The semi-analytical solution for transient electroosmotic flow through elliptic cylindrical microchannels is derived from the Navier-Stokes equations using the Laplace transform. The electroosmotic force expressed by the linearized Poisson-Boltzmann equation is considered the external force in the Navier-Stokes equations. The velocity field solution is obtained in the form of the Mathieu and modified Mathieu functions and it is capable of describing the flow behavior in the system when the boundary condition is either constant or varied. The fluid velocity is calculated numerically using the inverse Laplace transform in order to describe the transient behavior. Moreover, the flow rates and the relative errors on the flow rates are presented to investigate the effect of eccentricity of the elliptic cross-section. The investigation shows that, when the area of the channel cross-sections is fixed, the relative errors are less than 1% if the eccentricity is not greater than 0.5. As a result, an elliptic channel with the eccentricity not greater than 0.5 can be assumed to be circular when the solution is written in the form of trigonometric functions in order to avoid the difficulty in computing the Mathieu and modified Mathieu functions.

Electroosmotic force has been used extensively to manipulate fluid flow in a microfluidic system with various channel shapes, especially an elliptic cylinder. However, developing a computational domain and simulating fluid flow for a system involving an elliptic channel consumes a large amount of time. Moreover, the mathematical expression for the fluid velocity of electroosmotic flow in an elliptic channel may be given in the form of the Mathieu functions that have difficulty in achieving the numerical result. In addition, there is clear scientific evidence that confirms the slippage of fluid at the solid-fluid interface in a microscale system. In this study, we present the mathematical model of combined pressure-driven and electroosmotic flow through elliptic microchannels under the slip-fluid condition. From the practical point of view in fluidics, the effect of the eccentricity of the channel cross-section is investigated on the volumetric flow rate to overcome the difficulty. The results show that the substitution of the equivalent circular channel for an elliptic channel provides a valid flow rate under the situation that the areas of both channel cross-sections are equal and the eccentricity of the elliptic cross-section is less than 0.5. Additionally, the flow rate obtained from the substitution is more accurate when the slip length increases or the pressure-gradient-to-external-electric-field ratio decreases.