SPH simulations of integral fractional viscoelastic models

To capture specific characteristics of non-Newtonian fluids, fractional constitutive models have gained significant attention in recent years [1]. These models provide a simple yet effective framework to describe the complex viscoelastic behaviour of materials: they are particularly advantageous in capturing materials with a wide relaxation spectrum, such as those with a power-law behaviour in their relaxation processes [2]. The implementation of these models requires numerical frameworks for integral constitutive equations [3]. In this context, a Lagrangian Framework simplifies the required integration along streamlines compared to an Eulerian approach, where reconstructing flow history is necessary and computationally expensive. By employing a Lagrangian Smoothed Particle Hydrodynamics (SPH) method, we can efficiently track particle history, enabling us to solve integral constitutive models in an innovative manner that avoids the need for complex computational tasks. Our SPH-based integral fractional viscoelastic method has been validated using classical frameworks like Oldroyd-B and multi-mode Maxwell models, under benchmark flows such as Small Amplitude Oscillatory Shear (SAOS), Kolmogorov, and Poiseuille [4]. Building upon this foundation, we have extended our analysis to address non-linear flow scenarios. Complex geometries, such as the flow around a cylinder, have been analysed. Moreover, we focused on Large Amplitude Oscillatory Shear (LAOS), which represents a non-linear regime, implemented using a fractional K-BKZ integral model. Unlike SAOS, LAOS cannot be fully characterized by elastic and loss moduli alone, necessitating a deeper investigation through Fourier analysis to capture higher-order harmonics to describe the material's response under strong deformation [5]. References.[1] A. Jaishankar and G. H. McKinley, J. Rheo. 58, 1751 (2014). [2] Pan Yangb, Yee Cheong Lama, Ke-Qin Zhub, J. Non-Newtonian Fluid Mech. 165, 88 (2010). [3] X.-L. Luo and R. I. Tanner, International J. Numerical Methods in Engineering 25, 9 (1988). [4] Luca Santelli, Adolfo Vazquez-Quesada, and Marco Ellero. J. of Non-Newtonian Fluid Mechanics 105235 (2024) [5] K Hyun, M Wilhelm, CO Klein, KS Cho, JG Nam, KH Ahn, SJ Lee, RH Ewoldt, & GH McKinley. Progress in polymer science, 36(12), 1697-1753 (2011).