The motion of a spherical particle undergoing electrophoresis in weakly inertial or viscoelastic shear flow is quantified via asymptotic analysis. We are motivated by several experimental studies reporting cross-streamline migration of electrophoretic colloids in Poiseuille microchannel flow. Specifically, particles migrate in a Newtonian liquid to the center (walls) of a channel when their electrophoretic velocity is in the opposite (same) direction to (as) the flow. Here, we calculate that weak fluid inertia causes a leading-order cross-streamline lift force of magnitude $5.50\varepsilon \left|\zeta \right|a^3\rho \dot{\gamma }{E}^{\infty }/\mu$ for electrophoresis along the velocity axis of an unbounded simple shear flow, where $\zeta$ and $a$ denote the particle zeta potential and radius, respectively; $\rho$, $\mu$, and $\varepsilon$ are the fluid density, viscosity, and permittivity, respectively; $\dot{\gamma }$ is the shear rate of the ambient flow; and $E^{\infty }$ is the strength of the imposed electric field. This force acts to propel the sphere to shear streamlines that, in a frame translating with the particle, are directed reverse to the electrophoretic motion, which is consistent with the above-mentioned experiments. Other recent experiments have observed migration of electrophoretic particles in Poiseuille flow of a viscoelastic polymer solution: the migration direction is opposite to that in a Newtonian liquid. Here, we calculate a leading-order cross-streamline lift force of magnitude $7.07(1-3.33{\mathrm{\Psi }}_2/{\mathrm{\Psi }}_1)\varepsilon \left|\zeta \right|a{\mathrm{\Psi }}_1\dot{\gamma }{E}^{\infty }/\mu$ for electrophoresis in simple shear flow of a second-order fluid, where ${\mathrm{\Psi }}_1$ and ${\mathrm{\Psi }}_2$ are the first and second normal stress coefficients, respectively. The lift is toward streamlines moving in the direction of electrophoresis for ${\mathrm{\Psi }}_2/{\mathrm{\Psi }}_1<0.3$ (a reasonable assumption for polymeric liquids, where ${\mathrm{\Psi }}_2$ and ${\mathrm{\Psi }}_1$ are negative and positive, respectively), which is consistent with the experiments. Finally, an estimation of the magnitude of the lift forces and associated drift velocities further suggests that the cumulative effect of weak instantaneous inertia or viscoelasticity is responsible, or at least contributes appreciably, to the observed migration.

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• Electrokinetic flows
• Non-Newtonian fluids
• Charged colloids
• Navier-Stokes equation