Electrokinetic Power-Series Solution in Narrow Cylindrical Capillaries for All Zeta Potentials

Work from Rice and Whitehead showed the results of electrokinetic flow in a capillary tube under the assumption of low zeta potential 25 mV, limiting the approximation's usability. Further research conducted by Philip and Wooding provided an alternative solution that assumes high zeta potentials $>$ 25 mV and relies on Rice and Whitehead's solution for lower ranges. However, this solution is presented as a piecewise function, where the functions change based on the zeta potential and the $\kappa a$ parameter, introducing infinite values for the zeta potential and discontinuities in the derived functions. This paper aims to provide a singular equation solution to the full Poisson-Boltzmann equation for a long cylindrical capillary for all zeta potentials. This solution is a single, continuous, and finite function that produces exact results instead of approximations for all ranges of zeta potential. This exact solution is compared against published approximate solutions for large zeta potentials shown by comparing the large zeta potential approximation with the new exact solution. Important parameters such as volume transport and apparent viscosity were found to have errors of up to 9.76%-57.4%, respectively. The function $f(\kappa a,{\psi }_0,\beta )$ has errors of up to 10.5% compared to our full solution.