RESEARCH PAPER Hae-Won Choi Æ Marius Paraschivoiu Advanced hybrid-flux approach for output bounds of electro-osmotic flows: adaptive refinement and direct equilibrating strategies Received: 6 January 2005 / Accepted: 19 April 2005 / Published online: 15 November 2005 Ó Springer-Verlag 2005 Abstract Fluidic flow and species transport in integrated microfluidic devices can readily be simulated with com- putational fluid dynamics. Nevertheless, the common practice to evaluate the solution accuracy is to decrease the discretization size (i.e., mesh size) until the value of the quantity of interest (termed ‘‘output’’ here-in) does not change within a fixed tolerance. For systems of partial differential equations such as the ones needed to be solved for electro-osmotic flows, this procedure is inappropriate due to the resulting large computation cost when dealing with finer discretizations. Further- more, in a design environment, when investigating many geometries and flow configurations, the numerical uncertainty in the output may not allow the designer to select the best design. In this paper, we present a numerical technique that is particularly appropriate to provide certainty information for outputs of electro-os- motic microflows. The method uses an a-posteriori error estimation technique termed the bound method to pro- vide fast, inexpensive, and reliable bounds to the ‘‘out- put’’, therefore, alleviating the need to systematically run different meshes. To demonstrate the usefulness of the bound method, the electro-osmotic flow applied to the cross-intersection in microchannel configuration is analyzed. The bound method presented in this paper is also extend to use an adaptive refinement strategy to sharpen the bounds, the direct equilibrating strategy to calculate the ‘hybrid-flux’ very efficiently, and parallel local computations to speed up the fine h-mesh com- putations. Keywords Electro-osmotic flow Æ Bound method Æ Hybrid-flux Æ Direct equilibrating Æ Adaptive refinement Æ Parallel processing 1 Introduction Recent advances in micro- and nano-scale technologies reflect a significant interest and growth of miniature applications to be utilized for chemical, biomedical, and micro-electro-mechanical systems (MEMS) industries. Microfluidic devices or systems, manufactured by etch- ing or fabricated onto glass, quartz or polymers and covered with plates, can form a network of microchan- nels. Such devices and systems tend to be an integration of several micro-scale components in a single microchip as recently reviewed by Erickson and Li (2004). The fluidic flow and species transport in integrated micro- fluidic devices such as biochip and Lab-on-a-chip sys- tems are well controlled by electro-kinetic transport processes such as electro-osmosis or electrophoresis. These systems require no moving mechanical devices but solely utilize applied electric charges. The understanding of micro-scale fluidic and transport phenomena plays a fundamental role to properly model integrated micro- fluidic devices. The development of reliable and efficient computational tools is nevertheless still needed to im- prove accuracy and to reduce the computational cost. In this paper, we mainly focus on a numerical technique to accurately predict quantities of interest in the electro- osmotic microflow by providing bounds to these quan- tities. A number of numerical techniques have been devel- oped to simulate electro-osmotic microflows. Using fi- nite difference methods (FDM), Ermakov et al. (1998) investigated the focusing/mixing effects and Yang et al. H.-W. Choi Department of Mechanical and Industrial Engineering, University of Toronto, 5 King’s College Road, Toronto, ON, M5S3G8 Canada M. Paraschivoiu Department of Mechanical and Industrial Engineering, Concordia University, 1455 de Maisonneuve Slvd. W., Montreal, QC H3G 1M8, Canada E-mail: paraschi@ME.Concordia.CA Present address: H.-W. Choi (&) Scientific Computing Division, National Center for Atmospheric Research (NCAR), 1850 Table Mesa Drive, Boulder, CO 80305, USA E-mail: haewon@ucar.edu Microfluid Nanofluid (2006) 2: 154–170 DOI 10.1007/s10404-005-0059-2