Theoretical Studies of Cross-Stream Migration Non-Spherical Particles in an Arbitrary Quadratic Flow of a General Second-Order Fluid

Cross-stream particle migration in viscoelastic suspensions is essential in many biological as well as industrial processes, yet the effect of particle shape on this phenomenon is not well understood. In this work, we perform a theoretical study on the dynamics of an arbitrary-shaped particle in a quadratic flow of a general second-order fluid. In this analysis, we solve for the polymeric force and torque on a particle by performing a perturbation expansion on the fluid velocity and pressure field using the Weissenberg number (Wi) as a small parameter. The force and torque are evaluated to O(Wi) using the Lorentz reciprocal theorem. The total polymeric force and torque can be summed into two parts: (1) an analytical solution where ψ₁= -2ψ₂ (ψ₁ and ψ₂ being the first and second normal stress coefficients), and (2) a remaining part where ψ₁≠ -2ψ₂ that requires numerical evaluation. The total solution compares well to previous studies on spherical particles. We then apply the derived solutions to investigate the migration and orientation dynamics of ellipsoidal particles in different quadratic flow profiles (e.g., slit-like flows, tube-like flows). We will identify the key factors that governs the migration dynamics of the particles in different non-Newtonian fluids (dilute polymer solutions, emulsions, and colloidal dispersions) based on the ratio of normal stress coefficients. We finally compare the results of the general model and to previous models that made the co-rotational assumption of ψ₁= -2ψ₂, and comment on the effectiveness of the latter model in describing qualitative physics.