Skip to main content
SpringerLink
Log in

Principles of the Dimensional Analysis

  • Chapter
  • First Online:
Dimensional Analysis Beyond the Pi Theorem

Abstract

Nearly all scientists at conjunction with simplifying a differential equation have probably used dimensional analysis. Dimensional analysis (also called the factor-label method or the unit factor method) is an approach to problem that uses the fact that one can multiply any number or expression without changing its value. This is a useful technique. However, the reader should take care to understand that chemistry is not simply a mathematics problem. In every physical problem, the result must match the real world.

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 109.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 139.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 139.99
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

References

  1. G.I. Barenblatt, Dimensional Analysis (Gordon and Breach Science, New York, 1987)

    Google Scholar 

  2. Galileo Galilei, Discorsi e Dimostrazioni Matematiche intorno à due nuoue scienze Attenenti alla Mecanica & i Movimenti Locali (1638)

    Google Scholar 

  3. I. Stewart, Does God Play Dice (Penguin, London, 1989), p. 219

    MATH  Google Scholar 

  4. Tina Komulainen, Helsinki University of Technology, Laboratory of Process Control and Automation. tiina.komulainen@hut.fi

    Google Scholar 

  5. J. Sylvan Katz, The Self-Similar Science System. Research Policy 28, 501–517 (1999)

    Article  Google Scholar 

  6. C. Judd, Fractals ¨C Self-Similarity. http://www.bath.ac.uk/¡«ma0cmj/FractalContents.html. Accessed 16 Mar 2003

  7. N. Shiode, M. Batty, Power law distributions in real and virtual worlds. http://www.isoc.org/inet2000/cdproceedings/2a/2a_2.htm. Accessed 17 Mar 2003

  8. Charles Stuart University. The Hausdorff Dimension http://life.csu.edu.au/fractop/doc/notes/chapt2a2.html. Accessed 17 Mar 2003

  9. P. Krugman, The Self-Organizing Economy (Blackwell, Oxford, 1996)

    Google Scholar 

  10. G.I. Barenblatt, Scaling Phenomena in Fluid Mechanics, 1st edn. (Cambridge University Press, Cambridge, 1994)

    MATH  Google Scholar 

  11. O. Petruk, Approximations of the self-similar solution for blast wave in a medium with power-law density variation (Institute for Applied Problems in Mechanics and Mathematics NAS of Ukraine, Lviv, 2000)

    Google Scholar 

  12. G.I. Taylor, Proc. R. Soc. London A201, 159 (1950)

    Article  Google Scholar 

  13. L. Sedov, Prikl. Mat. Mekh. 10, 241 (1946)

    Google Scholar 

  14. L. Sedov, Similarity and Dimensional Methods in Mechanics (Academic, New York, 1959)

    MATH  Google Scholar 

  15. R.K. Merton, The Matthew effect in science. Science 159(3810), 56–63 (1968)

    Article  Google Scholar 

  16. R.K. Merton, The Matthew effect in science II. ISIS 79, 606–623 (1988)

    Article  Google Scholar 

  17. W.B. Krantz, Scaling Analysis in Modeling Transport and Reaction Processes (Wiley Interscience, Hoboken, 1939)

    Google Scholar 

  18. I. Proudman, J.R.A. Pearson, J. Fluid Mech 2, 237 (1957)

    Article  MathSciNet  Google Scholar 

  19. Harald Hance Olsen. Bukingham’s Pi Theorem. www.math.ntnu.no/~hanche/notes/buckingham/buckingham-a4.pdf

  20. G.I. Taylor, The formation of a blast wave by a very intense explosion. I. Theoretical discussion. Proc. Roy Soc. A 201, 159–174 (1950)

    Article  MATH  Google Scholar 

  21. G.I. Taylor, The formation of a blast wave by a very intense explosion. II. The atomic explosion of 1945. Proc. Roy Soc. A 201, 175–186 (1950)

    Article  MATH  Google Scholar 

  22. C.L. Dym, Principle of Mathematical Modeling. 2nd edn, (Elsevier, 2004)

    Google Scholar 

  23. L.D. Landau, E.M. Lifshitz, Fluid Mechanics (Pergamon Pres, New York, 1959). § 61

    MATH  Google Scholar 

  24. R. Illner, C.S. Bohun, S. McCollum, T. Van Roode, Mathematical Modeling, A Case Studies Approach (American Mathematical Society (AMS), 2005)

    Google Scholar 

  25. Y.B. Zel’dovich, Y.P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamics Phenomena (Dover Publication, New York, 2002)

