Modified Bernoulli Equation for Use with Combined Electro-Osmotic and Pressure-Driven Microflows

TL;DR

This paper extends the Bernoulli equation to electro-osmotic flows by incorporating electrical potential energy, providing a unified framework for analyzing combined electro-osmotic and pressure-driven microflows.

Contribution

It introduces a modified Bernoulli equation that includes electrical energy terms for electro-osmotic flows, linking fluid mechanics with electrokinetic effects.

Findings

Friction factor for pure EO flow varies inversely with Reynolds number based on Debye length.

Derived expressions for friction factor in combined EO and pressure-driven flow.

Validated the applicability of the modified Bernoulli equation to microflow scenarios.

Abstract

In this paper we present electro-osmotic (EO) flow within a more traditional fluid mechanics framework. Specifically, the modified Bernoulli equation (viz. the energy equation, the mechanical energy equation, the pipe flow equation, etc.) is shown to be applicable to EO flows if an electrical potential energy term is also included. The form of the loss term in the modified Bernoulli equation is unaffected by the presence of an electric field; i.e., the loss term still represents the effect of wall shear stress, which can be represented via a friction factor. We show that that the friction factor for pure EO flow (no applied pressure gradient) varies inversely with the Reynolds number based on the Debeye length of the electric double layer. Expressions for friction factor for combined laminar pressure-driven and EO flow are also given. These are shown to be functions of Reynolds number and geometry, as well as the relative strength of the applied electric field to the applied pressure gradient.