A multi-scale model for solute transport in a wavy-walled channel

  • H. F. Woollard
  • J. Billingham
  • O. E. Jensen
  • G. Lian
Open Access
Article

Abstract

This paper concerns steady flow and solute uptake in a wavy-walled channel, where the wavelength and amplitude of the wall are comparable to each other but are much shorter than the width of the channel. The problem has two primary asymptotic regions: a core region where the walls appear flat at leading order and a wall region where there is full interaction between advection, diffusion and uptake at the wavy wall. For weak wall uptake, the effective uptake from the core is shown to increase with wall waviness in proportion to surface area, whereas for stronger wall uptake, it is found that the uptake from the core can be reduced as the wall amplitude increases. Conditions are identified under which this approximation is uniformly valid in a full channel flow, accounting for inlet conditions, and a comprehensive survey of the asymptotic distributions of solute both along and across the channel is provided. It is also shown how this multiscale approach can readily be extended to account for channel walls with multiple lengthscales of spatial variation.

Keywords

Advection–diffusion Matched asymptotic expansions Solute uptake Viscous flow 

Notes

Acknowledgements

This work is supported by an EPSRC industrial CASE award with Unilever, made available through the Knowledge Transfer Network for Industrial Mathematics.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. 1.
    Kunert C, Harting J (2007) Roughness induced boundary slip in microchannel flows. Phys Rev Lett 99: 176001CrossRefADSGoogle Scholar
  2. 2.
    Neto C, Evans DR, Bonaccurso E, Butt HJ, Craig VSJ (2005) Boundary slip in Newtonian liquids: a review of experimental studies. Rep Prog Phys 68: 2859–2897CrossRefADSGoogle Scholar
  3. 3.
    Taylor GI (1971) A model for the boundary condition of a porous material. Part 1. J Fluid Mech 49: 319–326CrossRefADSGoogle Scholar
  4. 4.
    Richardson S (1971) A model for the boundary condition of a porous material. Part 2. J Fluid Mech 49: 327–336MATHCrossRefADSGoogle Scholar
  5. 5.
    Richardson S (1973) On the no-slip boundary condition. J Fluid Mech 59: 707–719MATHCrossRefADSGoogle Scholar
  6. 6.
    Hocking LM (1976) A moving fluid interface on a rough surface. J Fluid Mech 76: 801–817MATHCrossRefADSGoogle Scholar
  7. 7.
    Tuck EO, Kouzoubov A (1995) A laminar roughness boundary condition. J Fluid Mech 300: 59–70MATHCrossRefADSMathSciNetGoogle Scholar
  8. 8.
    Zhou H, Khayat RE, Martinuzzi RJ, Straatman AG (2002) On the validity of the perturbation approach for the flow inside weakly modulated channels. Int J Numer Methods Fluids 39: 1139–1159MATHCrossRefGoogle Scholar
  9. 9.
    Higdon JJL (1985) Stokes flow in arbitrary two-dimensional domains: shear flow over ridges and cavities. J Fluid Mech 159: 195–226MATHCrossRefADSMathSciNetGoogle Scholar
  10. 10.
    Shen C, Floryan JM (1985) Low Reynolds number flow over cavities. Phys Fluids 28: 3191–3202MATHCrossRefADSMathSciNetGoogle Scholar
  11. 11.
    Lecoq N, Anthore R, Cichocki B, Szymczak P, Feuillebois F (2002) Drag force on a sphere moving towards a corrugated wall. J Fluid Mech 513: 247–264CrossRefADSGoogle Scholar
