Ludwig Prandtl’s Boundary Layer Theory

Abstract

Boundary layer theory formally came into existence in Heidelberg, Germany at 11:30 am on August 12, 1904 when Ludwig Prandtl (1875–1953), a professor (and chair) of mechanics at the Technical University of Hanover (the youngest professor in Prussia according to Bodenschatz and Eckert [49]), gave a ten-minute talk to the Third International Congress of Mathematicians entitled “Über Flüssigkeitsbewegung bei sehr kleiner Reibung” (On Fluid Motion with Small Friction). (Figs. 1.1 and 1.2).

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Boundary layer theory formally came into existence in Heidelberg, Germany at 11:30 am on August 12, 1904 when Ludwig Prandtl (1875–1953), a professor (and chair) of mechanics at the Technical University of Hanover (the youngest professor in Prussia according to Bodenschatz and Eckert [49]), gave a ten-minute talk to the Third International Congress of Mathematicians entitled “Über Flüssigkeitsbewegung bei sehr kleiner Reibung” (On Fluid Motion with Small Friction). (Figs. 1.1 and 1.2). (Recall that Hilbert presented his famous list of twenty-three problems for the new century at the second ICM in Paris in 1900.) Prandtl’s resulting seven-page paper in the proceedings, Prandtl [399] [translated to English as NACA Memo No. 452 (1928)] states:

The physical processes in the boundary layer (Grenzschicht ) between fluid and solid body can be calculated in a sufficiently satisfactory way if it is assumed that the flow adheres to the walls, so that the total velocity there is zero—or equal to the velocity of the body. If the velocity is very small and the path of the fluid along the wall is not too long, the velocity will have again its usual value very near to the wall. In the thin transition layer (Übergangsschicht) the sharp change of velocity, in spite of the small viscosity coefficient, produces noticeable effects.

Figure 1.1:

Title page: Proceedings, Third International Congress of Mathematicians

Figure 1.2:

First page: L. Prandtl’s talk to International Congress, 1904

(Here, and below, quotations will usually be indented.)

Prandtl’s amazing scientific insight, evolving from an ultra-practical era of hydraulics, but less than a year after the Wright brothers’ flight, involves the basic concept of what would become singular perturbations. The governing Navier–Stokes equations

$$\displaystyle{\rho _{0}(\partial _{t}u + (u \cdot \nabla )u) + \nabla p = \frac{1} {\mbox{ Re }}\varDelta u,\ \ \nabla \cdot u = 0}$$

(cf., for background, Anderson [8], and Drazin and Riley [125]) reduce to Euler’s equations when the viscosity (1∕Re for the dimensionless Reynolds number) is zero, but the solutions do not uniformly reduce to those of Euler’s equations when the viscosity tends to zero. As Ting [483] summarized:

Prandtl’s boundary layer theory initiated a systematic procedure for joining local and global expansions to form uniformly-valid approximations.

This technique, later known as the method of “matched asymptotic expansions,” identifies the (local) boundary layer solution and the inviscid solution with the leading order “inner” and “outer” solutions (cf. Darrigol [110], Meier [314], Eckert [131], and O’Malley [373] for background material).

Prandtl’s student (and brother-in-law) Ludwig Föppl gave the following opinion in his personal memoirs:

In view of the importance of the work, I would like to point out its essentials. By that time, there had been no theoretical explanation for the drag experienced by a body in a flowing liquid or in air. The same applies to the lift on an airplane. Classical mechanics was either based on frictionless flow or, when friction was taken into account, mathematical difficulties were so enormous that hitherto no practical solution had been found. Prandtl’s idea that led out of the bottleneck was the assumption that a frictionless flow was everywhere with the exception of the region along solid boundaries. Prandtl showed that friction, however small, had to be taken into account in a thin layer along the solid wall. Since that time, this layer has been known as Prandtl’s boundary layer. With these simplifying assumptions, the mathematical difficulties $\ldots$ could be overcome in a number of practical cases.

(See Stewartson [474], regarding the troublesome d’Alembert’s paradox (having the implausible no drag conclusion) that Prandtl’s theory eliminated).

