Pulsatile Flows for Simplified Smart Fluids with Variable Power-Law: Analysis and Numerics

We study the fully-developed, time-periodic motion of a shear-dependent non-Newtonian fluid with variable exponent rheology through an infinite pipe Ω:=ℝ×Σ⊆ℝ^d, d∈{2,3}, of arbitrary cross-section Σ⊆ℝ^(d−1). The focus is on a generalized p(⋅)-fluid model, where the power-law index is position-dependent (with respect to Σ), i.e., a function p:Σ→(1,+∞). 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 p-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-differences/-elements discretization, consistent with our numerical experiments. Our approach, which unifies the treatment of all values of p(x)∈(1,+∞), x∈Σ, without requiring an auxiliary Newtonian term, provides new insights even in the constant exponent case. The theoretical findings are reviewed by means of numerical experiments.