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- Fabricius, John, et al.
(author)
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Asymptotic behaviour of Stokes flow in a thin domain with amoving rough boundary
- 2014
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In: Proceedings of the Royal Society. Mathematical, Physical and Engineering Sciences. - : The Royal Society. - 1364-5021 .- 1471-2946. ; 470:2167
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Journal article (peer-reviewed)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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| 2. |
- Tsandzana, Afonso Fernando
(author)
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Homogenization with applications in lubrication theory
- 2014
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Licentiate thesis (other academic/artistic)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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