Affiliations: Department of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand | Département de mathématiques et de statistique, Université de Montréal, Montréal, QC, Canada
Note: [] Address for correspondence: Robert G. Owens, Département de mathématiques et de statistique, Université de Montréal, CP 6128 succ. Centre-Ville, Montréal, QC H3C 3J7, Canada. Tel.: +1 514 343 2315; Fax: +1 514 343 5700; E-mail: owens@dms.umontreal.ca.
Abstract: In an earlier paper, Moyers-Gonzalez et al. [J. Fluid. Mech. 617 (2008), 327–354] used kinetic theory to derive a non-homogeneous haemorheological model and applied this to simulate the properties of steady flow of blood in a tube. By adjusting the tube haematocrit to match that of the experimental fitted curve of Pries et al. [Circ. Res. 67 (1990), 826–834] the authors showed that it was possible to quantitatively predict the apparent viscosity values presented in a later paper by Pries et al. [Am. J. Physiol. 263 (1992), 1770–1778]. In the present paper, it is the discharge haematocrit rather than the tube haematocrit that is prescribed. We further develop the predictive capacities of the original model of Moyers-Gonzalez et al. [J. Fluid. Mech. 617 (2008), 327–354] by introducing a cell-free peripheral layer next to the tube wall where, following the ideas of Sharan and Popel [Biorheology 38 (2001), 415–428], dissipation in this layer is accounted for by allowing the viscosity there to exceed that of plasma. Using both the apparent viscosity data of Pries et al. [Am. J. Physiol. 263 (1992), 1770–1778] and the relative tube haematocrit relation proposed by Sharan and Popel [Biorheology 38 (2001), 415–428], we predict the thickness of the cell-free layer and the relative viscosity in this layer. The predicted thickness of the cell-free layer as a function of both a pseudo-shear rate and the tube diameter for 45% haematocrit blood is shown to be in very close conformity with the experimental measurements of Reinke et al. [Am. J. Physiol. 253 (1987), 540–547]. With increasing discharge haematocrit the cell-free layer thickness is shown to decrease, as observed in several experimental papers [Bugliarello and Hayden, Trans. Soc. Rheol. VII (1963), 209–230, Bugliarello and Sevilla, Biorheology 7 (1970), 85–107, Soutani et al., Am. J. Physiol. 268 (1995), 1959–1965]. Our prediction of the relative viscosity in the cell-free layer shows a similar trend to that computed by Sharan and Popel [Biorheology 38 (2001), 415–428]. Finally, for sufficiently large pseudo-shear rates it is shown that the Deborah number (a non-dimensional relaxation time) may be taken to be a constant, thus greatly simplifying our haemorheological model and allowing for a partially analytic solution to the problem of steady non-homogeneous flow of blood in a tube.
Keywords: Cell-free peripheral layer, non-homogeneous flow, kinetic theory modelling
DOI: 10.3233/BIR-2010-0558
Journal: Biorheology, vol. 47, no. 1, pp. 39-71, 2010
