The use of computational modelling tools has become indispensable across many engineering applications. In computational fluid dynamics, problems involving viscoelastic fluid flows remain particularly challenging because these materials exhibit a broad range of complex and often counter-intuitive behaviours, such as shear-dependent viscosity, non-linear stress transport, elastic instabilities, and elasto-inertial turbulence. These phenomena are central to many applications, from biomedical microfluidic mixing devices, where elastic instabilities and turbulence are exploited to enhance mixing, to polymer processing technologies, where elasto-inertial turbulence makes process optimisation more challenging. Nevertheless, viscoelastic fluid flows remain poorly understood, especially regarding the fundamental mechanisms originating instabilities and turbulence. Numerical simulation is therefore a valuable tool to investigate the underlying physics these fascinating fluids.
The numerical simulation of viscoelastic fluid flows is especially demanding. In addition to the classical difficulties encountered in Newtonian fluid flows, such as the pressure-velocity coupling, further challenges arise, including the strong non-linearity of constitutive models, elastic instabilities at low Reynolds numbers, elasto-inertial turbulence at high Reynolds numbers, and the long-standing high Weissenberg number problem (HWNP). Indeed, the HWNP has been regarded as a key challenge in computational rheology, and developing robust and stable numerical methods for simulating highly elastic fluid flows has become a major research subject. Beyond these fundamental aspects of polymer rheology, industrial applications introduce additional complexity, being characterised by multiphysics, multi-phase, and multi-domain interactions, and intricate geometries. As a result, simulations of viscoelastic fluid flows are not only computationally expensive but also prone to numerical breakdowns, often failing to yield reliable and physically admissible solutions.
Industries frequently resort to model simplifications and coarse discretisations to reduce computational costs, which compromises both the physical meaning and the numerical accuracy of simulations, resulting in longer development cycles and higher costs. Developing efficient and reliable numerical methods and algorithms for viscoelastic fluid flow simulations therefore carries not only scientific relevance from the fundamental physical viewpoint but also substantial economic and environmental impact on the competitiveness and sustainability of industries.