Shear rate projection schemes for non-Newtonian fluids

• Published in: Computer methods in applied mechanics and engineering Vol. 354; pp. 620 - 636
• Main Authors: Deteix, J., Yakoubi, D.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.09.2019; Elsevier BV
• Subjects: Boundary conditions; Carreau; Computational fluid dynamics; Computer simulation; Cosmetics industry; Cylinders; Navier–Stokes equations; Newtonian fluids; Non Newtonian fluids; Non-Newtonian; Numerical methods; Production methods; Projection; Projection scheme; Rheological properties; Shear rate; Viscosity; Navier–Stokes equations; 35Q35; Carreau; 76D05; 65M60; Non-Newtonian; Projection scheme; Shear rate
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/j.cma.2019.06.006

The operator splitting approach applied to the Navier–Stokes equations gave rise to various numerical methods for the simulations of the dynamics of fluids. The separate work of Chorin and Temam on this subject gave birth to the so-called projection methods. The most basic of those schemes, the incremental and non-incremental variant (see Guermond et al. 2006) induce an artificial Neumann boundary condition on the pressure. The so-called rotational incremental pressure-correction scheme proposed by  Timmermans et al. (1996) gives a consistent equation for the pressure in case of a Newtonian fluids with homogeneous viscosity. In this work we propose a family of projection methods for generalized Newtonian fluids based on an extension of the rotational projection scheme. Called shear rate projections, these methods produce consistent pressure when applied to generalized Newtonian fluids. Accuracy of the methods will be illustrated using a manufactured solution. Numerical experiments for the flow past a cylinder, with a Carreau rheological model, will also be presented. •Construction of the shear rate projection, a new scheme for non Newtonian flows.•Numerical schemes based on 2nd order time difference and the finite element method.•Optimal temporal and spatial accuracy are obtained using a manufactured solution.•Improved accuracy in the approximation of the pressure is illustrated with two tests.