.
(författare)
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Pressure-driven flow in thin domains
- 2020
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Ingår i: Asymptotic Analysis. - : IOS Press. - 0921-7134 .- 1875-8576. ; 116:1, s. 1-26
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 27. |
- Fabricius, John
(författare)
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Stokes flow with kinematic and dynamic boundary conditions
- 2019
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Ingår i: Quarterly of Applied Mathematics. - : American Mathematical Society (AMS). - 0033-569X .- 1552-4485. ; 77:3, s. 525-544
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 28. |
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| 29. |
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| 30. |
- Haller, Elena
(författare)
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Pressure-driven flows in thin and porous domains
- 2021
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The present thesis is devoted to the derivation of Darcy's law for incompressible viscous fluid flows in perforated and thin domains by means of homogenization techniques. The problem of describing asymptotic flows in porous/thin domains occurs in the study of various physical phenomena such as filtration in sandy soils, blood circulation in capillaries, lubrication and heationg/cooling processes. In all such cases flow characteristics are obviously dependent of microstructure of the fluid domains. However, in the most of practical applications the significant role is played by average (or integral) quantities, such as permeability and macroscopic pressure. In order to obtain them there exist several mathematical approaches collectively referred to as homogenisation theory. This thesis consists of five papers. Papers I and V represent the general case of thin porous domains where both parameters ε - the period of perforation, and δ - the thickness of the domain, are involved. We assume that the flow is governed by the Stokes equation and driven by an external pressure, i.e. the normal stress is prescribed on a part of the boundary and no-slip is assumed on the rest of the boundary. Let us note that from the physical point of view such mixed boundary condition is natural whereas in mathematical context it appears quite seldom and raises therefore some essential difficulties in analytical theory. Depending on the limit value λ of mutual δ / ε -ratio, a form of Darcy's law appears as both δ and ε tend to zero. The three principal cases namely are very thin porous medium (λ =0), proportionally thin porous medium (0< λ<∞) and homogeneously thin porous medium (λ =∞). The results are obtained first by using the formal method of multiple scale asymptotic expansions (Paper I) and then rigorously justified in Paper V. Various aspects of such justification (a priori estimates, two-scale and strong convergence results) are done separately for porous media (Paper II) and thin domains (Paper III). The vast part of Papers II and III is devoted to the adaptation of already existing results for systems that satisfy to no-slip condition everywhere on the boundary to the case of mixed boundary condition mentioned above. Alternative justification approach (asymptotic expansion method accomplished by error estimates) is presented in Paper IV for flows in thin rough pipes.
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| 31. |
- Kalliorinne, Kalle, et al.
(författare)
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Application of topological optimisation methodology to infinitely wide slider bearings operating under compressible flow
- 2020
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Ingår i: Proceedings of the Institution of mechanical engineers. Part J, journal of engineering tribology. - : Sage Publications. - 1350-6501 .- 2041-305X. ; 234:7, s. 1035-1050
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Tidskriftsartikel (refereegranskat)abstract
- It has been over a century since the interest in inventing the optimal topology for bearings arose. A significant achievement was published by Lord Rayleigh, who found the step-bearing geometry which maximise the load-carrying capacity when the classical Reynolds equation is used to model thin film flow of an iso-viscous and incompressible fluid. Since then, new optimisation methods considering some variants of governing equations for finding the best possible bearings have surfaced, one of which will be presented in this paper. Here, two different formulations for compressible flow, i.e. ideal gas and constant bulk modulus compressibility, as well as the classical Reynolds formulation will be used in combination with the method of moving asymptotes for topological optimisation. All three of these problem formulations provide us with unique geometries, which either maximise the load-carrying capacity or minimise friction, for fluids with a wide variety of compressibility.