Prandtl’s father Alexander was a surveying and engineering professor at a Bavarian agricultural college in Weihenstephan, while his mother, the former Magdelene Ostermann, spent much of her life as a mental patient. Ludwig was the only child of three to survive birth. Both parents died before he was twenty-five. Though the family was Catholic, Prandtl didn’t practice his religion as an adult. His earlier education was in engineering at the Technical University of Munich, but Prandtl received his Ph.D. in mathematics in 1900 from the University of Munich for a thesis on the torsional instability of beams with an extreme depth-width ratio, under the supervision of Professor August Föppl of the Technical University. Prandtl worked for a year at Maschinenfabrik Augsburg-Nürnberg designing diffusers to increase the efficiency of wood-cutting machines, so the flow concerned consisted of wood shavings.

(We note that informative biographies of many mathematicians, though not Prandtl, are available on the MacTutor History of Mathematics archive [355] (St. Andrews University website.)

Officially named the Georg-August Universität Göttingen, the University of Göttingen was founded in 1734 by George II, King of Great Britain and Elector of Hanover. The geometer Felix Klein (1849–1925) hired both Prandtl and Carl Runge at Göttingen in fall 1904 from Hanover to begin an Institute for Applied Mathematics and Mechanics, as he sought to narrow the gulf between mathematics and technology in Göttingen. (Klein’s health had collapsed after intense competition with Henri Poincaré concerning automorphic functions, after which he became a university administrator and power broker in German mathematics (cf. Rowe [425], Hersh and John-Steiner [204], Gray [182], and Verhulst [502]). Klein came to Göttingen from Leipzig in 1886, hired David Hilbert from Königsberg in 1895, and earned the honorific title Geheimrat (similar to the British privy councillor). Klein especially appreciated Prandtl’s

strong power of intuition and great originality of thought with the expertise of an engineer and the mastery of the mathematical apparatus.

Anticipating more recent schemes, Klein founded the Göttingen Society for the Promotion of Applied Mathematics and Physics, allowing him to obtain and spend supplementary industrial funding for favored projects. Prandtl ultimately supervised 85 dissertations, published 1600 pages of technical papers, and directed an aeronautical proving ground (from 1907) and the Kaiser Wilhelm Institute for Fluid Mechanics (from 1925). Both organizations survive, though with changed names (cf. Oswatitsch and Wieghardt [383]). Prandtl ultimately published about 33 papers on boundary layers and turbulence and directed 31 theses on those subjects.

Theodore von Kármán (1881–1963) arrived as a graduate student in Göttingen in 1906, after receiving an undergraduate degree in Budapest in 1902. Gorn [177] reports:

Almost from the start, a thinly concealed rivalry developed between the 31-year-old mentor and the 25-year-old pupil. The Hungarian’s joie de vivre contrasted sharply with the shy, formal, pedantic habits of Prandtl.

von Kármán received his doctorate in 1908 for work on the buckling of columns and he served as an assistant to Prandtl (Privat-docent) until 1913, when he succeeded Hans Reissner as Professor of Aeronautics and Mechanics at the Technical University of Aachen. He served in the Austro-Hungarian army from 1915–1918 and left Aachen for the California Institute of Technology (or, more informally, Caltech) in 1930. In the (posthumous) autobiography von Kármán and Edson [241], von Kármán observes:

I came to realize that ever since I had come to Aachen, my old professor and I were in a kind of world competition. The competition was gentlemanly, of course. But it was first-class rivalry nonetheless, a kind of Olympic games, between Prandtl and me, and beyond that, between Göttingen and Aachen. The “playing field” was the Congress of Applied Mechanics. Our “ball” was the search for a universal law of turbulence.

The competition between von Kármán and Prandtl regarding turbulence is further highlighted in Leonard and Peters [285].

von Kármán’s former student Bill Sears (cf. Sears [444]) wrote:

Dr. von Kármán was a master of the à propos anecdote . He never forgot a joke, and always had one to illustrate most vividly and tellingly any situation in which he found himself. The result is that memories of von Kármán tend to become collections of anecdotes.

