Benchmarking of 2D and 3D numerical solvers for generalized Newtonian flows using semi-analytical solutions

The solution of realistic problems in Hemodynamics generally involves the solution of very large systems of linear and non-linear equations. These systems arise from the discretization of non-linear partial differential equations, coupling the conservation of mass and momentum with some constitutive equation expressing the relation between stresses and the history of deformation. Analytical solutions are usually not available, either because of the complex geometries of blood vessels or because of the complexity of the constitutive equations. Even considering a simple geometry like a straight tube, exact solutions are only known in the Newtonian case or for very simple shear-dependent fluids like the {\it power law fluids}. We present a general method for obtaining solutions for the flow of generalized Newtonian fluids in straigth tubes (2D and 3D), with arbitrary precision, and use these solutions to create benchmarks for numerical solvers. Some numerical examples are shown for the {\it Carreau-Yasuda} viscosity model, analyzing the performance of two numerical solvers (Adina and FreeFem++), using different finite element spaces.