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On the boundary layer structure near a highly permeable porous interface

Published online by Cambridge University Press:  31 May 2016

Mohit P. Dalwadi*
Affiliation:
Synthetic Biology Research Centre, University of Nottingham, University Park, Nottingham NG7 2RD, UK Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
S. Jonathan Chapman
Affiliation:
Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
Sarah L. Waters
Affiliation:
Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
James M. Oliver
Affiliation:
Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
*
Email address for correspondence: mohit.dalwadi@nottingham.ac.uk

Abstract

The method of matched asymptotic expansions is used to study the canonical problem of steady laminar flow through a narrow two-dimensional channel blocked by a tight-fitting finite-length highly permeable porous obstacle. We investigate the behaviour of the local flow close to the interface between the single-phase and porous regions (governed by the incompressible Navier–Stokes and Darcy flow equations, respectively). We solve for the flow in these inner regions in the limits of low and high Reynolds number, facilitating an understanding of the nature of the transition from Poiseuille to plug to Poiseuille flow in each of these limits. Significant analytical progress is made in the high Reynolds number limit, and we explore in detail the rich boundary layer structure that occurs. We derive general results for the interfacial stress and for the conditions that couple the flow in the outer regions away from the interface. We consider the three-dimensional generalization to unsteady laminar flow through and around a tight-fitting highly permeable cylindrical porous obstacle within a Hele-Shaw cell. For the high Reynolds number limit, we give the coupling conditions and interfacial stress in terms of the outer flow variables, allowing information from a nonlinear three-dimensional problem to be obtained by solving a linear two-dimensional problem. Finally, we illustrate the utility of our analysis by considering the specific example of time-dependent forced far-field flow in a Hele-Shaw cell containing a porous cylinder with a circular cross-section. We determine the internal stress within the porous obstacle, which is key for tissue engineering applications, and the interfacial stress on the boundary of the porous obstacle, which has applications to biofilm erosion. In the high Reynolds number limit, we demonstrate that the fluid inertia can result in the cylinder experiencing a time-independent net force, even when the far-field forcing is periodic with zero mean.

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Papers
Copyright
© 2016 Cambridge University Press 