Somewhat unfortunately, then, von Kármán’s colorful stories provide us information about Prandtl, though they may not necessarily be strictly accurate. (See Nickelson et al. [348], a recent biography of von Kármán.) von Kármán characterized Prandtl’s life as

particularly full of overtones of naïveté.

One often-quoted von Kármán story (reproduced in Lienhard [289]) reads:

In 1909, for example, Prandtl decided that he really ought to marry; but he didn’t know how to proceed. Finally, he wrote to Mrs. Föppl, asking for the hand of one of her daughters. But which one? Prandtl had not specified. At a family conference, the Föppls made the practical decision that he should marry the eldest daughter, Gertrude. He did and the marriage was apparently a happy one. Daughters were born in 1914 and 1917.

Klaus Gersten, editor of the latest edition of Schlichting’s Boundary Layer Theory [438], insists that Prandtl married the appropriate daughter since he was 34 and the two Föppl daughters were then 27 and 17.

Intrigued readers might recall von Neumann’s definition (cf. Beckenbach [34]):

It takes a Hungarian to go into a revolving door behind you and come out first.

See Horvath [216] and Dyson [129], however, regarding the extraordinary (largely Jewish) Hungarian contributions to twentieth-century mathematics.

In his history of aerodynamics, von Kármán [239] summarized Prandtl’s skills as follows:

His control of mathematical methods and tricks was limited$\ldots$. But his ability to establish systems of simplified equations which expressed the essential physical relations and dropped the nonessentials was unique. $\ldots$ Prandtl had so precise a mind that he could not make a statement without qualifying it. This is a mistake. To be effective a teacher must see that a beginner in science grasps the basic principle before he can be expected to understand the exceptions.

In an obituary of Prandtl (Busemann [61]), his former assistant Adolf Busemann, from NASA’s Langley Research Center, wrote:

According to his aim to make his sentences foolproof, they required a rewording, a re-phrasing, an explanation of the reasons for re-phasing and perhaps a discussion of some exceptions to the stated rule. It is quite obvious how different the results of such lectures might have been for beginners and for listeners with background. Seeing the details as clearly as Prandtl did, and never trying to circumvent difficulties by omitting them, made the lectures, seminars, colloquia a rich source of information for all his pupils, beside those fortunate few who walked with him home from work.

Certainly, Prandtl was not a great teacher and expositor (as his father-in-law (cf. Holton [210]). Likewise, there is no doubt that von Kármán was more charming and mathematically sophisticated (cf. von Kármán and Biot [240] and von Kármán’s [238] Gibbs lecture to the American Mathematical Society). Sears [444] wrote:

von Kármán never identified himself as a mathematician$\ldots$ always as an engineer. But it was clear to those of us who worked close to him that mathematics—applied mathematics—was his first love.

In his autobiography, von Kármán concluded:

In my opinion Prandtl unravelled the puzzle of some natural phenomena of tremendous importance and was deserving of a Nobel prize .

G. I. Taylor (1886–1975), the leading British fluid dynamicist of the twentieth century, wrote Prandtl in 1935 to say that it was insulting that Prandtl hadn’t been awarded a Nobel prize. (Details regarding Prandtl’s and Taylor’s nominations for a Nobel Prize (in physics) are given in Sreenivasan [471]). Taylor had, no doubt, promoted the honorary doctorate that Prandtl received from the University of Cambridge in 1936. See Batchelor [31] regarding Taylor and his very productive life. In his 1975 obituary of Taylor, Cavendish Professor Brian Pippard begins

Sir Geoffry Ingram Taylor$\ldots$ was one of the great scientists of our time and perhaps the last representative of that school of thought that includes Kelvin, Maxwell and Rayleigh, who were physicists, applied mathematicians and engineers—the distinction is irrelevant because their skill knew no such boundaries.

In summarizing fluid dynamics in the first half of the twentieth century, Sydney Goldstein [176] observed (in the first Annual Review of Fluid Mechanics):

In 1928 I asked Prandtl why he kept it so short, and he replied that he had been given ten minutes for his lecture at the Congress and that, being still quite young, he had thought he could publish only what he had time to say. The paper will certainly prove to be one of the most extraordinary papers of this century, and perhaps of many centuries.

