Modified Bernoulli Equation for Use with Combined Electro-Osmotic and   Pressure-Driven Microflows
  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. 
Thomas M. Adams, Aditya Raghunandan
physics.flu-dyn