Hostname: page-component-76fb5796d-dfsvx Total loading time: 0 Render date: 2024-04-25T21:03:37.563Z Has data issue: false hasContentIssue false
  • English
  • Français

Dispersion controlled by permeable surfaces: surface properties and scaling

Published online by Cambridge University Press:  19 July 2016

Bowen Ling ,
Alexandre M. Tartakovsky  and
Ilenia Battiato
Bowen Ling
Affiliation:
Mechanical and Aerospace Engineering Department, University of California San Diego, La Jolla, CA 92093, USA Mechanical Engineering Department, San Diego State University, San Diego, CA 92182, USA
Alexandre M. Tartakovsky
Affiliation:
Fundamental and Computational Sciences Directorate, Pacific Northwest National Laboratory, Richland, WA 99352, USA
Ilenia Battiato*
Affiliation:
Mechanical Engineering Department, San Diego State University, San Diego, CA 92182, USA
*
†Email address for correspondence: ibattiato@mail.sdsu.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Permeable and porous surfaces are common in natural and engineered systems. Flow and transport above such surfaces are significantly affected by the surface properties, e.g. matrix porosity and permeability. However, the relationship between such properties and macroscopic solute transport is largely unknown. In this work, we focus on mass transport in a two-dimensional channel with permeable porous walls under fully developed laminar flow conditions. By means of perturbation theory and asymptotic analysis, we derive the set of upscaled equations describing mass transport in the coupled channel–porous-matrix system and an analytical expression relating the dispersion coefficient with the properties of the surface, namely porosity and permeability. Our analysis shows that their impact on the dispersion coefficient strongly depends on the magnitude of the Péclet number, i.e. on the interplay between diffusive and advective mass transport. Additionally, we demonstrate different scaling behaviours of the dispersion coefficient for thin or thick porous matrices. Our analysis shows the possibility of controlling the dispersion coefficient, i.e. transverse mixing, by either active (i.e. changing the operating conditions) or passive mechanisms (i.e. controlling matrix effective properties) for a given Péclet number. By elucidating the impact of matrix porosity and permeability on solute transport, our upscaled model lays the foundation for the improved understanding, control and design of microporous coatings with targeted macroscopic transport features.