  12. 12.
    Miksis MJ, Davis SH (1994) Slip over rough and coated surfaces. J Fluid Mech 273: 125–139MATHCrossRefADSGoogle Scholar
  13. 13.
    Sarkar K, Prosperetti A (1996) Effective boundary conditions for Stokes flow over a rough surface. J Fluid Mech 316: 223–240MATHCrossRefADSGoogle Scholar
  14. 14.
    Tartakovsky DM, Xiu D (2006) Stochastic analysis of transport in tubes with rough walls. J Comput Phys 217: 248–259MATHCrossRefADSMathSciNetGoogle Scholar
  15. 15.
    Stroock AD, Dertinger SK, Whitesides GM, Ajdari A (2002) Patterning flows using grooved surfaces. Anal Chem 74: 5306–5312CrossRefGoogle Scholar
  16. 16.
    Wang CY (2003) Flow over a surface with parallel grooves. Phys Fluids 15: 1114–1121CrossRefADSGoogle Scholar
  17. 17.
    Moffatt HK (1964) Viscous and resistive eddies near a sharp corner. J Fluid Mech 18: 1–18MATHCrossRefADSGoogle Scholar
  18. 18.
    Wierschem A, Scholle M, Aksel N (2003) Vortices in film flow over strongly undulated bottom profiles at low Reynolds numbers. Phys Fluids 15: 426–435CrossRefADSMathSciNetGoogle Scholar
  19. 19.
    Scholle MI, Wierschem AI, Aksel NI (2004) Creeping films with vortices over strongly undulated bottoms. Acta Mech 168: 167–193MATHCrossRefGoogle Scholar
  20. 20.
    Pozrikidis C (1987) Creeping flow in two-dimensional channels. J Fluid Mech 180: 495–514CrossRefADSGoogle Scholar
  21. 21.
    Luchini P, Manzo F, Pozzi A (1991) Resistance of a grooved surface to parallel flow and cross-flow. J Fluid Mech 228: 87–109MATHADSGoogle Scholar
  22. 22.
    Park JS, Hyun JM (2002) The Stokes-flow friction on a wedge surface by the sliding of a plate. Fluid Dyn Res 30: 93–106CrossRefADSGoogle Scholar
  23. 23.
    Phan-Thien N (1982) Hydrodynamic lubrication of rough surfaces. Proc R Soc Lond A Mat 383: 439–446MATHCrossRefADSGoogle Scholar
  24. 24.
    Achdou Y, Pironneau O, Valentin F (1998) Effective boundary conditions for laminar flows over periodic rough boundaries. J Comput Phys 147: 187–218MATHCrossRefADSMathSciNetGoogle Scholar
  25. 25.
    Brady M, Pozrikidis C (1993) Diffusive transport across irregular and fractal walls. Proc R Soc Lond A Math 442: 571–583CrossRefADSGoogle Scholar
  26. 26.
    Blyth MG, Pozrikidis C (2003) Heat conduction across irregular and fractal-like surfaces. Int J Heat Mass Transf 46: 1329–1339MATHCrossRefGoogle Scholar
  27. 27.
    Fyrillas MM, Pozrikidis C (2001) Conductive heat transport across rough surfaces and interfaces between two conforming media. Int J Heat Mass Transf 44: 1789–1801MATHCrossRefGoogle Scholar
  28. 28.
    Neagu M, Bejan A (2001) Constructal placement of high-conductivity inserts in a slab: optimal design of roughness. J Heat Transf 123: 1184CrossRefGoogle Scholar
  29. 29.
    Horner M, Metcalfe G, Wiggins S, Ottino JM (2002) Transport enhancement mechanisms in open cavities. J Fluid Mech 452: 199–229MATHCrossRefADSGoogle Scholar
  30. 30.
    Laine-Pearson FE, Hydon PE (2006) Particle transport in a moving corner. J Fluid Mech 559: 379–390MATHCrossRefADSMathSciNetGoogle Scholar
  31. 31.
    Weber JEH (2005) Mean drift velocity in viscous flow over a corrugated bottom. Phys Fluids 17: 113102CrossRefADSGoogle Scholar
  32. 32.
