The homogenization process of the time dependent Reynolds equation describing compressible liquid flow

• To increase the hydrodynamic performance in different machine elements during lubrication, e.g. journal bearings and thrust bearings, it is important to understand the influence of surface roughness. In this connection one encounters different approaches commonly based on some form of the Reynolds equation. They may generally be divided into deterministic- and averaging- techniques. The former regards all surface roughness information and provides a detailed understanding of the local effects that arise. The latter method is suitable when investigating how the surface roughness affects performance of the machine element as a whole. Homogenization is a rigorous mathematical concept that when applied to a certain problem may be thought of as an averaging technique also providing information about local effects. In this work the compressible time dependent Reynolds equation is homogenized. Related problems have recently been analyzed by homogenization techniques under various assumptions. In the present paper the compressibility is modeled assuming a constant lubricant bulk modulus. The formal method of multiple scale expansion is used to derive a so-called homogenized equation and a numerical solution method to solve both the deterministic problem and the homogenized problem is implemented. The numerical results clearly show that the solution of the homogenized equation is a suitable approximation to the solution of the deterministic problem. It is also demonstrated that for small values of the roughness wavelength, the homogenization technique is superior, since the solution of the deterministic problem requires an extremely fine discretization mesh. More over, the solution of the time dependent homogenized problem may in some cases be reduced to solve a stationary problem that facilitates the solution process and interpretation of results.

Ämnesord

TEKNIK OCH TEKNOLOGIER  -- Maskinteknik -- Tribologi (hsv//swe)

ENGINEERING AND TECHNOLOGY  -- Mechanical Engineering -- Tribology (hsv//eng)

NATURVETENSKAP  -- Matematik -- Matematisk analys (hsv//swe)

NATURAL SCIENCES  -- Mathematics -- Mathematical Analysis (hsv//eng)

Nyckelord

Machine Elements

Maskinelement

Matematik

Mathematics

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