Cross-stream migration of non-spherical particles in a general second-order fluid flows

We have developed a theory to investigate the cross-stream migration of ellipsoids in a weakly viscoelastic fluid under various pressure-driven flow profiles (circular tube flows and slit flows). The viscoelastic fluid we investigate is a general second-order fluid characterized by Weissenberg number Wi$=\psi $1 $\gamma $/$\mu $ and constant $\alpha =\psi $2/$\psi $1, where the first and second normal stress coefficients are $\psi $1 and $\psi $2, $\gamma $ is the characteristic shear rate of the flow, and $\mu $ is the total viscosity. Considering the limit of weakly viscoelastic flow (Wi$\ll $1), we use perturbation theory and the reciprocal theorem to derive both the polymeric force and torque on a particle to O(Wi). Our theory is valid for an ellipsoid in any quadratic flow field, and have validated the theory for three cases: (a) sedimentation of a general ellipsoid in a second order fluid, (b) particle migration of a sphere in a pressure driven flow, and (c) particle migration of an ellipsoid in a pressure driven flow under the co-rotational limit ($\alpha =$ -1/2). After verification, we use our theory to compute how the first and second normal stresses affect the motion of a particle to towards the center of pressure driven flows, and discuss the orientation dynamics. Scaling results are presented for the particle migration speed.