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References

Auriault, J.-L. 2009 On the domain of validity of Brinkman’s equation. Trans. Porous Med. 79 (2), 215223.CrossRefGoogle Scholar
Beavers, G. S. & Joseph, D. D. 1967 Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30 (1), 197207.CrossRefGoogle Scholar
Bessiere, Y., Abidine, N. & Bacchin, P. 2005 Low fouling conditions in dead-end filtration: evidence for a critical filtered volume and interpretation using critical osmotic pressure. J. Membr. Sci. 264 (1), 3747.CrossRefGoogle Scholar
Bodoia, J. R. & Osterle, J. F. 1961 Finite difference analysis of plane Poiseuille and Couette flow developments. Appl. Sci. Res. 10 (1), 265276.CrossRefGoogle Scholar
Boyd, J. P. 2008 The Blasius function: computations before computers, the value of tricks, undergraduate projects, and open research problems. SIAM Rev. 50 (4), 791804.CrossRefGoogle Scholar
Brandt, A. & Gillis, J. 1966 Magnetohydrodynamic flow in the inlet region of a straight channel. Phys. Fluids 9, 690699.CrossRefGoogle Scholar
Carraro, T., Goll, C., Marciniak-Czochra, A. & Mikelić, A. 2015 Effective interface conditions for the forced infiltration of a viscous fluid into a porous medium using homogenization. Comput. Meth. Appl. Mech. Engng 292, 195220.CrossRefGoogle Scholar
Chung, C. A., Chen, C. W., Chen, C. P. & Tseng, C. S. 2007 Enhancement of cell growth in tissue-engineering constructs under direct perfusion: modeling and simulation. Biotechnol. Bioengng 97 (6), 16031616.CrossRefGoogle Scholar
Cummings, L. J., Sawyer, N. B. E., Morgan, S. P., Rose, F. & Waters, S. L. 2009 Tracking large solid constructs suspended in a rotating bioreactor: a combined experimental and theoretical study. Biotechnol. Bioengng 104 (6), 12241234.CrossRefGoogle Scholar
Cummings, L. J. & Waters, S. L. 2007 Tissue growth in a rotating bioreactor. Part II fluid flow and nutrient transport problems. Math. Med. Biol. 24 (2), 169208.CrossRefGoogle Scholar
Dalwadi, M. P.2014 Flow and nutrient transport problems in rotating bioreactor systems. PhD thesis, University of Oxford.Google Scholar
Duddu, R., Chopp, D. L. & Moran, B. 2009 A two-dimensional continuum model of biofilm growth incorporating fluid flow and shear stress based detachment. Biotechnol. Bioengng 103 (1), 92104.CrossRefGoogle ScholarPubMed
El Haj, A. J., Minter, S. L., Rawlinson, S. C. F., Suswillo, R. & Lanyon, L. E. 1990 Cellular responses to mechanical loading in vitro. J. Bone Mineral. Res. 5 (9), 923932.CrossRefGoogle ScholarPubMed
Gabouev, A. I., Schultheiss, D., Mertsching, H., Köppe, M., Schlote, N., Wefer, J., Jonas, U. & Stief, C. G. 2003 In vitro construction of urinary bladder wall using porcine primary cells reseeded on acellularized bladder matrix and small intestinal submucosa. Intl J. Artif. Organs 26 (10), 935942.CrossRefGoogle ScholarPubMed
Jaasma, M. J., Plunkett, N. A. & O’Brien, F. J. 2008 Design and validation of a dynamic flow perfusion bioreactor for use with compliant tissue engineering scaffolds. J. Biotechnol. 133 (4), 490496.CrossRefGoogle ScholarPubMed
Kaviany, M. 2012 Principles of Heat Transfer in Porous Media. Springer.Google Scholar
Lei, X.-H., Ning, L.-N., Cao, Y.-J., Liu, S., Zhang, S.-B., Qiu, Z.-F., Hu, H.-M., Zhang, H.-S., Liu, S. & Duan, E.-K. 2011 NASA-approved rotary bioreactor enhances proliferation of human epidermal stem cells and supports formation of 3D epidermis-like structure. PLoS ONE 6 (11), e26603.CrossRefGoogle ScholarPubMed
Levy, T. & Sanchez-Palencia, E. 1975 On boundary conditions for fluid flow in porous media. Intl J. Engng Sci. 13 (11), 923940.CrossRefGoogle Scholar
McCarthy, A. A., O’Shea, D. G., Murray, N. T., Walsh, P. K. & Foley, G. 1998 Effect of cell morphology on dead-end filtration of the dimorphic yeast Kluyveromycesmarxianus Var. marxianus NRRLy2415. Biotechnol. Prog. 14 (2), 279285.CrossRefGoogle Scholar
Nabovati, A., Llewellin, E. W. & Sousa, A. C. M. 2009 A general model for the permeability of fibrous porous media based on fluid flow simulations using the lattice Boltzmann method. Compos. A Appl. Sci. Manufacturing 40 (6), 860869.CrossRefGoogle Scholar
