Effects of rough boundary on the heat transfer in a thin-film flow

Abstract

In this Note, a heat flow through a rough thin domain filled with fluid (lubricant) is studied. The domainʼs thickness is considered as the small parameter ε, while the roughness is defined by a periodical function with a period of order ε2. We assume that the lubricant is cooled by the exterior medium and we describe the heat exchange on the rough part of the boundary by Newtonʼs cooling law. Depending on the magnitude of the heat transfer coefficient with respect to ε, we obtain three different macroscopic models via formal asymptotic analysis. We identify the critical case explicitly acknowledging both roughness-induced effects and the effects of the surrounding medium on heat transfer at main order. We illustrate the obtained results by some numerical simulations.