Interfacial fluid flow for systems with anisotropic roughness

Abstract

I discuss fluid flow at the interface between solids with anisotropic roughness. I show that the Bruggeman effective medium theory and the critical junction theory give nearly the same results for the fluid flow conductivity. This shows that, in most cases, the surface roughness observed at high magnification is irrelevant for fluid flow problems such as the leakage of static seals, and fluid squeeze-out. The effective medium theory predicts that the fluid flow conductivity vanishes at the relative contact area A/A 0 = 0.5 independent of the anisotropy. However, the effective medium theory does not solve the elastic contact mechanics problem but is based on a purely geometric argument. Thus, for anisotropic roughness the contact area may percolate at different values of A/A 0 depending on the direction. We discuss how this may be taken into account in the effective medium and critical junction theories.

Comments on the Theory of Fluid Flow Between Solids 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 and 1∕ along the two principal fluid flow directions, as also found in the effective medium theory [1], and in exact numerical calculations [2], close to the percolation threshold.…”

“…For randomly rough surfaces, the flow conductivity eff can be calculated approximately using the Bruggeman effective medium [1,4] or the simpler critical junction theory [3]. These theories were originally developed for isotropic roughness but have been generalized to surfaces with anisotropic roughness, where the roughness is characterized by the Peklenik number [5,6] .…”

“…7 in Ref. [1]. Results were obtained using the effective medium (em) theory for = 1∕4 (red curve) and = 4 (blue curve).…”

“…This was attempted in Ref. [1] but unfortunately, a factor was placed in the wrong place as described in Ref. [2].…”

“…. From these equations, one can show that at percolation [1] Thus, given A p ∕A 0 obtained from (exact) numerical simulations, if we choose then the (modified) effective medium theory will result in a flow conductivity which vanishes when the (normalized) contact area reaches the value A p ∕A 0 where the contact area percolates.…”