Nield, D. A. & Bejan, A. 2006 Convection in Porous Media. Springer.Google Scholar
O’Dea, R. D., Osborne, J. M., El Haj, A. J., Byrne, H. M. & Waters, S. L. 2013 The interplay between tissue growth and scaffold degradation in engineered tissue constructs. J. Math. Biol. 67 (5), 11991225.CrossRefGoogle ScholarPubMed
O’Dea, R. D., Waters, S. L. & Byrne, H. M. 2009 A multiphase model for tissue construct growth in a perfusion bioreactor. Math. Med. Biol. 133.Google Scholar
Pathi, P., Ma, T. & Locke, B. R. 2005 Role of nutrient supply on cell growth in bioreactor design for tissue engineering of hematopoietic cells. Biotechnol. Bioengng. 89 (7), 743758.CrossRefGoogle ScholarPubMed
Pazzano, D., Mercier, K. A., Moran, J. M., Fong, S. S., DiBiasio, D. D., Rulfs, J. X., Kohles, S. S. & Bonassar, L. J. 2000 Comparison of chondrogensis in static and perfused bioreactor culture. Biotechnol. Prog. 16 (5), 893896.CrossRefGoogle ScholarPubMed
Picioreanu, C., Van Loosdrecht, M. C. M. & Heijnen, J. J. 2001 Two-dimensional model of biofilm detachment caused by internal stress from liquid flow. Biotechnol. Bioengng. 72 (2), 205218.3.0.CO;2-L>CrossRefGoogle ScholarPubMed
Pierre, J., Gemmiti, C. V., Kolambkar, Y. M., Oddou, C. & Guldberg, R. E. 2008 Theoretical analysis of engineered cartilage oxygenation: influence of construct thickness and media flow rate. Biomech. Model. Mechanobiol. 7 (6), 497510.CrossRefGoogle ScholarPubMed
Prandtl, L. 1905 Über Flüssigkeitsbewegung bei sehr kleiner Reibung. In Proceedings of the Third International Mathematic Congress, Heidelberg, 1904 (ed. Krazer, A.), pp. 484491.Google Scholar
Purcell, E. M. 1977 Life at low Reynolds number. Am. J. Phys. 45 (1), 311.CrossRefGoogle Scholar
Schlichting, H. & Gersten, K. 2000 Boundary-layer Theory. Springer.CrossRefGoogle Scholar
Šimáček, P. & Advani, S. G. 1996 Permeability model for a woven fabric. Polym. Compos. 17 (6), 887899.CrossRefGoogle Scholar
Su, G. N.-C., Hidaka, M., Kimura, Y. & Yamamoto, G. 2003 In situ collagen gelation: a new method for constructing large tissue in rotary culture vessels. In Vitro Cell. Dev. Biol.-Animal 39 (8–9), 368374.2.0.CO;2>CrossRefGoogle ScholarPubMed
Sucosky, P., Osorio, D. F., Brown, J. B. & Neitzel, G. P. 2004 Fluid mechanics of a spinner-flask bioreactor. Biotechnol. Bioengng. 85 (1), 3446.CrossRefGoogle ScholarPubMed
Telgmann, U., Horn, H. & Morgenroth, E. 2004 Influence of growth history on sloughing and erosion from biofilms. Water Res. 38 (17), 36713684.CrossRefGoogle ScholarPubMed
Thompson, B. W. 1968 Secondary flow in a Hele-Shaw cell. J. Fluid Mech. 31 (2), 379395.CrossRefGoogle Scholar
Van der Bruggen, B., Vandecasteele, C., Van Gestel, T., Doyen, W. & Leysen, R. 2003 A review of pressure-driven membrane processes in wastewater treatment and drinking water production. Environ. Prog. 22 (1), 4656.CrossRefGoogle Scholar
Van Dyke, M. 1970 Entry flow in a channel. J. Fluid Mech. 44 (4), 813823.CrossRefGoogle Scholar
Van Dyke, M. 1975 Perturbation Methods in Fluid Dynamics. Parabolic Press.Google Scholar
Waters, S. L., Cummings, L. J., Shakesheff, K. M. & Rose, F. 2006 Tissue growth in a rotating bioreactor. Part I mechanical stability. Math. Med. Biol. 23 (4), 311337.CrossRefGoogle Scholar
Whittaker, R. J., Booth, R., Dyson, R., Bailey, C., Chini, L. P., Naire, S., Payvandi, S., Rong, Z., Woollard, H., Cummings, L. J. et al. 2009 Mathematical modelling of fibre-enhanced perfusion inside a tissue-engineering bioreactor. J. Theor. Biol. 256 (4), 533546.CrossRefGoogle ScholarPubMed
Wilson, S. D. R. 1971 Entry flow in a channel. Part 2. J. Fluid Mech. 46 (4), 787799.CrossRefGoogle Scholar
Zhao, F., Pathi, P., Grayson, W., Xing, Q., Locke, B. R. & Ma, T. 2005 Effects of oxygen transport on 3-d human mesenchymal stem cell metabolic activity in perfusion and static cultures: experiments and mathematical model. Biotechnol. Prog. 21 (4), 12691280.CrossRefGoogle ScholarPubMed
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