Skip to main content
SpringerLink
Log in

Asymptotic Methods For PDE Problems In Fluid Mechanics and Related Systems With Strong Localized Perturbations In Two-Dimensional Domains

  • Chapter
Asymptotic Methods in Fluid Mechanics: Survey and Recent Advances

Part of the book series: CISM Courses and Lectures ((CISM,volume 523))

Abstract

The method of matched asymptotic expansions is a powerful systematic analytical method for asymptotically calculating solutions to singularly perturbed PDE problems. It has been successfully used in a wide variety of applications (cf. Kevorkian and Cole (1993), Lagerstrom (1988), Dyke (1975)). However, there are certain special classes of problems where this method has some apparent limitations.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  • C. Bandle and M. Flucher. Harmonic radius and concentration of energy; hyperbolic radius and liouville’s equation. SIAM Review, 38(2):191–238, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  • P. C. Bressloff, D. Earnshaw, and M. J. Ward. Diffusion of protein receptors on a cylindrical dendritic membrane with partially absorbing traps. SIAM J. Appl. Math., 68(5):1223–1246, 2008.

    Article  MATH  MathSciNet  Google Scholar 

  • A. Cheviakov and M. J. Ward. Optimizing the fundamental eigenvalue of the laplacian in a sphere with interior traps. Mathematical and Computer Modeling, to appear, 2010.

    Google Scholar 

  • A. Cheviakov, M. J. Ward, and R. Straube. An asymptotic analysis of the mean first passage time for narrow escape problems: Part ii: The sphere. SIAM J. Multiscale Modeling, to appear, 2010.

    Google Scholar 

  • J. Choi, D. Margetis, T. M. Squires, and M. Z. Bazant. Steady advection-diffusion around finite absorbers in two-dimensional potential flows. J. Fluid Mech., 536:155–184, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  • D. Coombs, R. Straube, and M. J. Ward. Diffusion on a sphere with localized traps: Mean first passage time, eigenvalue asymptotics, and fekete points. SIAM J. Appl. Math., 70(1):302–332, 2009.

    Article  MATH  MathSciNet  Google Scholar 

  • A. M. Davis and S. G. Llewellyn-Smith. Perturbation of eigenvalues due to gaps in two-dimensional boundaries. Proc. Roy. Soc. A, 463:759–786, 2007.

    Article  MATH  MathSciNet  Google Scholar 

  • W. W. Dijkstra and M. E. Hochstenbach. Numerical approximation of the logarithmic capacity. CASA Report 08-09, Technical U. Eindhoven, 2008.

    Google Scholar 

  • M. Van Dyke. Perturbations Methods in Fluid Mechanics. Parabolic Press, 1975.

    Google Scholar 

  • J. E. Fletcher. Mathematical modeling of the microcirculation. Math Biosciences, 38:159–202, 1978.

    Article  MATH  MathSciNet  Google Scholar 

  • I. Imai. On the asymptotic behavior of viscous fluid flow at a great distance from a cylindrical body, with special reference to filon’s paradox. Proc. Roy. Soc. A, 208:487–516, 1951.

    Article  MATH  Google Scholar 

  • S. Kaplun. Low reynolds number flow past a circular cylinder. J. Math. Mech., 6(5):52–60, 1957.

    Google Scholar 

  • J. B. Keller and M. J. Ward. Asymptotics beyond all orders for a low reynolds number flow. J. Engrg. Math., 30(1–2):253–265, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  • J. Kevorkian and J. Cole. Multiple Scale and Singular Perturbation Methods. Applied Mathematical Sciences, Vol. 114, Springer-Verlag, 1993.

    Google Scholar 

  • T. Kolokolnikov, M. Titcombe, and M. J. Ward. Optimizing the fundamental neumann eigenvalue for the laplacian in a domain with small traps. Europ. J. Appl. Math., 16(2):161–200, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  • A. Krogh. The number and distribution of capillaries in muscles with calculations of the oxygen pressure head necessary for supplying the tissue. J. Physiology (London), 52:409–415, 1919.

    Google Scholar 

  • M. C. Kropinski, M. J. Ward, and J. B. Keller. A hybrid asymptotic-numerical method for calculating low reynolds number flows past symmetric cylindrical bodies. SIAM J. Appl. Math., 55(6):1484–1510, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  • P. A. Lagerstrom. Matched Asymptotic Expansions. Applied Mathematical Sciences, Vol. 76, Springer-Verlag, 1988.

    Google Scholar 

  • C. Lange and H. Weinitschke. Singular perturbations of elliptic problems on domains with small holes. Stud. Appl. Math., 92(1):55–93, 1994.

    MATH  MathSciNet  Google Scholar 

  • S. H. Lee and L. G. Leal. Low reynolds number flow past cylindrical bodies of arbitrary cross-sectional shape. J. Fluid Mech., 164:401–427, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  • MATLAB. Partial Differential Equations Toolbox, User’s Guide. The Math-Works, Inc., 1996.

    Google Scholar 

  • M. Matthews and J. Hill. Flow around nanospheres and nanocylinders. Q. J. Mech. Appl. Math., 59(2):191–210, 2006.

