Homogenization of the transient Reynolds equation

State‐of‐the‐art magnetic storage devices have head‐to‐disk distances of about 300 Angstrom, for which compressibility, slip‐flow and roughness effects are significant. Since the head and the disk are in relative motion, the air‐gap thickness when both surfaces are rough varies rapidly in both space and time. A rigorous homogenization of the transient compressible Reynolds equation appropriate for such situation is presented in this article. If ε is the roughness length and p_ε the pressure field for that roughness, the existence of p₀∈L²(0,T,H¹(Ω)) such that p_ε→p₀ strongly in L²(Ω×]0,T[) when ε→0 is proved.

A homogenized problem for p₀ is introduced together with a uniqueness result under remarkably weak assumptions, i.e., p₀∈L²(0,T,H¹(Ω)) and ∂p₀/∂t∈L²(0,T,H⁻¹(Ω)). Interestingly, no time derivatives appear in the auxiliary local problems, which are thus computed as in the steady state case. The role of the time variable is to parameterize the relative positions of the roughness shapes, and the homogenized coefficients result from averaging all such positions. To our knowledge, this is the first rigorous treatment of lubrication problems accounting for roughness on both surfaces.