More specifically, the prominent French fluid dynamicist Paul Germain [168] wrote:

Prandtl appears to be the first visionary discoverer of what we may, now, call fluid dynamics inspired by asymptotics . $\ldots$ One must stress that forty lines only are sufficient to Prandtl for delivering the essentials of a number of great discoveries: the boundary layer concept itself, the equations which rule it and how they may be used, their self-similar solutions, the basic law that the boundary layer goes like the square root of the viscosity. It is impossible to announce such major achievements in a shorter way.

There were certainly roots of singular perturbations and the boundary layer concept in much nineteenth-century scientific literature. In his fluid mechanics textbook, Prandtl [401], Prandtl called an 1881 paper by a Danish physicist, L. Lorenz, the

first paper on boundary layers.

Few would agree today. Indeed, the reference has been dropped in the surviving text, Oertel [357]. Van Dyke [492] notes Laplace’s and Rayleigh’s work on the meniscus, Stokes’ work on the drag on a sphere, Hertz’s work on elastic bodies in contact, Maxwell’s measurement of viscosity, Helmholtz and Kirchhoff’s work on circular-disc capacitors, and Rayleigh and Love’s work on thin shells. Additional historical perspective can be found in Bloor [47]. Indeed, Bloor [46] includes another list of precedents.

Hans Reissner’s late son, Eric, a long-time applied math professor at MIT, used to insist that the edge effect in thin shell theory (cf. Reissner [415]) paralleled fluid dynamical boundary layer analysis, though naturally interchanging the inner and outer terminology. Von Kármán, likewise, reported that the elder Reissner told him:

People attribute to Prandtl and to you discoveries which neither of you ever made

(cf. Friedrichs [159] for further related comments). We note that Frank [157] describes singular perturbations in elasticity theory, using the perspective of operator theory. Also note Andrianov et al. [9] and the classical reference Gol’denveizer [173]. In Bloor [46], the sociologist argues convincingly that boundary layer theory is a “social construct .”

One fascinating section of Schlichting and Gersten [438] quotes extensively from Prandtl’s lectures from the winter semester of 1931–1932 on intuitive and useful (anschlauliche und nützliche) mathematics, describing the oscillations of a very small mass with damping. He considered the asymptotic solution of the two-point boundary value problem

$$\displaystyle{ \epsilon y^{{\prime\prime}} + y^{{\prime}} + y = 0,\ \ \ 0 \leq x \leq 1 }$$

(1.1)

with y(0) and y(1) prescribed and with ε being a small positive parameter , reminiscent of the square root of the reciprocal of the physically relevant Reynolds number. Cole [92] considers the corresponding initial value problem (cf. Chap. 3 below). From here onwards, readers should realize that we will always take the symbol ε to be such a quantity. To emphasize this, we write

$$\displaystyle{0 <\epsilon \ll 1.}$$

The small parameters we encounter will typically be dimensionless, resulting from scaling a physical model (cf. Lin and Segel [291] and Holmes [208]). Our analyses will typically begin as boundary value problems, skipping both the very important initial modeling stage as well as the mathematical solution’s ultimate physical reinterpretation.

The differential equation (1.1) can be solved, following Euler, by setting

$$\displaystyle{y = e^{\lambda x}}$$

where the constant λ satisfies the characteristic polynomial

$$\displaystyle{\epsilon \lambda ^{2} +\lambda +1 = 0.}$$

The series expansions for its two roots

$$\displaystyle{-\nu (\epsilon ) \equiv -\frac{(1 -\sqrt{1 - 4\epsilon })} {2\epsilon } = -(1 +\epsilon +\ldots )}$$

and

$$\displaystyle{-\frac{\kappa (\epsilon )} {\epsilon } \equiv -\left (\frac{1 + \sqrt{1 - 4\epsilon }} {2\epsilon } \right ) = -\frac{1} {\epsilon } (1 -\epsilon -\epsilon ^{2}+\ldots )}$$

converge for ε < 1∕4, following the binomial theorem. Linearity implies that the general solution to the differential equation is a linear combination

$$\displaystyle{ y(x,\epsilon ) = e^{-\nu (\epsilon )(x-1)}c_{ 1}(\epsilon ) + e^{-\kappa (\epsilon )x/\epsilon }c_{ 2}(\epsilon ) }$$

