Skip to main content
SpringerLink
Log in
Book cover

A Celebration of Mathematical Modeling pp 199–220Cite as

Physical Insight, Mathematical Modeling and Asymptotics

  • Chapter

  • 489 Accesses

Abstract

We review several problems in fluid mechanics to demonstrate how physical insight of a problem leads to a simplified mathematical model, which specifies the choice of the proper scalings, the small parameters and the expansion scheme. The mathematical model in turn gives a more precise description of the insight. It defines the conditions under which the model is not applicable and the necessary refinement of the model. The model allows for a systematic derivation of the leading and higher order system of equations, identifies the initial and or/boundary conditions needed for the simplified system and derives the missing condition(s) of the leading order system of equations, if any, from the compatibility condition(s) on the next order. Special emphasis is placed on the importance of formulating the correct mathematical model relevant to the physical insight and the understanding of the physical implications.

This is a preview of subscription content, 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   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Bauer, F., Garabedian, P., Korn, D. and Jameson, A., Supercritical Wing Sections II, Lecture Notes in Economics and Mathematical Systems 108, Springer-Verlag, New York, (1975).

    Google Scholar 

  • Birkhoff, G. and Zarantonello, E. H., (1957) Jets, Wakes and Cavities, Academic Press, NY.

    Google Scholar 

  • Caflisch, R. E., Miksis, M. J., Papanicolaou, G. C. and Ting, L., (1985), Effective equations for wave propagation in a bubbly liquid, J. Fluid Mech., 153, 259–273.

    Article  MATH  Google Scholar 

  • Callegari, A. and Ting, L., (1978), Motion of a curved vortex filament with decaying vortical core and axial velocity, SIAM J. Appl. Math., 35, pp. 148–175.

    Article  MathSciNet  MATH  Google Scholar 

  • Courant, R. and Friedrichs, K. O., (1967) Supersonic Flow and Shock Waves, John Wiley, NY.

    Google Scholar 

  • Ferri, A., Libby, P. A., Bloom, M. and Zakkay, V., (1955) Development of Polytechnic Institute of Brooklyn Hypersonic Facilities, WADC TN-695, Wright Air Force Development Center, Dayton, OH.

    Google Scholar 

  • Friedrichs, K. O., (1953) Special Topics in Fluid Dynamics. New York Univ., New York.

    Google Scholar 

  • Friedrichs, K. O., (1955) Asymptotic phenomena in mathematical physics. Bull. Amer. Math. Soc. 61, 485–504.

    Article  MathSciNet  MATH  Google Scholar 

  • Hunter, J. K. and Keller, J. B., (1984) Weak shock diffraction, Wave Motion, 6, 79–89.

    Article  MathSciNet  MATH  Google Scholar 

  • Hunter, J. K., (1995) Asymptotic equations for nonlinear hyperbolic waves, Surveys in Appl. Math., 2, Eds.: J. B. Keller, D. McLaughlin, and G. Papanicolaou, Plenim Press.

    Google Scholar 

  • Kevorkian, J. and Cole, J. D., (1980) Perturbation Methods in Applied Mathematics, Springer-Verlag, NY.

    Google Scholar 

  • Keller, J. B. (1958), A geometrical theory of diffraction, in Proc. 8th Symp. Appl. Math., AMS, McGraw-Hill, NY, 153–175.

    Google Scholar 

  • Keller, J. B., (1980) Darcy’s law for flow in porous media and the two-space method. In Sternberg, R. L., Kalmowski, A. J. and Papadakis, J. S., editors: Nonlinear Partial Differential Equations in Engineering and Applied Sciences, Marcel Dekker, NY, 429–443.

    Google Scholar 

  • Keller, J. B., (1983) The breaking of liquid films and threads, Phys. Fluids, 26, 3451–3453.

    Article  MATH  Google Scholar 

  • Keller, J. B. (1985) One hundred years of diffraction theory. IEEE Trans. Antennas and Prop., 33, 123–127.

    Google Scholar 

  • Keller, J. B. and Blank, A., (1951) Diffraction and reflection of pulses by wedges and corners, Comm. Pure Appl. Math., 4, 317–328.

    Article  MathSciNet  Google Scholar 

  • Keller, J. B. and Geer, J., (1973) Flows of thin streams with free boundary. J. Fluid Mech. 59, 417–432.

