A finite element method for a microstructure-based model of blood

This paper presents the first flows computed in non-trivial geometries while accounting for the contribution of the red cells to the Cauchy stress using the haemorheological model of Owens and Fang (J. Non-Newtonian Fluid Mech. 2006; 140:57-70; Biorheology 2006; 43:637-660). In this model, the local shear viscosity is determined in terms of both the local shear-rate and the average rouleau size, with the latter being the solution of an advection-reaction equation. The model describes the viscoelastic, shear-thinning and hysteresis behaviour of flowing blood, and includes non-local effects in the determination of the blood viscosity and stresses. This rheological model is first briefly derived. A finite element method is next presented, extending the DEVSS method of Fortin and coworkers (Comput. Methods Appl. Mech. Engrg. 2000; 189:121-139; J. Non-Newtonian Fluid Mech. 1995; 60:27-52; Comput. Methods Appl. Mech. Engrg. 1997; 143:79-95) to the solution of this Oldroyd-B type model but with a non-constant Deborah number. A streamline upwind Petrov-Galerkin approach is also adopted in the discretization of the constitutive equation and the microstructure evolution equation. The numerical results presented begin with validation of our finite element scheme in a coaxial rheometer both for a prescribed homogeneous velocity field and for a fully two-dimensional calculation including the solution of the linear momentum equation. The consideration of blood flow in a straight channel allows us to test convergence of our numerical scheme. Finally, we present results for an aneurytic channel under both steady and pulsatile flow conditions. Comparisons are made with the results from an equivalent Newtonian fluid. Our choice of material parameters leads to only weakly elastic effects but noticeable differences are seen between the Newtonian and non-Newtonian flows, especially in the pulsating case. Copyright (C) 2011 John Wiley & Sons, Ltd.