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| 32. |
- Manjate, Salvador
(författare)
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On pressure-driven Hele-Shaw flow
- 2022
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The present licentiate thesis is devoted to the rigorous derivation of the equations governing thin-film flow of incompressible Newtonian and non-Newtonian fluids. More precisely, we consider flow in a generalized Hele-Shaw cell, which is a thin three-dimensional domain confined between two surfaces connected by cylindrical obstacles of various shapes.Thin-film flows arise naturally in several applications. For instance, it is commonly used when the domain itself has different characteristic lengths in different directions, i.e. when the domain is a thin layer or a slender tube. Mathematically, the flow is described by a set of partial differential equations in a thin domain which depends on a small parameterε, e.g. the ratio of two characteristic lengths. By letting ε tend to zero, one can obtain a better understanding of the properties of solutions of such equations. In this limit process, all variables involved (e.g. velocity and pressure) depend on ε and the resulting limit problem yields a simplified model of the flow. There exist several mathematical techniques that have been developed to deal with such problems, e.g. asymptotic expansions, two-scale convergence for thin domains, etc.The scientific results in this thesis are presented in two papers (I and II) and a complimentary appendix. The results are discussed in a more general context in an introduction which also gives an overview of the subject. In both papers, we assume that the flow is governed by the Stokes system posed in a generalized Hele-Shaw cell satisfying a mixed boundary condition. The so-called no-slip and no-penetration conditions require that the velocity vanishes on the solid surfaces of the cell. This condition is complemented by the normal stress condition on the lateral boundary which is defined by an external pressure. Physically this means that the motion of the fluid is caused by the external pressure gradient, which acts in a direction parallel to the surfaces. One of the main objectives of this thesis is to develop a rigorous mathematical description of pressure-driven flow in thin domains.In paper I, we consider Hele-Shaw flow of an incompressible Newtonian fluid. The results are based on the formal asymptotic expansion method, i.e. by introducing a small parameter ε representing the thickness of the domain, rescaling the problem to a fixed domain, and considering solutions in the form of power series of ε. It is shown that the leading term of the velocity satisfies the so-called Poiseuille-law, i.e. the velocity is a linear function of the pressure gradient, whereas the leading pressure term satisfies the generalized Hele-Shaw equation. The main result is the construction of an approximate solution, which is justified by estimating the L2-norm of the error, i.e. the difference between the exact solution and the approximation.In paper II, the situation is similar to that of paper I, but the fluid obeys a more general constitutive relationship between the stress and the shear rate. More precisely, the functional relationship between the viscosity and the symmetrical part of the velocity gradient is given by a power-law. We develop techniques of functional analysis and calculus of variations in order to justify theorems concerning the existence and uniqueness of weak solutions of the corresponding Stokes system. The nonlinear Poiseuille-law, i.e. the limit velocity and the limit pressure gradient follow a power-law, is derived by using a two-scale convergence procedure and monotonicity arguments. Finally, uniqueness and regularity results for the solution of the limit problem are proved.
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| 33. |
- Tsandzana, Afonso Fernando, 1969-
(författare)
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Homogenization of some new mathematical models in lubrication theory
- 2016
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- We consider mathematical modeling of thin film flow between two rough surfaces which are in relative motion. For example such flows take place in different kinds of bearings and gears when a lubricant is used to reduce friction and wear between the surfaces. The mathematical foundations of lubrication theory is given by the Navier--Stokes equation, which describes the motion of viscous fluids. In thin domains several approximations are possible which lead to the so called Reynolds equation. This equation is crucial to describe the pressure in the lubricant film. When the pressure is found it is possible to predict vorous important physical quantities such as friction (stresses on the bounding surfaces), load carrying capacity and velocity field.In hydrodynamic lubrication the effect of surface roughness is not negligible, because in practical situations the amplitude of the surface roughness are of the same order as the film thickness. Moreover, a perfectly smooth surface does not exist in reality due to imperfections in the manufacturing process. Therefore, any realistic lubrication model should account for the effects of surface roughness. This implies that the mathematical modeling leads to partial differential equations with coefficients that will oscillate rapidly in space and time. A direct numerical computation is therefore very difficult, since an extremely dense mesh is needed to resolve the oscillations due to the surface roughness. A natural approach is to do some type of averaging.In this PhD thesis we use and develop modern homogenization theory to be able to handle the questions above. Especially, we use, develop and apply the method based on the multiple scale expansions and two-scale convergence. The thesis is based on five papers (A-E), with an appendix to paper A, and an extensive introduction, which puts these publications in a larger context.In Paper A the connection between the Stokes equation and the Reynolds equation is investigated. More precisely, the asymptotic behavior as both the film thickness and wavelength of the roughness tend to zero is analyzed and described. Three different limit equations are derived. Time-dependent equations of Reynolds type are obtained in all three cases (Stokes roughness, Reynolds roughness and high frequency roughness regime). In paper C we extend the work done in Paper A where we compare the roughness regimes by numeric computations for the stationary case.In paper B we present a mathematical model that takes into account cavitation, surfaces roughness and compressibility of the fluid. We compute the homogenized coefficients in the case of unidirectional roughness.In the paper D we derive a mathematical model of thin film flow between two close rough surfaces, which takes into account cavitation, surface roughness and pressure dependent density. Moreover, we use two-scale convergence to homogenize the model. Finally, in paper E we prove the existence of solutions to a frequently used mathematical model of thin film flow, which takes cavitation into account.
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