    Article  MathSciNet  MATH  Google Scholar 

  • Keller, J. B., King, A. and Ting, L. (1995) Blob formation, Phys. Fluids, 7, 226–228.

    Article  MathSciNet  MATH  Google Scholar 

  • Keller, J. B. and Lewis, R. M., (1964) Asymptotic method for partial differential equations: the reduced wave equation and Maxwell’s equation, Rep. EM-194, CIMS, NYU. Also in Surveys in Appl. Math, 1, 1995, Plenum, NY.

    Google Scholar 

  • Keller, J. B. and Ting, L., (1951) Approximate equations for large scale atmospheric motions, Preprint, Inst, for Math. Mech., NYU.

    Google Scholar 

  • Keller, J. B. and Ting, L., (1966) Periodic vibrations of systems governed by nonlinear partial differential equations, Comm. Pure and Appl. Math, 19, 371–420.

    Article  MathSciNet  MATH  Google Scholar 

  • Keller, J. B. and Ting, L., (1993) Singularities of semilinear waves, Comm. Pure and Appl. Math, 46, 341–352.

    Article  MathSciNet  MATH  Google Scholar 

  • Keller, J. B. and Weitz, M. L., (1957) A theory of thin jets. Proc. 9th Intern. Congr. Appl. Mech. 1, 316–323, Univ. Brussels, Belgium.

    Google Scholar 

  • Knio, O, Ting, L. and Klein, R. (2003) Theory of compressible vortex filaments, in K. J. Bathe, editor, Proc. Second MIT Conf. on Comput. Fluid and Solid Mech, 2003, 1, 971–973, Elserver, NY.

    Google Scholar 

  • Krause, E.,(1967) Numerical solutions of the boundary equations. AIAA J., 5, 1231–1237.

    Google Scholar 

  • Lamb, H., (1932) Hydrodynamics, 6th ed. Cambridge Univ. Press, NY.

    MATH  Google Scholar 

  • Libby, P. A. and Economos, C., (1963) A flame zone model for chemical reaction in a laminar boundary layer with application to the injection of hydrogen-oxygen mixtures, Intern. J. Heat and Mass Transfer, 6, 113–128.

    Article  Google Scholar 

  • Lighthill, M. J., (1949) A technique for rendering approximate solutions to physical problems uniformly valid, and The shock strength in supersonic conical flow, Phil. Mag., Series 7, 40, 1179–1201 and 1201–1221.

    Google Scholar 

  • Messiter, A. F., (1970) Boundary layer near the trailing edge of a flat plate, SIAM J. Appl. Math., 18, 241–257.

    Article  MATH  Google Scholar 

  • Miksis, M. J. and Ting, L., (1983) Surface wave induced by an impinging jet, Phys. Fluids, 26, 2378–2384.

    Article  MathSciNet  MATH  Google Scholar 

  • Miksis, M. J. and Ting, L., (1992) Effective equations for multiphase flows–waves in a bubbly liquid. In Hutchinson, J. W. and Wu, T. Y., editors: Advances in Applied Mechanics, 28, Academic Press, NY, 142–256.

    Google Scholar 

  • Morikawa, G. k., (1960) Geostrophic vortex motion, J. Meteorol., 17, 148–158.

    Article  Google Scholar 

  • Peters, N. (1979) Premixed burning in diffusion flames–the flame zone model of Libby and Economos, Intern. J. of Heat and Mass Transfer, 22, 691–703.

    Article  MATH  Google Scholar 

  • Rauch, J. B. and Reed, M. C., (1982) Singularities produced by nonlinear interaction of three progressing waves: Examples, Comm. Partial Diff. Eqs. 7, 1117–1133.

    MathSciNet  MATH  Google Scholar 

  • Lord Rayleigh, (1877) 1 st Ed., Theory of Sound, reprinted, Dover Publ., NY.

    Google Scholar 

  • Schlichting, H. and Gersten, G., (1999) Boundary Layer Theory, 8th Ed., Springer-Verlag, NY.

    Google Scholar 

  • Stoker, J. J., (1957) Water Waves. John Wiley, NY.

    MATH  Google Scholar 

  • Ting, L., (1959) On the mixing of two parallel streams. J. Math. Phys., 38, 153–165.

