Comments on the Theory of Fluid Flow Between Solids with Anisotropic Roughness
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COMMENT
Comments on the Theory of Fluid Flow Between Solids with Anisotropic Roughness B. N. J. Persson1,2 Received: 14 October 2020 / Accepted: 11 November 2020 © The Author(s) 2020
Abstract I consider fluid flow at the interface between solids with random roughness. For anisotropic roughness, obtained by stretching isotropic roughness in the x-direction by a factor of 𝛾 1∕2 and in the y-direction with a factor of 𝛾 −1∕2 , I give an argument for why the flow conductivity in the critical junction theory should be proportional to 𝛾 in the x-direction and proportional to 1∕𝛾 in the y-direction. Keywords Leakage · Fluid flow · Roughness · Seals · Fluid flow factor The influence of the surface roughness on the fluid flow dynamics is a complex topic. However, if there is a separation of length scales the problem can be simplified: if R >> 𝜆0 , where R is the (smallest) length characterizing the macroscopic shape of the bodies and 𝜆0 is the longest (relevant) surface roughness component, then it is possible to eliminate (integrate out) the surface roughness and obtain effective fluid flow equations involving solid bodies with smooth surfaces (no roughness). The effective fluid flow equations depend on two quantities determined by the surface roughness, usually denoted fluid flow factors. These factors depend on the average surface separation ū , which will vary throughout the nominal contact region; ū is the local interfacial surface separation u(x, y) averaged over the surface roughness. The aim of this short communication is to add some new results for fluid flow between surfaces with anisotropic roughness. In particular, I will argue that the most narrow constrictions in the critical junction theory of fluid flow can be treated as square-like pores even for surfaces with anisotropic roughness. Only if this is the case will the effective flow conductivity (in the critical junction theory) scale as 𝛾 and 1∕𝛾 along the two principal fluid flow directions,
* B. N. J. Persson b.persson@fz‑juelich.de 1
PGI-1, FZ Jülich, Jülich, EU, Germany
Multiscale Consulting, Wolfshovener str 2, 52428 Jülich, Germany
2
as also found in the effective medium theory [1], and in exact numerical calculations [2], close to the percolation threshold. We consider the simplest fluid flow problems, which include the leakage of static seals [3, 4] and the squeeze-out of fluids between elastic solids. For these applications, the roughness enter only via one function, namely, the pressure flow factor 𝜙p (̄u) (in general a 2 × 2 tensor) or, equivalently, the (effective) fluid flow conductivity 𝜎eff defined by the equation
𝐉̄ = − 𝜎eff ∇̄p
(1)
where p̄ = ⟨p(x, y)⟩ is the fluid pressure and 𝐉̄ = ⟨𝐉(x, y)⟩ is the two-dimensional (2D) fluid flow current, both averaged over the surface roughness (ensemble averaging). The flow conductivity 𝜎eff is a 2 × 2 matrix (tensor). From the fluid flow conductivity, one can calculate the pressure flow factor using
𝜎eff =
ū 3 𝜙 12𝜂 p
(2)
where 𝜂 is the fluid viscosity. For randomly rough surface
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