(1.2)

for any constants c ₁(ε) and c ₂(ε). (We shifted the first exponent for later convenience.) To satisfy the boundary conditions, c ₁ and c ₂ must satisfy the linear system

$$\displaystyle{\begin{array}{*{10}c} y(0)& = e^{\nu (\epsilon )}c_{ 1}(\epsilon ) + c_{2}(\epsilon ) \\ y(1)&= c_{1}(\epsilon ) + e^{-\kappa (\epsilon )/\epsilon }c_{2}(\epsilon ). \end{array} }$$

Since the system is nonsingular for ε small and positive, Cramer’s rule uniquely determines

$$\displaystyle{c_{1}(\epsilon ) = \frac{\left\vert \begin{array}{lll} y(0)& 1 \\ y(1) & e^{\frac{-\kappa (\epsilon )} {\epsilon}} \end{array} \right\vert} {\left\vert \begin{array}{lll} e^{\nu (\epsilon )} & 1 \\ 1 &e^{-\frac{^{\kappa (\epsilon)}} {\epsilon}} \end{array} \right\vert} \quad \mbox{and}\quad c_{2}(\epsilon ) = \frac{\left\vert \begin{array}{lll} e^{\nu (\epsilon )} & y(0) \\ 1 &y(1) \end{array} \right\vert} {\left\vert \begin{array}{ll} e^{\nu (\epsilon )} & 1 \\ 1 &e^{-\frac{\kappa (\epsilon)} {\epsilon}} \end{array} \right\vert}}$$

and thereby the displacement y(x, ε). (Estrada and Kanwal [144] provides a distributional approach to solving such problems.) Neglecting multiples of e ^{−κ(ε)∕ε} (which we will later formally declare to be asymptotically negligible), we get the asymptotic limit

$$\displaystyle{ y(x,\epsilon ) \sim A(x,\epsilon ) + B(x,\epsilon )e^{-x/\epsilon }. }$$

(1.3)

(The tilde, representing asymptotic equality, will be defined in Chap. 2.) Here the outer expansion

$$\displaystyle\begin{array}{rcl} A(x,\epsilon )& \equiv & e^{-\nu (\epsilon )(x-1)}c_{ 1}(\epsilon ) {}\\ & \sim & e^{1-x}(1 +\epsilon (1 - x)+\ldots )y(1) {}\\ \end{array}$$

satisfies the given differential equation $\epsilon A^{{\prime\prime}} + A^{{\prime}} + A = 0$ and the terminal condition $A(1,\epsilon ) \sim y(1)$ as a formal power series

$$\displaystyle{ A(x,\epsilon ) = A_{0}(x) +\epsilon A_{1}(x)+\ldots }$$

(1.4)

in ε with

$$\displaystyle{\begin{array}{*{10}c} A_{0}(x)& = e^{1-x}y(1),\\ & \\ A_{1}(x)&= e^{1-x}(1 - x)y(1),\end{array} }$$

etc. Setting like coefficients of ε to zero in the differential equation and the boundary condition for A, we’d successively need $A_{0}^{{\prime}} + A_{0} = 0,\ A_{0}(1) = y(1)$; $A_{1}^{{\prime}} + A_{1} + A_{0}^{{\prime\prime}} = 0,\ A_{1}(1) = 0$; etc. Thus, the limiting solution A ₀(x) for x > 0 satisfies a first-order differential equation and the terminal condition. We will later learn that the reason that this boundary condition is selected for the outer expansion A is because the characteristic polynomial has a large negative (i.e., stable) root when $\epsilon \rightarrow 0$. The related supplemental term B(x, ε)e ^−x∕ε in (1.3) provides boundary layer behavior near x = 0, i.e. nonuniform convergence of the solution y from y(0) to A(0, ε) in an “ε-thick” right-sided neighborhood of x = 0. Since $(Be^{-x/\epsilon })^{{\prime}} = \left (B^{{\prime}}-\right.$ $\left.\frac{B} {\epsilon } \right )e^{-x/\epsilon }$ and $(Be^{-x/\epsilon })^{{\prime\prime}} = \left (B^{{\prime\prime}}-\frac{2} {\epsilon } B^{{\prime}} + \frac{B} {\epsilon ^{2}} \right )e^{-x/\epsilon }$, this boundary layer correction will satisfy the given linear equation (1.1) and the implied initial condition if the factor B satisfies the initial value problem

