Spatial decay estimate in the problem of entry flow for a Bingham fluid filling a pipe.

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. Our main result is an explicit estimate which establishes the rate of exponential decay, with axial distance z 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, the flow tends to the Poiseuille flow in the energy norm as z tends to +infinity. Moreover we give an upper bound for the perturbation energy through the data of the problem.