Numerical simulation of steady Bingham flow through an eccentric annular cross-section by distributed Lagrange multiplier/fictitious domain and augmented Lagrangian methods

This paper deals with the numerical simulation of the uniaxial steady flow of a viscoplastic fluid through an eccentric annular cross-section. The situation considered here refers to mud flow in drilling operations or cement slurry flow in cementing process in the oil and gas industry. The rheological model employed is the classical isothermal Bingham model. The steady solution of the governing equations is obtained through a transient algorithm that allows to compute solutions at an imposed flow rate. Mathematical difficulties related to the Bingham model are overcome thanks to the use of a Lagrange multiplier and an augmented Lagrangian/Uzawa method. The inner cylinder is treated as a fictitious domain (FD) in which a zero velocity constraint is imposed. The velocity constraint is relaxed by the introduction of a distributed Lagrange multiplier (DLM). The governing equations are discretized using a finite element method with triangular elements. The resulting numerical method highlights strong and robust convergence properties. The use of the DLM/FD method enables to study the influence of the eccentricity very conveniently. In particular, we show the effect of the mesh refinement on the solution accuracy in relation with the use of the DLM/FD method. Results obtained underline the effect of the eccentricity on the flow pattern. Finally, we propose an engineering response surface methodology (RSM) approach to predict the pressure drop.