Global dynamic modes of peristaltic-ciliary flow of a Phan–Thien–Tanner hybrid nanofluid model

In this paper, the tools of dynamical systems theory are applied to examine the streamline patterns and their local and global bifurcations for ciliary-induced peristalsis of non-Newtonian fluid (blood) with the suspension of hybrid nanoparticles (Cu–Ag/Phan–Thien–Tanner based fluid) in a tube with heat source effect. The thermodynamics of this model are recently described by Ali et al. (Biomech Model Mechanobiol 20:2393–2412, 2021), where the fluid flows through a tube whose inner walls are considered to be ciliated with small hair-like structures. However, our novel approach allows us to create a complete picture of the model’s overall dynamic behavior in terms of bifurcation point analysis exhibiting qualitatively different flow modes. Special attention is paid to the computing, analysis and simulation of equilibrium points in terms of capturing the global dynamics, such as evaluating the heteroclinic bifurcation, which is used to identify trapping phenomena in response to biological characteristics such as wave amplitude, Weissenberg and wave numbers. The main novelty here is the ability to control the position of the equilibrium points in the domain of interest, allowing one to identify global bifurcations that reflect key dynamic properties of the model. Based on the advantages of this technique, the maximum trapping volume and symmetric trapping zones adjacent to the walls are determined as a novel result. We also show that as the solid volume fraction of copper and the Brinkman number increases, the isotherm patterns become more distorted. Our findings highlight a novel class of complex behavior that governs transitions between qualitatively different modes and trapping phenomena.