Error estimates for a viscosity-splitting scheme in time applied to non-Newtonian fluid flows

A time fractional-step method is presented for numerical solutions of the incompressible non-Newtonian fluids for which the viscosity is non-linear depending on the shear-rate magnitude according to a generic model. The method belongs to a class of viscosity-splitting procedures and it consists of separating the convection term and incompressibility constraint into two time steps. Unlike the conventional projection methods, the viscosity is not dropped in the last step allowing to enforce the full original boundary conditions on the end-of-step velocity which eliminates any concerns about the numerical boundary layers. We carry out a rigorous error analysis and provide a full first-order error estimate for both the velocity and pressure solutions in the relevant norms. Numerical results are presented for two test examples of non-Newtonian fluid flows to demonstrate the theoretical analysis and confirm the reliability of this viscosity-splitting scheme