$$\displaystyle{ \epsilon B^{{\prime\prime}}- B^{{\prime}} + B = 0,\ \ \ B(0,\epsilon ) = y(0) - A(0,\epsilon ). }$$

(1.5)

Introducing the formal series expansion

$$\displaystyle{ B(x,\epsilon ) = B_{0}(x) +\epsilon B_{1}(x)+\ldots }$$

(1.6)

and equating successive coefficients in (1.5), we naturally require that they satisfy

$$\displaystyle{\begin{array}{*{10}c} B_{0}^{{\prime}}& = B_{0},\ \ B_{0}(0) = y(0) - ey(1),\\ & \\ B_{1}^{{\prime}}&= B_{1} + B_{0}^{{\prime\prime}},\ \ B_{1}(0) = -A_{1}(0) = ey(1),\end{array} }$$

etc., which upon integration agrees with the convergent power series expansion in ε of the exact solution (presuming ε < 1∕4), i.e.

$$\displaystyle{B(x,\epsilon ) = e^{-(\kappa (\epsilon )-1)x/\epsilon }(y(0) - e^{\nu (\epsilon )}y(1)).}$$

Simply compare (1.2) and (1.3) for the unique c _j s found, neglecting e ^−κ∕ε. Our termwise solution for A and B will be generalized to the regular perturbation method in the next chapter. Beware, however, the sum (1.3) is not an exact solution (cf. Howls [219]), though asymptotic (a term we’ll carefully define in Chap. 2).

Readers should note the important role played by the rapidly decaying function

$$\displaystyle{e^{-x/\epsilon }}$$

as ε → 0⁺ on x ≥ 0. (See Fig. 1.3.) It converges nonuniformly from the value 1 at x = 0 to the trivial limit at each x > 0 as $\epsilon \rightarrow 0^{+}$ (quite like the limiting idealized discontinuous unit Heaviside step function). Setting ε = 0 is meaningless when x = 0, but it provides the trivial limit obtained for any fixed x > 0. Also note how helpful it would be to immediately assume that the asymptotic solution of (1.1) has the additive form (1.3), a plunge that we will confidently take in our final chapter.

Figure 1.3:

The function e ^−x∕ε for x ≥ 0 and ε = 1, 1∕10, 1∕100

More generally, note that the sum

$$\displaystyle{e^{-x/\epsilon } + e^{-\epsilon x},\ \ x \geq 0}$$

(like its second term) has the power series expansion

$$\displaystyle{1 -\epsilon x + \frac{1} {2}\epsilon ^{2}x^{2}+\ldots }$$

on any interval $0 <\delta \leq x \leq B < \infty$, but the sum converges nonuniformly to 2 at x = 0 and to 0 at $x = \infty$.

Finally, observe that Prandtl used the model (1.1) 10 years before K.-O. Friedrichs considered the nearly equivalent problem

$$\displaystyle{\nu f_{yy} =\alpha -f_{y},\ \ 0 \leq y \leq 1,\ \ f(0) = 0,\ \ f(1) = 1}$$

at Brown University’s Program of Advanced Instruction and Research in Mechanics (cf. von Mises and Friedrichs [322]). For a fixed constant α, its exact solution

$$\displaystyle{f(y,\nu ) = (1-\alpha )\frac{(1 - e^{-y/\nu })} {(1 - e^{-1/\nu })} +\alpha y}$$

features an initial layer of nonuniform convergence near y = 0 as $\nu \rightarrow 0^{+}$. Indeed, the outer limit is 1 +α(−1 + y) for y > 0 and the nonuniform initial layer behavior is described by the correction term (α − 1)e ^−y∕ν (i.e., by completely neglecting the asymptotically negligible term ε ^−1∕ν).

