Abstract
We study the fully-developed, time-periodic motion of a shear-dependent non-Newtonian fluid with variable exponent rheology through an infinite pipe [Formula: see text], [Formula: see text], of arbitrary cross-section [Formula: see text]. The focus is on a generalized [Formula: see text]-fluid model, where the power-law index is position-dependent (with respect to [Formula: see text]), i.e., a function [Formula: see text]. We prove the existence of time-periodic solutions with either assigned time-periodic flow-rate or pressure-drop, generalizing known results for the Navier–Stokes and for [Formula: see text]-fluid equations.
In addition, we identify explicit solutions, relevant as benchmark cases, especially for electro-rheological fluids or, more generally, ‘smart fluids’. To support practical applications, we present a fully-constructive existence proof for variational solutions by means of a fully-discrete finite-difference/-element discretization, consistent with our numerical experiments. Our approach, which unifies the treatment of all values of [Formula: see text], [Formula: see text], without requiring an auxiliary Newtonian term, provides new insights even in the constant exponent case. The theoretical findings are illustrated by means of numerical experiments.