Simulation techniques for viscoelastic fluids with zero solvent viscosity based on three field approaches
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Date
2024
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Abstract
Solving viscoelastic fluid flow problems is challenging task due to their complex behavior, which involves the coupling of elastic and viscous stresses in a highly nonlinear manner. Additionally, numerically simulating pure polymer melts is particularly challenging due to the absence of solvent contributions to viscosity in the standard viscoelastic model. The absence of a diffusive operator in the momentum equation prevents the problem from being addressed in a decoupled manner and imposes limitations on the application of solution methods, generally rendering multigrid solvers impractical with a monolithic approach. This thesis aims to present a finite element method for solving the two-dimensional three-field Stokes flow for pure polymer melts, using the Elastic Viscous Stress Splitting (EVSS) and Tensor Stokes formulation. The formulation is expressed in terms of velocity, pressure, and the stress tensor. Both the EVSS and Tensor Stokes formulations help to reintroduce velocity coupling into the momentum equation by applying a change of variables in the standard viscoelastic formulation. This approach enables the problem to be handled in a decoupled manner and facilitates the application of multigrid solution methods using a monolithic approach. Nevertheless, this change of variables introduces additional terms with second-order velocity derivatives in the convective part of the constitutive equation for stress. To address this, the four-field approach is often employed, which includes the deformation tensor as an additional field to manage higher-order derivatives. The convective term is reformulated by taking into account the divergence-free nature of the velocity field, shifting the higher-order derivatives to the test function in the weak formulation thus
maintaining the problem size to three field. The velocity, pressure, and stress are discretized using the higher-order
disc stable FEM triplet (Q2,P1 ,Q3). The proposed scheme is evaluated using the Oldroyd-B, Giesekus, and PTT
exponential fluids, employing both decoupled and monolithic solution approaches. Numerical results are obtained for a four-to-one curved contraction, for highly viscoelastic fluids with the aim to achieve results at relatively large values of the relaxation parameter λ and observe the shear-thinning effect w.r.t. the relaxation parameter λ.
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Keywords
Elastic Viscous Stress Splitting, Tensor Stokes formulation, Viscoelastic fluids, Pure polymer melts, Finite Element Method, Decoupled and monolithic approaches, Shear thinning effect, Multigrid