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HISTORY OF PHYSICS: ON THE PRANDTL BOUNDARY LAYER

The following points are made by John D. Anderson Jr (Physics Today 2005 December):

1) During the week of 8 August 1904, a small group of mathematicians and scientists gathered in picturesque Heidelberg, Germany, known for its baroque architecture, cobblestone streets, and castle ruins that looked as if they were still protecting the old city. Home to Germany's oldest university, which was founded in 1386, Heidelberg was a natural venue for the 3rd International Mathematics Congress. One of the presenters at the congress was Ludwig Prandtl (1875-1953), a 29-year-old professor at the Technische Hochschule (equivalent to a US technical university) in Hanover. Prandtl's presentation was only 10 minutes long, but that was all the time needed to describe a new concept that would revolutionize the understanding and analysis of fluid dynamics. His presentation, and the subsequent paper that was published in the congress's proceedings one year later, introduced the concept of the boundary layer in a fluid flow over a surface. In 2005, concurrent with the World Year of Physics celebration of, among other things, Albert Einstein and his famous papers of 1905, we should also celebrate the 100th anniversary of Prandtl's seminal paper. The modern world of aerodynamics and fluid dynamics is still dominated by Prandtl's idea. By every right, his boundary-layer concept was worthy of the Nobel Prize. He never received it, however; some say the Nobel Committee was reluctant to award the prize for accomplishments in classical physics.

2) Archimedes (287-212 BC) introduced some basic ideas in fluid statics, and Leonardo da Vinci (1452 1519) observed and drew sketches of complex flows over objects in streams. But a quantitative physical and mathematical understanding of fluid flow began -- haltingly -- only when Isaac Newton (1642 1727) devoted Book II of his Principia Mathematica (1687) exclusively to the examination of fluid dynamics and fluid statics. Efforts to obtain a mathematical formulation of a fluid flow took shape during the century following the publication of the Principia with the contributions of Daniel Bernoulli (1700 82), Jean le Rond d'Alembert (1717 83), and Leonhard Euler (1707 83) -- all well-known heavy hitters in classical physics.

3) Of the three, Euler was the most instrumental in conceptualizing the mathematical description of a fluid flow. He described flow in terms of spatially varying three-dimensional pressure and velocity fields and modeled the flow as a continuous collection of infinitesimally small fluid elements. By applying the basic principles of mass conservation and Newton's second law, Euler obtained two coupled, nonlinear partial differential equations involving the flow fields of pressure and velocity. Although those Euler equations were an intellectual breakthrough in theoretical fluid dynamics, obtaining general solutions of them was quite another matter. Moreover, Euler did not account for the effect of friction acting on the motion of the fluid elements that is, he ignored viscosity.

4) It was another hundred years before the Euler equations were modified to account for the effect of internal friction within a flow field. The resulting equations, a system of even more elaborate nonlinear partial differential equations now called the Navier Stokes equations, were first derived by Claude-Louis Navier in 1822, and then independently derived by George Stokes in 1845. To this day, those equations are the gold standard in the mathematical description of a fluid flow, and no one has yet obtained a general analytical solution of them.

5) Against this backdrop, along came Prandtl and his seminal presentation at Heidelberg. The companion paper, entitled "Über Flüssigkeitsbewegung bei sehr kleiner Reibung" ("On the Motion of Fluids with Very Little Friction"), was only eight pages long, but it would prove to be one of the most important fluid-dynamics papers ever written.[2,3] Prandtl's paper gave the first description of the boundary-layer concept. He theorized that an effect of friction was to cause the fluid immediately adjacent to the surface to stick to the surface -- in other words, he assumed the no-slip condition at the surface -- and that frictional effects were experienced only in a boundary layer, a thin region near the surface. Outside the boundary layer, the flow was essentially the inviscid flow that had been studied for the previous two centuries. [4,5]

References (abridged):

1. See for example J. D. Anderson Jr, Introduction to Flight, 5th ed., McGraw-Hill Higher Education, Boston (2005)

2. For a more thorough discussion see J. D. Anderson Jr, A History of Aerodynamics, Cambridge U. Press, New York (1998)

3. L. Prandtl, in Verhandlungen des dritten internationalen Mathematiker-Kongresses in Heidelberg 1904, A. Krazer, ed., Teubner, Leipzig, Germany (1905), p. 484. English trans. in Early Developments of Modern Aerodynamics, J. A. K. Ackroyd, B. P. Axcell, A. I. Ruban, eds., Butterworth-Heinemann, Oxford, UK (2001), p. 77

4. S. Goldstein, in Annual Review of Fluid Mechanics, vol. 1, W. R. Sears, M. Van Dyke, eds., Annual Reviews, Palo Alto, CA (1969), p. 1

5. P. R. H. Blasius, Z. Math. Phys. 56, 1 (1908)

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