    Article  MATH  MathSciNet  Google Scholar 

  • M. Matthews and J. Hill. Asymptotic analysis of the viscous micro/nano pump at low reynolds number. J. Engrg. Math., 63(2–4):279–292, 2009.

    Article  MATH  MathSciNet  Google Scholar 

  • A. Mikelić and M. Primicerio. A diffusion-consumption problem for oxygen in a living tissue perfused by capillaries. Nonlinear Diff. Eq. Appl., 13(3): 349–367, 2006.

    Article  MATH  Google Scholar 

  • L. M. Milne-Thomson. Theoretical Aerodynamics. Dover Publications Inc., 1958.

    Google Scholar 

  • S. Ozawa. Singular variation of domains and eigenvalues of the laplacian. Duke Math. J., 48(4):767–778, 1981.

    Article  MATH  MathSciNet  Google Scholar 

  • S. Pillay, M. J. Ward, A. Peirce, and T. Kolokolnikov. An asymptotic analysis of the mean first passage time for narrow escape problems: Part i: Two-dimensional domains. SIAM J. Multiscale Modeling, to appear, 2010.

    Google Scholar 

  • A. S. Popel. Theory of oxygen transport to tissue. Critical Reviews in Biomedical Engrg., 17:257–321, 1989.

    Google Scholar 

  • I. Proudman and J. Pearson. Expansions at small reynolds number for the flow past a sphere and a circular cylinder. J. Fluid Mech., 2:237–262, 1957.

    Article  MATH  MathSciNet  Google Scholar 

  • K. B. Ranger. Explicit solutions of the steady two-dimensional navier-stokes equations. Stud. Appl. Math., 94(2):169–181, 1995.

    MATH  MathSciNet  Google Scholar 

  • T. Ransford. Potential Theory in the Complex Plane. London Mathematical Society Student Texts Vol. 28, Cambridge University Press, 1995.

    Google Scholar 

  • K. Shintani, A. Umemura, and A. Takano. Low reynolds number flow past an elliptic cylinder. J. Fluid Mech., 136:277–289, 1983.

    Article  MATH  Google Scholar 

  • L. A. Skinner. Generalized expansions for slow flow past a cylinder. Q. J. Mech. Appl. Math., 28(3):333–340, 1975.

    Article  MATH  Google Scholar 

  • R. Straube and M. J. Ward. Intraceulluar signalling gradients arising from multiple compartments: A matched asymptotic expansion approach. SIAM J. Appl. Math., 70(1):248–269, 2009.

    Article  MATH  MathSciNet  Google Scholar 

  • M. Titcombe and M J. Ward. Convective heat transfer past small cylindrical bodies. Stud. Appl. Math., 99(1):81–105, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  • M. Titcombe and M J. Ward. Summing logarithmic expansions for elliptic equations in multiply-connected domains with small holes. Canadian Appl. Math. Quart., 7(3):313–343, 1999.

    MATH  MathSciNet  Google Scholar 

  • M. Titcombe and M J. Ward. An asymptotic study of oxygen transport from multiple capillaries to skeletal muscle tissue. SIAM J. Appl. Math., 60(5):1767–1788, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  • M. Titcombe, M J. Ward, and M. C. Kropinski. A hybrid asymptotic-numerical solution for low reynolds number flow past an asymmetric cylindrical body. Stud. Appl. Math., 105(2):165–190, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  • D. J. Tritton. Experiments on the flow past a circular cylinder at low reynolds numbers. J. Fluid Mech., 6:547–567, 1959.

    Article  MATH  Google Scholar 

  • J. Veysey and N. Goldenfeld. Simple viscous flows: From boundary layers to the renormalization group. Reviews of Modern Physics, 79:883–927, 2007.

    Article  MATH  MathSciNet  Google Scholar 

  • M. J. Ward, W. D. Henshaw, and J. B. Keller. Summing logarithmic expansions for singularly perturbed eigenvalue problems. SIAM J. Appl. Math., 53(3):799–828, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  • M. J. Ward and J. B. Keller. Strong localized perturbations of eigenvalue problems. SIAM J. Appl. Math., 53(3):770–798, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  • A. J. Ward-Smith. Internal Fluid Flow. Clarendon Press, Oxford, 1980.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada

    Michael J. Ward

  2. Department of Mathematics, Simon Fraser University, Burnaby, B.C., Canada

    Michael J. Ward & Mary-Catherine Kropinski

Authors
  1. Michael J. Ward
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mary-Catherine Kropinski
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Vienna University of Technology, Austria

    Herbert Steinrück

Rights and permissions

Reprints and permissions

Copyright information

© 2010 CISM, Udine

About this chapter

Cite this chapter

Ward, M.J., Kropinski, MC. (2010). Asymptotic Methods For PDE Problems In Fluid Mechanics and Related Systems With Strong Localized Perturbations In Two-Dimensional Domains. In: Steinrück, H. (eds) Asymptotic Methods in Fluid Mechanics: Survey and Recent Advances. CISM Courses and Lectures, vol 523. Springer, Vienna. https://doi.org/10.1007/978-3-7091-0408-8_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-7091-0408-8_2

  • Publisher Name: Springer, Vienna

  • Print ISBN: 978-3-7091-0407-1

  • Online ISBN: 978-3-7091-0408-8

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation