Comment on “Migration of an electrophoretic particle in a weakly inertial or viscoelastic shear flow”

Abstract

A recent article [Khair and Kabarowski Phys. Rev. Fluids 5, 033702 (2020)] has studied the cross-streamline migration of electrophoretic particles in unbounded shear flows with weak inertia or viscoelasticity. That work compares their results with those reported in two of our previous studies [Choudhary et al. J. Fluid Mech. 874, 856 (2019); J. Fluid Mech. 898, A20 (2020)] and reports a disagreement in the derived analytical expressions. In this comment, we resolve this discrepancy. For viscoelastic flows, we show that Khair and Kabarowski have not accounted for a leading order surface integral of polymeric stress in their calculation of first-order viscoelastic lift. When this integral is included, the resulting migration velocity matches exactly with that reported in our work [J. Fluid Mech. 898, A20 (2020)]. This qualitatively changes the migration direction that is reported by Khair and Kabarowski for viscoelastic flows. For inertial flows, we clarify that Khair and Kabarowski find the coefficient of lift to be $1.75\ensuremath{\pi}$ compared to $2.35\ensuremath{\pi}$ in our previous work [J. Fluid Mech. 874, 856 (2019)]. We show that this difference occurs because Khair and Kabarowski accurately include the effect of a rapidly decaying $\ensuremath{\sim}O(1/{r}^{4})$ velocity field (a correction to the stresslet field $\ensuremath{\sim}1/{r}^{2}$), which was neglected in our work.