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The extensional flow of a thin sheet of incompressible, transversely isotropic fluid

Published online by Cambridge University Press:  01 June 2008

J. EDWARD F. GREEN  and
AVNER FRIEDMAN
J. EDWARD F. GREEN
Affiliation:
Mathematical Biosciences Institute, The Ohio State University, Columbus, Ohio, 43210USA email: egreen@mbi.osu.edu, afriedman@mbi.osu.edu
AVNER FRIEDMAN
Affiliation:
Mathematical Biosciences Institute, The Ohio State University, Columbus, Ohio, 43210USA email: egreen@mbi.osu.edu, afriedman@mbi.osu.edu
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Abstract

Motivated by the aim of modelling the mechanical behaviour of biological gels (such as collagen gels) which have a fibrous microstructure, we consider the extensional flow of a thin two-dimensional film of incompressible, transversely isotropic viscous fluid. Neglecting inertia, and the effects of gravity and surface tension, leading-order equations are derived from a perturbation expansion of the full flow problem in powers of the (small) inverse aspect ratio. The existence and uniqueness of the solution of the reduced system of equations for small times is then proven. Special cases, in which the solution may be determined explicitly, are considered and we discuss the physical interpretation of the results.

Type
Papers
Information
European Journal of Applied Mathematics , Volume 19 , Issue 3 , June 2008 , pp. 225 - 257
Copyright
Copyright © Cambridge University Press 2008

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References

[1]Allen, S. J. & DeSilva, C. N. (1966) A theory of transversely isotropic fluids. J. Fluid Mech. 24 (4), 801–821.CrossRefGoogle Scholar
[2]Barocas, V. H. & Tranquillo, R. T. (1997) An anistropic biphasic theory of tissue-equivalent mechanics: The interplay among cell traction, fibrillar network deformation, fibril alignment and cell contact guidance. J. Biomech. Eng. 119, 137–145.CrossRefGoogle Scholar
[3]Chen, X. & Friedman, A. (2003) A free boundary problem for a elliptic-hyperbolic system: An application to tumour growth. SIAM J. Math. Anal. 35 (4), 974–986.CrossRefGoogle Scholar
[4]Cook, J. (1995) Mathematical Models for Dermal Wound Healing: Wound Contraction and Scar Formation. PhD thesis, University of Washington.Google Scholar
[5]Cummings, L. J. (2004) Evolution of a thin film of nematic liquid crystal with anisotropic surface energy. Euro. J. Appl. Math. 15, 651–677.CrossRefGoogle Scholar
[6]Cummings, L. J. & Howell, P. D. (1999) On the evolution of non-axisymmetric viscous fibres with surface tension, inertia and gravity. J. Fluid Mech. 389, 361–389.CrossRefGoogle Scholar
[7]Dewynne, J., Ockendon, J. R. & Wilmott, P. (1992) A systematic derivation of the leading-order equations for extensional flows in slender geometries. J. Fluid Mech. 244, 323–338.CrossRefGoogle Scholar
[8]Ericksen, J. L. (1960) Transversely isotropic fluids. Colloid and Polym. Sci. 173 (2), 117–122.Google Scholar
[9]Green, J. E. F. (2006) Mathematical Modelling of Cell Aggregation in Liver Tissue Engineering. PhD thesis, University of Nottingham, UK.Google Scholar
[10]Howell, P. D. (1994) Extensional Thin Layer Flows. PhD thesis, University of Oxford, UK.Google Scholar
[11]Howell, P. D. (1996) Models for thin viscous sheets. Eur. J. Appl. Math. 7, 321–343.CrossRefGoogle Scholar
[12]Korff, T. & Augustin, H. G. (1999) Tensional forces in fibrillar extracellular matrices control directional capillary sprouting. J. Cell Sci. 112, 3249–3258.CrossRefGoogle ScholarPubMed
[13]Lee, M. E. M. & Ockendon, H. (2005) A continuum model for entangled fibres. Euro. J. Appl. Math. 16, 145–160.CrossRefGoogle Scholar
[14]Murray, J. D. (1993) Mathematical Biology. 2nd ed., Springer-Verlag, New York.CrossRefGoogle Scholar
[15]Myers, T. G. (1998) Thin films with high surface tension. SIAM Rev. 40 (3), 441–462.CrossRefGoogle Scholar
[16]Oster, G. F., Murray, J. D. & Harris, A. K. (1983) Mechanical aspects of mesenchymal morphogenesis. J. Embryol. Exp. Morph. 78, 83–125.Google ScholarPubMed
[17]Rogers, T. G. (1989) Squeezing flow of fibre-reinforced viscous fluids. J. Engng. Math. 23, 81–89.CrossRefGoogle Scholar
[18]Sawhney, R. K. & Howard, J. (2002) Slow local movements of collagen fibres by fibroblasts drive the rapid global self-organisation of collagen gels. J. Cell Biology, 157 (6), 1083–1091.CrossRefGoogle Scholar
[19]Spencer, A. J. M. (1972) Deformations of Fibre-Reinforced Materials. Oxford University Press. Oxford, 128 pp.Google Scholar
[20]Spencer, A. J. M. (1997) Fibre-streamline flows of fibre-reinforced viscous fluids. Euro. J. Appl. Math. 8, 209–215.CrossRefGoogle Scholar
[21]Spencer, A. J. M. (2000) Theory of fabric-reinforced viscous fluids. Composites: Part A. 31, 1311–1321.CrossRefGoogle Scholar
[22]Tosin, A., Ambrosi, D. & Preziosi, L. (2006) Mechanics and chemotaxis in the morphogenesis of vascular networks. Bull. Math. Biol. 68 (7), 1819–1836.CrossRefGoogle ScholarPubMed
[23]Tranquillo, R. T. & Murray, J. D. (1993) Mechanistic model of wound contraction. J. Surg. Res. 55, 233–247.CrossRefGoogle Scholar