Obtaining the shear rate profile of steady laminar tube flow of Newtonian and non-Newtonian fluids from nuclear magnetic resonance imaging and laser Doppler velocimetry data

Abstract

The problem of extracting shear rate profiles from experimentally measured velocity profiles in steady laminar tube flow is formulated as an integral equation of the first kind to reveal its ill-posed nature. A procedure, based on Tikhonov regularization, is then developed to solve this problem. The performance of the procedure is assessed by applying it to the measured velocity profile of a number of fluids of increasing rheological complexity. These velocity profiles, taken from recent literature, are representative of what is currently achievable with nuclear magnetic resonance imaging and laser Doppler techniques. The shear rate profiles thus obtained are compared against those obtained by existing methods of treating this problem. For velocity profiles where the associated pressure gradient is available, the outcome of Tikhonov regularization is converted into a viscosity versus shear rate relationship and compared against that obtained by direct viscometric measurement. The results show that Tikhonov regularization is a reliable method and has the added advantage of not requiring the assumption of a rheological constitutive equation. It can therefore be applied with equal ease to non-Newtonian, including those with yield stress, as to Newtonian fluids.