A finite-volume based conservative level-set method is presented to numerically solve the non-Newtonian multiphase flow problems. One set of governing equations is written for the whole domain, and different phases are treated with variable material and rheological properties. Main challenging areas of numerical simulation of multiphase non-Newtonian fluids, including tracking of the interface, mass conservation of the phases, small timestep problems encountered by non-Newtonian fluids, numerical instabilities regarding the high Weissenberg Number Problem (HWNP), instabilities encouraged by low solvent to polymer viscosity ratio in viscoelastic fluids and instabilities encountered by surface tensions are addressed and proper numerical treatments are provided in the proposed method. The numerical method is validated for different types of non-Newtonian fluids, e.g. shear-thinning, shear-thickening and viscoelastic fluids using structured and unstructured meshes. The proposed numerical solver is capable of readily adopting different constitutive models for viscoelastic fluids to different stabilization approaches. The constitutive equation is solved fully coupled with the flow equations. The method is validated for non-Newtonian single-phase flows against the analytical solution of start-up Poiseuille flow and the numerical solutions of well-known lid-driven Cavity problem. For multiphase flows, impact of a viscoelastic droplet problem, non-Newtonian droplet passing through a contraction-expansion, and Newtonian/non-Newtonian drop deformation suspended in Newtonian/non-Newtonian matrix imposed to shear flow are solved, and the results are compared with the related analytical, numerical and experimental data.
