Analytical solution of the transient electro-osmotic flow of a generalized fractional Maxwell fluid in a straight pipe with a circular cross-section

In this study, we examine the transient electro-osmotic flow of a generalized Maxwell fluid with a fractional derivative in a narrow capillary tube. Using the integral transform method, analytical expressions are derived for the electric potential and transient velocity profile by solving the linearized Poisson–Boltzmann equation and the Navier–Stokes equation. We show that the distribution and establishment of the velocity comprises two parts: the steady and unsteady parts. We demonstrate the effects of the relaxation time, fractional derivative parameter, and the Debye–Hückel parameter on the generation of flow in a graphical manner and we analyze them numerically. The velocity overshoot and oscillation are observed and discussed.