Spatial decay estimate in steady motion of a micropolar fluid in a pipe

Abstract

The purpose of this paper is to deal with the entry flow problem for a micropolar fluid in steady motion in a semi-infinite cylindrical pipe. The problem to be studied is «end effect» involving comparison between two motions: the Poiseuille flow (basic flow) and another flow with the same velocity flux Φ. Our main result is an explicit estimate which establishes the rate of exponential decay, with axial distancez from the entry, of the dissipation energy of the perturbation thus providing a qualitative description of the flow development. We find that, under mild hypothesis on the asymptotic behaviour of the fields and a smallness assumption on the velocity flux Φ, the flow tends to the Poiseuille flow in the energy norm asz→+∞. Moreover we give an upper bound for the perturbation energy through the data of the problem.