Estimation of viscosity profiles using velocimetry data from parallel flows of linearly viscous fluids: application to microvascular haemodynamics

Abstract

An approach is presented that uses velocimetry data to estimate accurately the spatial distribution of viscosity in steady laminar parallel flows of incompressible linearly viscous fluids. The approach is generally applicable to Newtonian fluids with spatially varying viscosity or to particle-suspension flows where a non-uniform distribution of the particles contributes to spatial variations in the local effective viscosity of the suspension. Emphasis is placed on the application of these methods to steady axisymmetric blood flow in cylindrical glass capillary tubes and microvessels. In this context, the spatial variations in viscosity over the vessel cross-section are predicted where it is assumed that the rheological properties associated with a heterogeneous red blood cell suspension can be well approximated by a continuous generalized linearly viscous fluid having a spatially non-uniform viscosity. For such a fluid, an expression for the viscosity profile over the vessel cross-section is derived that satisfies the conservation principles of mass and momentum and depends upon the a priori determined velocity distribution, which is extracted from fluorescent micro-particle image velocimetry data obtained from microvessels in vivo. These profiles provide useful information about dynamic, kinematic and rheological properties of the flow that include expressions for the axial pressure-gradient component, the local shear stress distribution, and the relative apparent viscosity. In microvessels, the effect of the glycocalyx surface layer on the vessel wall is also accounted for in the analysis by modelling the layer as a uniformly thick porous medium. Velocimetry data are presented from in vivo measurements made in venules after the application of a light-dye treatment to degrade the glycocalyx. Results reveal that these methods are sufficiently sensitive to detect a reduction in glycocalyx thickness of ${\sim}\,0.3\,\umu{\rm m}$, which represents a fractional decrease in thickness of ${\sim}\,60$–70% when compared with results from a separately published data set obtained from venules having an intact glycocalyx.