A finite element method for a non-Newtonian dilute polymer fluid

Abstract

We study the discretisation of a uniaxial (rank-one) reduction of   the Oldroyd--B model for dilute polymer solutions, in which the   conformation tensor is represented as $\sig=\vec b\otimes\vec   b$. Building on structural analogies with magnetohydrodynamics   (MHD), we formulate a finite element framework compatible with the   de~Rham complex, so that the discrete velocity is exactly   divergence-free. The spatial discretisation combines an   interior-penalty treatment of viscosity with upwind transport to   control stress layers, and we prove inf--sup conditions for the   mixed pairs. For time-stepping, we design an IMEX scheme that is   linear at each step and show well-posedness of the fully discrete   problem together with a discrete energy law mirroring the continuum   dissipation for the unmodified reduced model. Numerical experiments   include a manufactured-solution test and canonical benchmarks   comprising lid-driven cavity, pipe-with-cavity, $4{:}1$ planar   contraction and constrained flow past a cylinder. The computations   demonstrate spatial accuracy and resolve sharp stress gradients and   corner singularities while using the reduced number of constitutive   unknowns provided by the uniaxial model. The results indicate that   de~Rham-compatible discretisations coupled with energy-stable IMEX   time integration provide a structure-preserving framework for   computations at elevated elasticity. ​