JFM classification

Low-Reynolds-number flows: Low-Reynolds-number flows Micro-/Nano-fluid dynamics: Micro-/Nano-fluid dynamics Low-Reynolds-number flows: Porous media
Type
Papers
Information
Journal of Fluid Mechanics , Volume 801 , 25 August 2016 , pp. 13 - 42
Check for updates
Copyright
© 2016 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Al-Chidiac, M., Mirbod, P., Andreopoulos, Y. & Weinbaum, S. 2009 Dynamic compaction of soft compressible porous materials: experiments on air–solid phase interaction. J. Porous Media 12 (11), 1019–1035.Google Scholar
Aris, R. 1956 On the dispersion of a solute in a fluid flowing through a tube. Proc. R. Soc. Lond. 235 (1200), 67–77.Google Scholar
Battiato, I. 2012 Self-similarity in coupled Brinkman/Navier–Stokes flows. J. Fluid Mech. 699, 94–114.Google Scholar
Battiato, I. 2014 Effective medium theory for drag-reducing micro-patterned surfaces in turbulent flows. Eur. Phys. J. E 37 (19).Google Scholar
Battiato, I., Bandaru, P. & Tartakovsky, D. M. 2010 Elastic response of carbon nanotube forests to aerodynamic stresses. Phys. Rev. Lett. 105, 144504.Google Scholar
Battiato, I. & Rubol, S. 2014 Single-parameter model of vegetated aquatic flows. Water Resour. Res. 50 (8), 6358–6369.Google Scholar
Battiato, I. & Vollmer, J. 2012 Flow-induced shear instabilities of cohesive granulates. Phys. Rev. E 86, 031301.Google Scholar
Beavers, G. S. & Joseph, D. D. 1967 Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30 (01), 197–207.Google Scholar
Bodin, J., Delay, F. & De Marsily, G. 2003 Solute transport in a single fracture with negligible matrix permeability. 1. Fundamental mechanisms. Hydrogeol. J. 11 (4), 418–433.CrossRefGoogle Scholar
Boso, F. & Battiato, I. 2013 Homogenizability conditions of multicomponent reactive transport processes. Adv. Water Resour. 62, 254–265.Google Scholar
Bouquet, L. & Lauga, E. 2011 A smooth future? Nat. Mater. 10, 334–337.Google Scholar
Brenner, H. 1987 Transport Processes in Porous Media. McGraw-Hill.Google Scholar
Cui, J., Daniel, D., Grinthal, A., Lin, K. & Aizenberg, J. 2015 Dynamic polymer systems with self-regulated secretion for the control of surface properties and material healing. Nat. Mater. 14 (8), 790–795.Google Scholar
Davis, A. M. J. & Lauga, E. 2010 Hydrodynamic friction of fakir-like superhydrophobic surfaces. J. Fluid Mech. 661, 402–411.Google Scholar
Deck, C. P., Ni, C., Vecchio, K. S. & Bandaru, P. R. 2009 The response of carbon nanotube ensembles to fluid flow: applications to mechanical property measurement and diagnostics. J. Appl. Phys. 106 (7), 74304.Google Scholar
Dejam, M., Hassanzadeh, H. & Chen, Z. 2014 Shear dispersion in a fracture with porous walls. Adv. Water Resour. 74, 14–25.Google Scholar
Ghisalberti, M. 2009 Obstructed shear flows: similarities across systems and scales. J. Fluid Mech. 641, 51–61.Google Scholar
Gilroy, S. & Jones, D. L. 2000 Through form to function: root hair development and nutrient uptake. Trends Plant Sci. 5, 56–60.CrossRefGoogle ScholarPubMed
Goharzadeh, A., Khalili, A. & Jørgensen, B. B. 2005 Transition layer thickness at a fluid–porous interface. Phys. Fluids 17 (5), 057102.CrossRefGoogle Scholar
Gray, W. G. & Miller, C. T. 2005 Thermodynamically constrained averaging theory approach for modeling flow and transport phenomena in porous medium systems: 1. Motivation and overview. Adv. Water Resour. 28 (2), 161–180.CrossRefGoogle Scholar
Griffiths, I. M., Howell, P. D. & Shipley, R. J. 2013 Control and optimization of solute transport in a thin porous tube. Phys. Fluids 25 (3), 033101.Google Scholar
Gruenberger, A., Probst, C., Heyer, A., Wiechert, W., Frunzke, J. & Kohlheyer, D. 2013 Microfluidic picoliter bioreactor for microbial single-cell analysis: fabrication, system setup, and operation. J. Vis. Exp. (82), e50560.Google Scholar
Horne, R. N. & Rodriguez, F. 1983 Dispersion in tracer flow in fractured geothermal systems. Geophys. Res. Lett. 10 (4), 289–292.CrossRefGoogle Scholar
Hornung, U. 1997 Homogenization and Porous Media. Springer.Google Scholar
Hou, X., Hu, Y., Grinthal, A., Khan, M. & Aizenberg, J. 2015 Liquid-based gating mechanism with tunable multiphase selectivity and antifouling behaviour. Nature 519, 70–73.CrossRefGoogle ScholarPubMed
Kazezyılmaz-Alhan, C. M. 2008 Analytical solutions for contaminant transport in streams. J. Hydrol. 348 (3), 524–534.Google Scholar
Lauga, E. & Stone, H. A. 2003 Effective slip in pressure-driven Stokes flow. J. Fluid Mech. 489, 55–77.CrossRefGoogle Scholar
Le Bars, M. & Worster, M. G. 2006 Interfacial conditions between a pure fluid and a porous medium: implications for binary alloy solidification. J. Fluid Mech. 550, 149–173.Google Scholar
