Abstract
Choosing a suitable model and determining its associated parameters from fitting to experimental data is fundamental for many problems in biomechanics. Models of shear-thinning complex fluids, dating from the work of Bird, Carreau, Cross and Yasuda, have been applied in highly-cited computational studies of hemodynamics for several decades. In this manuscript we revisit these models, first to highlight a degree of uncertainty in the naming conventions in the literature, but more importantly to address the problem of inferring model parameters by fitting to rheology experiments. By refitting published data, and also by simulation, we find large, flat regions in likelihood surfaces that yield families of parameter sets which fit the data equally well. Despite having almost indistinguishable fits to experimental data these varying parameter sets can predict very different flow profiles, and as such these parameters cannot be used to draw conclusions about physical properties of the fluids, such as zero-shear viscosity or relaxation time of the fluid, or indeed flow behaviours. We verify that these features are not a consequence of the experimental data sets through simulations; by sampling points from the rheological models and adding a small amount of noise we create a synthetic data set which reveals that the problem of parameter identifiability is intrinsic to these models.
Highlights
Many complex fluids exhibit shear rate-dependent viscosity; suspensions, fluids of biological importance such as blood, and biological polymers, such as mucus, are typically shear-thinning, i.e. their viscosity reduces with increasing shear rate
Several models have been proposed for this behaviour and have been studied intensively; we will focus on a class of models which relate shear viscosity to shear rate via nonlinear algebraic equations, in particular the formulations of Cross, Bird, Carreau and Yasuda, and their subsequent application to blood rheology
We address two significant issues – first, an inconsistency in the literature regarding naming of models, and more importantly, some significant difficulties which appear in determining model parameters through least squares fitting
Summary
Many complex fluids exhibit shear rate-dependent viscosity; suspensions, fluids of biological importance such as blood, and biological polymers, such as mucus, are typically shear-thinning (pseudo-plastic), i.e. their viscosity reduces with increasing shear rate. The earliest model of Ostwald (1925) and de Waele (1923) is based on a power-law dependence of viscosity on shear rate; limitations of this simple model include its singularity at zero shear rate and inability to capture high shear rate dependency when compared to empirical data. In a study of polystyrene fluids, Yasuda (1979) modified this formulation to include a further parameter a to describe better the low shear to power-law transition region: This model has four free parameters ðl0; k; n; aÞ, and implies a zero viscosity limit as shear rate tends to infinity. We refer to Eqs. (1)–(4) as the Cross-1965, Carreau-1972, Yasuda-1979 and BCCY-1987 (Bird-Cross-Carreau-Yasuda) models respectively
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