In explaining his basic approach when elected an honorary member of the German Physical Society (upon his retirement), Prandtl [402] stated:

When the complete mathematical problem looks hopeless, it is recommended to enquire what happens when one essential parameter of the problem reaches the limit zero. It is assumed that the problem is strictly solvable when the parameter is set to zero from the start and that for very small values of the parameter a simplified approximate solution is possible. Then it must be checked whether the limit process and the direct way lead to the same solution. Let the boundary condition be chosen so that the answer is positive. The old saying “Natura non facit saltus” (Nature does not make sudden jumps) decides the physical soundness of the solution: in nature the parameter is arbitrarily small, but it never vanishes. Consequently, the first way (the limiting process) is the physically correct one!

(Quite analogously, Paulsen [387] characterizes a singular perturbation problem by a fundamental difference or change in behavior as a parameter ε changes from 0 to a positive number.)

An unusual situation arises when solving algebraic equations that depend analytically on a small parameter. There, singular perturbations occur when the solution involves a Laurent expansion (cf. Avrachenkov et al. [17]).

The extremely slow acceptance of Prandtl’s boundary layer theory was noted by Dryden [126] and Darrigol [110]. Bloor [47] explains that the British, in particular, stuck too long to the inviscid fluid dynamics of Rayleigh (from 1876). They were ultimately convinced to accept Prandtl’s theory, however.

Tani [479] observed:

• (i) In the 1905 paper, the essentials of boundary layer theory were compressed into two and a half pages, largely descriptive and extremely curtailed in expression. It is quite certain that the paper was very difficult to understand at the time, making its spread very sluggish.

• (ii) Prandtl’s idea was so much ahead of the times.

• (iii) The genesis of the boundary layer theory stood in sublime isolation: nothing similar had ever been suggested before, and no publications on the subject followed except for a small number of papers due to Prandtl’s students for almost two decades.

One might well conclude that boundary layer theory had a slow viscous flow out of Göttingen (cf. O’Malley [373]).

Only a brief mention of boundary layer theory appeared in the fifth edition of the preeminent English-language textbook Lamb’s Hydrodynamics [278]. There was no mention of the theory in the third and fourth editions of 1906 and 1916. G. I. Taylor [480], however, reported:

When I returned to Cambridge in 1919, I aimed to bridge the gap between Lamb and Prandtl.

Sydney Goldstein’s two-volume Modern Developments in Fluid Mechanics [175], later reprinted by Dover, was very influential in propagating ideas about the boundary layer. Goldstein had taken on the substantial editorial task of highlighting modern developments upon the death of Sir Horace Lamb in 1934. Lamb finally treated boundary layers extensively in the sixth edition of Hydrodynamics [278]. Prandtl’s Wilbur Wright Memorial Lecture [400] to the Royal Aircraft Society in 1927 made a substantial impact in Britain, together with the monograph Glauert [170] on airfoils and airscrews (i.e., wing shapes and propellers). Prandtl’s student, Hermann Schlichting’s 1941–1942 lectures from Braunschweig lived on as mimeographed versions until they were revised and published by G. Braun of Karlsruhe as Grenzschicht-Theorie in 1951. Current 8th editions [438] in German and English, at least, are still published by Springer. Schlichting, indeed, became director of Prandtl’s proving ground in 1957 (cf. Schlichting [437]).

Before January 11, 1933, when Adolf Hitler was appointed German chancellor by President Hindenburg, Göttingen was a Mecca for mathematicians worldwide (cf. MacLane [304]). Richard Courant (1888–1972) became director of the Mathematics Institute (Wissenschaftlicher Führer) in 1922 (succeeding Klein, who had retired in 1913). The German influence, especially in pure mathematics, had become dominant. Thus, it was natural that the early work of the prominent Japanese mathematician Mitio Nagumo be published in German and that he would visit Göttingen from 1932 to 1934. His lifetime work was greatly influenced by that stay. A chemist’s question motivated his 1939 paper on initial value problems for singularly perturbed second-order ordinary differential equations (cf. Yamaguti et al. [530]). As Sibuya observed there,

when this paper was written, singular perturbations didn’t exist.

