Viscoelastic phase separation: Well-posedness and numerical analysis
Date issued
Authors
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
License
Abstract
Viscoelastic phase separation describes dynamically asymmetric demixing of polymer solutions after a deep quench, i.e., after a sudden decrease in the temperature.
The dynamic asymmetry of the polymer solution gives rise to new and more complex phenomena than the standard phase separation of binary fluids. Coupled with the incomplete timescale separation, the process forms a complex multiscale problem.
Phenomenological continuum mechanical models for standard phase separation, e.g., model H, are insufficient to capture the enriched dynamics observed in experiments for viscoelastic phase separation. Hence, a key difficulty is the derivation of more complex models which resemble experimental data and preserve fundamental principles of physics, e.g., conservation of mass, momentum, and the second law of thermodynamics.
For a suitable phenomenological model, we consider mathematical well-posedness of the problem, i.e., existence, uniqueness and stability of solutions. The models are complex nonlinear parabolic systems of partial differential equations with an energy-dissipative structure based on a non-convex free energy functional. The key difficulties arise due to a strongly nonlinear cross-diffusive coupling of one subsystem and a logarithmic type of free energy for another subsystem.
We prove the global-in-time existence of dissipative weak solutions in two and three space dimensions using the energy method. Additionally, we employ relative energy methods to derive an abstract stability result. As an application, this approach yields the weak-strong uniqueness principle for dissipative weak solutions.
For the numerical approximation and the corresponding error analysis, it is suitable to derive numerical methods which preserve the second law of thermodynamics also on the discrete level. Key difficulties are the correct discretisation of the convective terms on the discrete level and suitable time integration methods for the non-convex energy.
For the semi-discretisation of a reduced model, we consider conforming inf-sup finite elements in space and analyse the corresponding semi-discrete problem. The thermodynamic properties are preserved by the Galerkin method, hence using a discrete version of the nonlinear stability estimate allows us to deduce the optimal second-order accuracy in a transparent and structured way.
In the fully discrete case, we employ a variational time discretisation via a Petrov-Galerkin method on the semi-discrete system. Time-discrete thermodynamic structure is preserved and together with a fully discrete stability estimate the corresponding error analysis is derived. This allows us to deduce here the optimal second-order accuracy in space and time using realistic smoothness assumptions. Theoretical error estimates of the semi-discrete and fully discrete scheme are illustrated by a series of numerical experiments.