Asymptotic behaviour of Stokes flow in a thin domain with a moving rough boundary
John Fabricius, Yulia Koroleva, Afonso Fernando Tsandzana, Peter Wall · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences · 2014
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 .