Li, X. M., Reinhoudt, D. & Crego-Calama, M. 2007 What do we need for a superhydrophobic surface? A review on the recent progress in the preparation of superhydrophobic surfaces. Chem. Soc. Rev. 36, 1350–1368.Google Scholar
Liu, C., Shang, J., Kerisit, S., Zachara, J. M. & Zhu, W. 2013 Scale-dependent rates of uranyl surface complexation reaction in sediments. Geochim. Cosmochim. Acta 105, 326–341.CrossRefGoogle Scholar
Lloyd, F. E. 1942 The Carnivorous Plants. Read Books Ltd.Google Scholar
Marmur, A. 2004 The lotus effect: superhydrophobicity and metastability. Langmuir 20, 3517–3519.Google Scholar
Marschner, H. & Dell, B. 1994 Nutrient uptake in mycorrhizal symbiosis. Plant Soil 159 (1), 89–102.CrossRefGoogle Scholar
Maruf, S. H., Rickman, M., Wang, L. IV, Mersch, J., Greenberg, A. R., Pellegrino, J. & Ding, Y. 2013a Influence of sub-micron surface patterns on the deposition of model proteins during active filtration. J. Membr. Sci. 444, 420–428.Google Scholar
Maruf, S. H., Wang, L., Greenberg, A. R., Pellegrino, J. & Ding, Y. 2013b Use of nanoimprinted surface patterns to mitigate colloidal deposition on ultrafiltration membranes. J. Membr. Sci. 428, 598–607.Google Scholar
Mikelic, A., Devigne, V. & Duijn, C. J. Van 2006 Rigorous upscaling of the reactive flow through a pore, under dominant Peclet and Damkohler numbers. SIAM J. Math. Anal. 38 (4), 1262–1287.Google Scholar
Nepf, H., Ghisalberti, M., White, B. & Murphy, E. 2007 Retention time and dispersion associated with submerged aquatic canopies. Water Resour. Res. 43 (4), W04422.Google Scholar
Nepf, H. M. 2012 Flow and transport in regions with aquatic vegetation. Annu. Rev. Fluid Mech. 44 (1), 123–142.Google Scholar
Nikora, V., Goring, D., McEwan, I. & Griffiths, G. 2001 Spatially averaged open-channel flow over rough bed. J. Hydraul. Engng.Google Scholar
Ogata, A. & Banks, R. B. 1961 A solution of the differential equation of longitudinal dispersion in porous media. US Geol. Surv. Prof. Pap; 411-A.Google Scholar
Ou, J., Perot, B. & Rothstein, J. P. 2004 Laminar drag reduction in microchannels using ultrahydrophobic surfaces. Phys. Fluids 16 (12), 4635–4643.Google Scholar
Papke, A. & Battiato, I. 2013 A reduced complexity model for dynamic similarity in obstructed shear flows. Geophys. Res. Lett. 40, 1–5.Google Scholar
Reichert, P. & Wanner, O. 1991 Enhanced one-dimensional modeling of transport in rivers. J. Hydraul. Engng 117 (9), 1165–1183.CrossRefGoogle Scholar
Rothstein, J. P. 2010 Slip on superhydrophobic surfaces. Annu. Rev. Fluid Mech. 42, 89–109.CrossRefGoogle Scholar
Roubinet, D., Dreuzy, J.-R. & Tartakovsky, D. M. 2012 Semi-analytical solutions for solute transport and exchange in fractured porous media. Water Resour. Res. 48 (1), W01542.Google Scholar
Scholz, I., Bückins, M., Dolge, L., Erlinghagen, T., Weth, A., Hischen, F., Mayer, J., Hoffmann, S., Riederer, M. & Riedel, M. 2010 Slippery surfaces of pitcher plants: nepenthes wax crystals minimize insect attachment via microscopic surface roughness. J. Expl Biol. 213, 1115–1125.Google Scholar
Stroock, A. D., Dertinger, S. K. W., Ajdari, A., Mezic, I., Stone, H. A. & Whitesides, G. M. 2002 Chaotic mixer for microchannels. Science 295, 647–651.Google Scholar
Stroock, A. D. & Whitesides, G. M. 2003 Controlling flows in microchannels with patterned surface charge and topography. Acc. Chem. Res. 36, 597–604.Google Scholar
Sudicky, E. A. & Frind, E. O. 1982 Contaminant transport in fractured porous media: analytical solutions for a system of parallel fractures. Water Resour. Res. 18 (6), 1634–1642.CrossRefGoogle Scholar
Tang, D. H., Frind, E. O. & Sudicky, E. A. 1981 Contaminant transport in fractured porous media: analytical solution for a single fracture. Water Resour. Res. 17 (3), 555–564.Google Scholar
Taylor, G. 1953 Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. Lond. 219, 186–203.Google Scholar
Valdes-Parada, F. J., Ochoa-Tapia, J. A. & Alvarez-Ramirez, J. 2009 Validity of the permeability Carman–Kozeny equation: a volume averaging approach. Physica A 388 (6), 789–798.Google Scholar
Weinbaum, S., Zhang, X., Han, Y., Vink, H. & Cowin, S. C. 2003 Mechanotransduction and flow across the endothelial glycocalyx. Proc. Natl Acad. Sci. 100 (13), 7988–7995.Google Scholar
Weinman, S. T. & Husson, S. M. 2016 Influence of chemical coating combined with nanopatterning on alginate fouling during nanofiltration. J. Membr. Sci. 513, 146–154.Google Scholar
Whitaker, S. 1999 The Method of Volume Averaging. Kluwer.Google Scholar
Wu, Y.-S., Ye, M. & Sudicky, E. A. 2010 Fracture-flow-enhanced matrix diffusion in solute transport trhough fractured porous media. Trans. Porous Med. 81 (1), 21–34.Google Scholar
Ybert, C., Barentin, C., Cottin-Bizonne, C., Joseph, P. & Bocquet, L. 2007 Achieving large slip with superhydrophobic surfaces: scaling laws for generic geometries. Phys. Fluids 19, 123601.Google Scholar