Taylor–Aris dispersion induced by axial variation in velocity profile in patterned microchannels
Graphical abstract
The concentration profile of a solute plug flowing through a microchannel patterned with pillars is narrower for the Cassie–Baxter state and broader for the Wenzel state.
Introduction
Hydrodynamic dispersion characterizes mass transport in processes involving diffusion coupled with flow. Examples of such processes include physiological mechanisms such as gas exchange in lungs (Hennessy, 1974) and blood (Ye et al., 2006), flow through porous media (Brenner, 1980, Brenner and Edwards, 1993) and analytical techniques such as chromatography, capillary electrophoresis and electrochromatography (Gas and Kenndler, 2000). Dispersion is favourable in processes requiring enhanced mixing (Wunderlich et al., 2014). However, it is undesirable for analytical techniques which require sharper resolution in terms of distinct peaks for constituent components (Gas and Kenndler, 2000).
Dispersion occurs through combined effects of advection and molecular diffusion (Taylor, 1953, Aris, 1956). When a solute pulse is injected into a flow stream, it stretches under the influence of advection, giving rise to sharp concentration gradients transverse to the flow direction. The extent of stretching affects dispersion, thereby rendering laminar flow prone to higher dispersion and plug flow, the limiting case for lowest dispersion (Yan et al., 2010). The concentration gradient created through advection is smeared through diffusion, which in turn inhibits the axial spreading. Overall, dispersion is directly proportional to advection and inversely proportional to diffusion. For a given diffusivity, dispersion depends only on the flow characteristics.
Due to different flow characteristics, dispersion is sensitive to the cross section of the conduit. It is characterized through the dispersion coefficient. For circular tubes (Brenner and Edwards, 1993, Taylor, 1953, Aris, 1956), the dispersion coefficient is a function of the Peclet number based on diameter. For rectangular channels, the dispersion coefficient varies with the aspect ratio of the cross section (Doshi et al., 1978, Chatwin and Sullivan, 1982, Guell et al., 1987, Parks and Romero, 2007). The dependence on aspect ratio is counterintuitive in several cases. For example, the dispersion coefficient in parallel flat plate differs remarkably from the asymptotic case of a rectangular microchannel with very high aspect ratio (Doshi et al., 1978, Chatwin and Sullivan, 1982). Also, in the case of smoothly varying cross section such as ellipsoid inscribed in rectangular cross section, dispersion coefficient depends on a Peclet number based on channel width rather than height (Guell et al., 1987). A typical way of limiting dispersion in pressure-driven flows is tailoring the cross-sectional shape of the microchannel (Dutta and Leighton, 2001, Vikhansky, 2009, Ajdari et al., 2006, Bontoux et al., 2006).
An example of factors affecting dispersion is the secondary flows that can supress the dispersion through an enhanced transverse diffusion (Bouquain et al., 2012, Adrover, 2013). Secondary flows may arise from the primary flow due to geometry effects such as periodic sized apertures (Bouquain et al., 2012) or in curved microchannels due to effects of centrifugal force (Deans flow) (Johnson and Kamm, 1986). Alternatively they may be engineered using electro-osmosis (Zholskovskij and Masliyah, 2004, Zhao and Bau, 2007, Rubinstein and Zaltzman, 2013) where its characteristic plug flow profile inhibits dispersion drastically.
Another factor affecting dispersion, yet unexplored, is the axial flow variation caused by patterns of periodic pillars and gaps in flow direction. Such patterns occur naturally in superhydrophobic surfaces such as lotus leaf, duck feathers (Lauga and Stone, 2003, Wenzel, 1936, Cassie and Baxter, 1944), may result from wall asperities (Lauga and Stone, 2003) or be fabricated purposefully in several microfluidic devices. Liquid placed over the pattern exists in two distinct limiting configurations: Cassie Baxter state where the gaps are filled with air pockets and Wenzel state when the gaps are filled by the liquid. Due to the free slip over the air pocket (air–liquid interface), liquid flow over patterns under Cassie state incurs an overall lower pressure drop (Choi et al., 2003, de Gennes et al., 2003, Joseph et al., 2006, Lee et al., 2008, Roach et al., 2008, Pihl et al., 2014, Sun et al., 2005, Feng et al., 2002). Due to a greater cross section over the gap zone under Wenzel case state, liquid flow over patterns experiences expansion and contraction, resulting in flow circulation/stagnant zones in the gap zones. Both flow features: slippage under Cassie state (Pihl et al., 2014) and recirculation/stagnant zones in Wenzel state, can potentially influence the transverse diffusion and dispersion.
The present work focuses on dispersion of solutes during laminar flow over patterned micro-channels both under Cassie Baxter and Wenzel State. The dispersion is characterized based on the solute concentration distribution as a function of Peclet number. Effects of geometrical parameters on the spreading of a solute pulse through the patterned micro-channels are analyzed numerically. The dispersion coefficient is calculated using the Aris method of moments.
Section snippets
Dispersion regimes
Fig. 1 illustrates dispersion of a solute pulse injected into a laminar flow of carrier liquid inside a cylindrical tube. The initial phase constitutes the pre-asymptotic regime when solute at the peripheral zones diffuse towards the central core in a transient manner with the concentration profile shape developing. A transition to the Aris regime occurs when the effects of diffusion encompasses the cross section leading to a quasi steady state in transverse direction. By this time the shape of
Formulation
The flow is assumed to be under steady state with a parabolic profile imposed at the inlet. The dispersion of an initial patch of liquid injected at the inlet is treated as a transient process. Since the entry effects of flow for subsequent gaps and pillars are crucial, the Navier Stokes equation applied retains the convective term unlike in Stokes flow. ls is chosen as the characteristic length in place of the diameter of the smooth channel. Using ls and other characteristic quantities,
Results and discussion
To discern even the mildest effects of convection, the flow and concentration are characterized initially at very low Peclet number (Pep=5). The effects seen at low Peclet number is expected to be enhanced at higher Peclet number with increase in the convection effects and hence the effects of slip.
Conclusion
The dispersion of solute during laminar flow over patterned micro-channel, both under Cassie Baxter and Wenzel states is analyzed. The dispersion coefficient is calculated using the Aris method of moments. The effects of geometry and flow conditions on the dispersion are quantified. The Cassie state is marked by a sharper and narrower spread compared to the smooth channel. The peak and the sharpness increase with decrease in η, the ratio of the pillar to the gap lengths and decrease in radius
Acknowledgement
The authors gratefully acknowledge the financial support provided by the Indian Institute of Technology Kharagpur (Sanction Letter no.: IIT/SRIC/ATDC/CEM/2013-14/118, dated 19.12.2013).
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