Least‐squares formulation to solve non‐Newtonian fluid flow and application of data assimilation in 2D

Abstract

In this contribution, we present an approach to model steady flow of incompressible non-Newtonian fluids with data assimilation into the numerical solution based on the least-squares finite element method (LSFEM). The assimilation of data (e.g. experimental or analytical) into numerical simulations offers promising possibilities when examining complex problems. Potential applications include the enhancement of numerical models using measured data or the completion of experimental data using numerical methods, e.g. the determination of non-measured quantities such as pressure. In particular for the field of fluid mechanics, the implemented LSFEM has some theoretical advantages compared to the well-known (mixed) Galerkin finite element method, since it is not restricted to the LBB-condition. Additionally, it results in a minimization problem with symmetric positive (semi-)definite equation systems also for differential equations with non-self-adjoint operators. A further advantage in this context is that the assimilation of data can easily be performed by adding a term to the least-squares (LS) functional such that it does not significantly increase the computational cost. The approach of data assimilation is shown by solving steady flow of a non-Newtonian fluid through a channel with a smooth contraction.