Analytical solutions for mass transport in hydrodynamic focusing by considering different diffusivities for sample and sheath flows

The fluid flow and mass transfer characteristics in two-dimensional hydrodynamic focusing are theoretically investigated by considering different physical properties for the sample and sheath flows. Adopting a single-domain formulation, which assigns the region variable physical properties, three-dimensional analytical solutions are obtained for species transport under hydrodynamically fully developed conditions. In addition, simplified analytical solutions are derived assuming a uniform velocity field appropriate to electrokinetic focusing. The results show that the normalized overall mean velocity is an increasing function of the height to width ratio and a decreasing function of the sheath to sample viscosity ratio. The dependence of this normalized mean velocity on the sheath to sample flow-rate ratio is, however, non-monotonic: it grows with the flow-rate ratio when the sample fluid is more viscous than the sheath fluid, whereas the opposite is true when the sheath fluid is more viscous. Moreover, although an increase in either the viscosity or flow-rate ratios results in creating a smaller value of the normalized focused width, varying the channel aspect ratio may lead to either thinner or thicker focused regions, depending on the viscosity ratio. The inspection of the mass transport characteristics reveals that only the viscosity ratio and the Peclet number can significantly alter the mixing length. Surprisingly, the minimum mixing length in the presence of significant axial diffusion is achieved for a single-phase flow. Finally, the dimensionless mixing length is reduced by increasing the Peclet number, although the changes are negligible when this parameter is above 10. This threshold is only true when the Peclet number is calculated based on the higher of the sample and sheath diffusion coefficients.