    Smith JW, Gowen RA (1965) Heat transfer efficiency in rough pipes at high Prandtl number. A I Ch E J 11: 941–943Google Scholar
  33. 33.
    Sobey IJ (1980) On flow through furrowed channels. Part 1. J Fluid Mech 96: 1–26MATHCrossRefADSGoogle Scholar
  34. 34.
    Russ G, Beer H (1997) Heat transfer and flow field in a pipe with sinusoidal wavy surface- I. Numerical investigation. Int J Heat Mass Transf 40: 1061–1070MATHCrossRefGoogle Scholar
  35. 35.
    Wang CC, Chen CK (2002) Forced convection in a wavy-wall channel. Int J Heat Mass Transf 45: 2587–2595MATHCrossRefGoogle Scholar
  36. 36.
    Das PK, Mahmud S, Tasnim SH, Islam A (2003) Effect of surface waviness and aspect ratio on heat transfer inside a wavy enclosure. Int J Numer Method H 13: 1097–1122MATHCrossRefGoogle Scholar
  37. 37.
    Nevard J, Keller JB (1997) Homogenization of rough boundaries and interfaces. SIAM J Appl Math 57: 1660–1686MATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Szumbarski J, Floryan JM (1999) A direct spectral method for determination of flows over corrugated boundaries. J Comput Phys 153: 378–402MATHCrossRefMathSciNetADSGoogle Scholar
  39. 39.
    Amirat Y, Bodart O, De Maio U, Gaudiello A (2004) Asymptotic approximation of the solution of the Laplace equation in a domain with highly oscillating boundary. SIAM J Math Anal 35: 1598–1616MATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Amidon GL, Kou J, Elliott RL, Lightfoot EN (1980) Analysis of models for determining intestinal wall permeabilities. J Pharm Sci 69: 1369–1373CrossRefGoogle Scholar
  41. 41.
    Wilding I (2000) Site-specific drug delivery in the gastrointestinal tract. Crit Rev Ther Drug Carrier Syst 17: 557–620Google Scholar
  42. 42.
    Wacher VJ, Salphati L, Benet LZ (2001) Active secretion and enterocytic drug metabolism barriers to drug absorption. Adv Drug Dely Rev 46: 89–102CrossRefGoogle Scholar
  43. 43.
    Stephens RH, Tanianis-Hughes J, Higgs NB, Humphrey M, Warhurst G (2002) Region-dependent modulation of intestinal permeability by drug efflux transporters: In vitro ttudies in mdr1a (-/-) mouse intestine. J Pharm Exp Ther 303: 1095–1101CrossRefGoogle Scholar
  44. 44.
    Žakelj S, Šturm K, Kristl A (2006) Ciprofloxacin permeability and its active secretion through rat small intestine in vitro. Int J Phar 313: 175–180CrossRefGoogle Scholar
  45. 45.
    Burton PS, Goodwin JT, Vidmar TJ, Amore BM (2002) Predicting drug absorption: how nature made it a difficult problem. J Pharm Exp Ther 303: 889–895CrossRefGoogle Scholar
  46. 46.
    Stoll BR, Batycky RP, Leipold HR, Milstein S, Edwards DA (2000) A theory of molecular absorption from the small intestine. Chem Eng Sci 55: 473–489CrossRefGoogle Scholar
  47. 47.
    Grotberg JB, Jensen OE (2004) Biofluid mechanics in flexible tubes. Annu Rev Fluid Mech 36: 121–147CrossRefADSMathSciNetGoogle Scholar
  48. 48.
    Abramowitz M, Stegun IA (eds) (1965) Handbook of Mathematical Functions. Dover Publications, New YorkGoogle Scholar

Copyright information

© The Author(s) 2008

Authors and Affiliations

  • H. F. Woollard
    • 1
  • J. Billingham
    • 1
  • O. E. Jensen
    • 1
  • G. Lian
    • 2
  1. 1.School of Mathematical SciencesUniversity of NottinghamNottinghamUK
  2. 2.Unilever R&D ColworthSharnbrook, BedfordUK

Personalised recommendations