Traceless decomposition of Johnson-Segalman viscoelastic models for blood flow simulations

The flow of blood (and some other biological fluids) can in certain regimes quite accurately be described by viscoelastic models of Oldroyd-type. Of course, in the case of blood flow these models should be properly generalized to account for shear-thinning effects. This has been done e.g. in previous works by Bodnar, Sequeira and co-workers in [1]-[5]. This contribution presents an alternative reformulation of the Oldroyd-type models describing viscoelastic fluids. For simplicity we only use the constant viscosity model to explain the main ideas, however the extension for shear-thinning variable viscosity generalizations is not only possible, but is trivial. It is well known that the Newtonian part of the stress tensor has only deviatoric (traceless) part, i.e. its trace is vanishing due to incompressibility (divergence-free constraint). The spherical part of the stress tensor is in Newtonian case removed to pressure. This behavior is however not shared by the viscoelastic extra stress tensor. In the above mentioned classical formulation the trace of the elastic extra-stress tensor is non-vanishing. The aim of this contribution is to reformulate the model in the way that separates the deviatoric and spherical parts of the elastic stress tensor. It will be shown that it is possible to reformulate the model in the form where a modified governing system will be derived for the deviatoric part of the extra stress and another equation for a new quantity called elastic pressure. This new system is equivalent to the classical one (in terms of solution), but has some advantages over it. The new formulation allows for easier quantification of fluid-to-wall forcing in fluid-structure interaction problems. This alternative formulation also seems to be more suitable in studying and understanding the high Weissenberg number problem which is one of the most challenging problems of the non-Newtonian fluid mechanics. References [1] Bodnar, T., Rajagopal, K., & Sequeira, A.: Simulation of the Three-Dimensional Flow of Blood Using a Shear-Thinning Viscoelastic Fluid Model. Mathematical Modelling of Natural Phenomena, vol. 6, no. 5: (2011) pp. 1–24. [2] Bodnar, T. & Sequeira, A.: On the Blood Flow Simulations in Axisymmetric Aneurysm Using Generalized Oldroyd-B Model. In: Topical Problems of Fluid Mechanics 2009, (pp. 13–16). Institute of Thermomechanics AS CR, Prague (2009). [3] Bodnar, T. & Sequeira, A.: Numerical Study of the Significance of the Non-Newtonian Nature of Blood in Steady Flow Through a Stenosed Vessel. In: Advances in Mathematical Fluid Mechanics (edited by R. Rannacher & A. Sequeira), (pp. 83–104). Springer Verlag (2010). [4] Bodnar, T., Sequeira, A., & Prosi, M.: On the Shear-Thinning and Viscoelastic Effects of Blood Flow under Various Flow Rates. Applied Mathematics and Computation, vol. 217, no. 11: (2011) pp. 5055–5067. [5] Pirkl, L., Bodnar, T., & Tuma, K.: Viscoelastic Fluid Flows at Moderate Weissenberg Numbers Using Oldroyd Type Model. In: AIP Conference Proceedings, vol. 1389, (pp. 102–105). American Institute of Physics (2011). ISBN 978-0-7354-0956-9