    Google Scholar 

  26. G. Guderley, Starke kugelige und zylindrische Verdichtungsstosse in der Nahe des Kugelmittelpunktes bzw. der Zylinderachse. Luftfahrt-Forsch 19, 302–312 (1942)

    MathSciNet  MATH  Google Scholar 

  27. D.S. Butler, Converging spherical and cylindrical shocks. Armament Research Establishment report 54/54 (1954)

    Google Scholar 

  28. J.H. Lau, M.M. Kekez, G.D. Laugheed, P. Savic, Spherically Converging Shock Waves in Dense Plasma Research, in Proceeding of 10th International Symposium on Shock Tube and Waves, 1979, p. 386

    Google Scholar 

  29. J. Nuckolls, L. Wood, A. Thiessen, G. Zimmerman, Laser compression of matter to super-high densities: thermonuclear (CRT) applications. Nature 239 (1972)

    Google Scholar 

  30. I.I. Glass, D. Sagie, Application of Explosive-Driven Implosions to Fusion. Physics of Fluids 25, 269–270 (1982)

    Article  Google Scholar 

  31. F. Winterberg, The Physical Principles of Thermonuclear Explosive Devices (Fusion Energy Foundation Frontiers of Science Series, New York, 1981)

    Google Scholar 

  32. L. Woljer, Supernova Remnants. Ann. Rev. Astr. 10, 129 (1972)

    Article  Google Scholar 

  33. I.I. Glass, S.P. Sharma, Production of Diamonds From Graphite using Explosive-Driven Implosions. AIAA Journal 14(3), 402–404 (1976)

    Article  Google Scholar 

  34. K.P. Stanyukovich, Unsteady Motion of Continuous Media (Academic, New York, 1960)

    Google Scholar 

  35. S. Ashraf, Approximate Analytic Solution of Converging Spherical and Cylindrical Shocks with Zero Temperature Gradient in the Rear Flow Field. Z. angew. Math. Phys. 6(6), 614 (1973)

    MATH  Google Scholar 

  36. S. Yadegari, Self-similarity. http://www-crca.ucsd.edu/¡«syadegar/MasterThesis/node25.html. Accessed 16 Mar 2003

  37. B.J. Carr, A.A. Coley, Self-similarity in general relativity. http://users.math.uni-potsdam.de/~oeitner/QUELLEN/ZUMCHAOS/selfsim1.htm. Accessed 16 Mar 2003

  38. V.J. Skoglund, Similitude, Theory and Applications (International Textbook Company, Scranton, 1967)

    Google Scholar 

  39. H. Schlichting, Boundary Layer Theory, 4th edn. (McGraw-Hill Book Company, New York, 1960)

    MATH  Google Scholar 

  40. G.I. Barenblatt, ‘Scaling’ Cambridge Texts in Applied Mathematics (Cambridge University Press, Cambridge, 2006)

    Google Scholar 

  41. A. Sakurai, On the problem of shock wave arriving at the edge of gas. Commun. Pure. Appl. Math 13, 353 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  42. P.L. Sachdev, S. Ashraf, Strong shock with radiation near the surface of a star. Phys. Fluids 14, 2107 (1971)

    Article  Google Scholar 

  43. J. von Neumann, Blast Waves Los Alamos Science Laboratory Technical Series, vol. 7 (Los Alamos, 1947).

    Google Scholar 

  44. L.I. Sedov, Similarity and Dimensional Methods in Mechanics (Academic, New York, 1969). Chap. IV

    MATH  Google Scholar 

  45. E. Waxman, D. Shvarts, Second-type self-similar solutions to the strong explosion problem. Phys. Fluids A 5, 1035 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  46. P. Best, R. Sari, Second-type self-similar solutions to the ultra-relativistic strong explosion problem. Phys. Fluids 12, 3029 (2000)

    Article  MATH  Google Scholar 

  47. R.’m. Sari, First and second type self-similar solutions of implosions and explosions containing ultra relativistic shocks. Physics of Fluids 18, 027106 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  48. R.D. Blandford, C.F. McKee, Fluid dynamics of relativistic blast waves. Phys. Fluids 19, 1130 (1976)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Galaxy Advanced Engineering, Inc., San Mateo, California, USA

    Bahman Zohuri

Authors
  1. Bahman Zohuri
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Zohuri, B. (2017). Principles of the Dimensional Analysis. In: Dimensional Analysis Beyond the Pi Theorem. Springer, Cham. https://doi.org/10.1007/978-3-319-45726-0_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-45726-0_1

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-45725-3

  • Online ISBN: 978-3-319-45726-0

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation