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

Abstract

The aim of this paper is to deal with the classical entry flow problem for a Bingham fluid in steady motion in a cylindrical pipe of arbitrary cross section and of finite or semi-infinite length. The problem to be treated is “end effect” involving comparison between two motions: the Poiseuille-Bingham flow (basic flow) and another flow with the same flux. Our main result is an explicit estimate which establishes the rate of decay, with axial distance from the entry, of a suitable energy functional thus providing a qualitative description of the flow development. The decay estimates are derived using differential inequality techniques. If the pipe is finite, we deduce that the energy of the perturbation decays exponentially provided the Reynold's number R satisfies a smallness condition involving the Bingham number B, the pressure gradient of the Poiseuille-Bingham flow, and the geometrical features of the cross-section. More precisely we find the exponential decay if R < R B where R B is an increasing function in B. We show that for a Bingham fluid the range of R is larger than the range for a Newtonian fluid with the same features as it is known from a phenomenological point of view. If the pipe is semi-infinite we obtain, under a mild asymptotic hypothesis on the stretching tensor, that the energy of the perturbation is finite and it decays like In z/z 2 as z → + ∞.