    MathSciNet  MATH  Google Scholar 

  • Ting, L., (1960) Boundary layer over a flat plate in the presence of a shear flow. Phys. Fluids, 3, (1960), 78–81.

    Article  MathSciNet  MATH  Google Scholar 

  • Ting, L., (1965) On the initial conditions for boundary layer equations. J. Math. Phys., 44, 353–367.

    MathSciNet  Google Scholar 

  • Ting, L., (2000a) Boundary layer theory to matched asymptotics, ZAMM, 80, 845–855.

    Article  MathSciNet  MATH  Google Scholar 

  • Ting, L., (2000b) Appropriate boundary conditions simulating flows around a body at high Reynolds number, ISBN: 84–89925–70–4, CD–ROM, Proc. Euro. Congr. on Comput. Methods in Appl. Sc. and Engrg, 2000.

    Google Scholar 

  • Ting, L. and Gunzburger, M., (1970) Diffraction of shock waves by a moving thin wing, J. Fluid Mechanics, 42, 585–608.

    Article  MATH  Google Scholar 

  • Ting, L. and Keller, J. B., (1974) Planing of a flat plate at high Froude number, Phys. Fluids, 17, 1080–1086.

    Article  MATH  Google Scholar 

  • Ting, L. and Keller, J. B., (1977) Radiation from the open end of a cylindrical or conical pipe and scattering from the end of a rod or slab, J. Acoust. Soc. Am. 61, 1438–1444.

    Article  MATH  Google Scholar 

  • Ting, L. and Keller, J. B., (1990) Slender jets and thin sheets with surface tension, SIAM J. Appl. Math., 50, 1533–1546.

    Article  MathSciNet  MATH  Google Scholar 

  • Ting, L. and Keller, J. B., (2000) Weak diffracted shocks near singular rays, Methods and Appl. of Analysis, 7, 565–576.

    MathSciNet  MATH  Google Scholar 

  • Ting, L. and Klein, R., (199 1) Viscous Vortical Flows, Lecture Notes in Physics 374, Springer-Verlag, NY.

    Google Scholar 

  • Ting, L. and Ling, G. C., (1983), Studies of the motion and core structure of a geostrophic vortex, Proc. 2nd Asian Congr. on Fluid Mech., 900–905, Science Press, Beijing, China.

    Google Scholar 

  • Ting, L. and Tung, C., (1965), On the motion and decay of a vortex in a nonuniform stream, Phys. Fluids 8, 1038–1051.

    Article  MathSciNet  Google Scholar 

  • Van Dyke, M., (1975) Perturbation Methods in Fluid Mechanics, The Parabolic Press, Stanford, CA.

    Google Scholar 

  • von Mises, R. and Friedrichs, K. O., (1946) Fluid Dynamics, Brown University Lecture Notes. Reprinted as Applied Mathematical Sciences, 5, Springer-Verlag, NY, 1971.

    Google Scholar 

  • Whitham, G. B., (1974) Linear and Nonlinear Waves, John Wiley, NY.

    MATH  Google Scholar 

  • Van Wijngaarden, L., (1972) One dimensional flow of liquids containing bubbles, Ann. Rev. Fluid Mech., 4. 369–396.

    Article  Google Scholar 

  • Zahalak, G. I. and Myers, M. K., (1974) Conical flow near singular rays, J. Fluid Mech., 63, 537–561.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, NY, NY, 10012, USA

    Ting Lu

Authors
  1. Ting Lu
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Aerospace Engineering, Technion — Israel Institute of Technology, Haifa, Israel

    Dan Givoli

  2. Department of Mathematics, University of Basel, Basel, Switzerland

    Marcus J. Grote

  3. Department of Mathematics, Stanford University, Stanford, California, USA

    George C. Papanicolaou

Rights and permissions

Reprints and permissions

Copyright information

© 2004 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Lu, T. (2004). Physical Insight, Mathematical Modeling and Asymptotics. In: Givoli, D., Grote, M.J., Papanicolaou, G.C. (eds) A Celebration of Mathematical Modeling. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0427-4_11

Download citation

  • DOI: https://doi.org/10.1007/978-94-017-0427-4_11

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-6526-1

  • Online ISBN: 978-94-017-0427-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation