| 1. |
- 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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| 2. |
- 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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| 3. |
- 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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| 4. |
- Tsandzana, Afonso Fernando, 1969-
(författare)
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Homogenization of a mathematical model for cavitation in thin film flow
- 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 a contribution to mathematical modeling of thin film flow between two surfaces which are in relative motion. In particular such flows are important in lubrication theory. For many shapes of the surfaces and boundary conditions the pressure in the fluid will be so low that the continuous fluid film ruptures and air bubbles are formed. This phenomenon is known as cavitation and have a huge impact on the hydrodynamic performance. We derive a mathematical model of thin film flow between two close surfaces which takes into account cavitation, surface roughness and pressure dependent density. Moreover, we use two-scale convergence to homogenize the model. In addition, we compute the coefficients of the homogenized equation for a simple class of functions that describe the film thickness.
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| 5. |
- 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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| 6. |
- Tsandzana, Afonso Fernando
(författare)
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Homogenization with applications in lubrication theory
- 2014
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- In this licentiate thesis we study some mathematical problems in hydrodynamic lubrication theory. It is composed of two papers (A and B) and a complementary appendix. Lubrication theory is devoted to fluid flow in thin domains. The main purpose of lubrication is to reduce friction and wear between two solid surfaces in relative motion. 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 leads 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 different important physical quantities such as friction (stresses on the bounding surfaces), load carrying capacity and velocity field.In many practical situations the surface roughness amplitude and the film thickness are of the same order. Therefore, any realistic model should account for the effect of surface roughness. This implies that the mathematical modelling leads to partial differential equations with coefficients that will oscillate rapidly in space and time due to the relative motion of the surfaces. A direct numerical analysis is very difficult since an extremely fine mesh is required to describe the different scales. One method which has proved successful to handle such problems is to do some averaging (asymptotic analysis). The branch in mathematics which has been developed for this purpose is called homogenization.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 ε and wavelength μ of the roughness tend to zero is analyzed and described. The results are obtained using the formal method of multiple scale expansion. The limit equation depends on how fast the two small parameters ε and μ go to zero relative to each other. 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 B we present a mathematical model in hydrodynamic lubrication that takes into account cavitation (formation of air bubbles), surface roughness and compressibility of the fluid. We compute the homogenized coefficients in the case of unidirectional roughness. A one-dimensional problem describing a step bearing is also solved explicitly and by numerical methods.
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