(The term hadn’t been defined.) One 1959 paper by Nagumo (on partial differential equations) also concerns singular perturbations. However, the later critical use of differential inequalities to analyze singularly perturbed two-point boundary value problems (cf. Brish [60], Chang and Howes [76], and De Coster and Habets [112]) is directly based on Nagumo’s use of upper and lower solutions (or bounding functions) from 1937 onwards. Nickel [347]’s mathematical treatment of boundary layer theory extensively used differential inequalities. The effective use of maximum principles for singularly perturbed differential equations is closely related (cf. Eckhaus and de Jager [138] and Dorr et al. [124]).

Somewhat similarly, the Chinese mathematician Yu-Why Chen came to study at Göttingen in 1928 at the urging of an earlier student of Courant. In a telephone conversation with this author, about 1989, Chen (then living in Amherst, Massachusetts) confirmed that Courant had suggested a thesis topic to him on an ordinary differential equation model featuring boundary layer behavior, because he wished to encourage a mathematical analysis of Prandtl’s boundary-layer phenomena. (Chen also said no one had asked him about his thesis in over 50 years.) Chen was granted his doctorate in 1935 for a thesis coinciding with the Compositio Mathematica paper, Tschen [485]. His thesis referees at Göttingen were Franz Rellich and Gustav Herglotz, since Courant had been dismissed because he was a Jew. Chen’s asymptotics work is largely forgotten, though it overlaps a bit with Wolfgang Wasow’s more influential New York University thesis of 1942 (which we will consider in Chap. 3). Such nostrification, or not properly recognizing work done elsewhere, may be inadvertent, though it is sometimes claimed to be quite common at certain introverted math centers, however prominent.

Courant had convinced the Rockefeller Foundation’s International Education Board (or IEB) to spend $275,000 to build a new Mathematical Institute in Göttingen in 1929, balancing the Institut Henri Poincaré it had already funded in Paris. Just before then, the new Cambridge PhD Sydney Goldstein had visited Prandtl for a year as a Rockefeller Research Fellow. There Goldstein studied numerical boundary layer calculations, a topic extensively developed later by his Manchester and Cambridge colleague, D. R. Hartree.

According to Siegmund-Schultze [460], the IEB’s rationale for funding Courant’s institute was

The Board would not be interested in $\ldots$ housing or even helping to house the mathematical department in more agreeable quarters, unless thereby there was a practical certainty that greater and much greater usefulness to a group of sciences would result. In fact, the new mathematical institute was finally erected on Bunsenstrasse in Göttingen close to the physical and chemical institutes and the aeronautical institutes of Ludwig Prandtl. These scientists in time now had better access to the mathematicians and to the mathematical library. Although Prandtl was not involved in the negotiations process, and at one point was mentioned in a slightly disparaging way as “more of an engineer than a physicist” (yet a professor in the University mathematics department), his institutes and the technical sciences are expressly included in the “group of sciences” which were of interest to the IEB’s Trowbridge. In fact this was completely in the tradition of Felix Klein who had called Prandtl to Göttingen in 1904 in order to enrich its mathematical environment. Trowbridge’s report includes a longer passage on his visit to Prandtl’s AVA (aerodynamic proving ground) with its wind tunnel, and mentions the new and more theoretical Kaiser-Wilhelm-Institut.

At the opening ceremony, Hilbert said:

realizing the idea of building the institute was a great and difficult task entailing various smaller problems. Courant treated each of these with the same love and devotion, always knowing how to find and cajole the most suitable and understanding man for the task

(cf. Bergmann et al. [42]). (Readers may recall that the original, very influential, thousand-page, two-volume Methoden der mathematischen Physik [102] was published in 1924 and 1937 as a collaborative effort at the university, based on Hilbert’s lectures, with Courant as editor.)

Applied mathematics was also being simultaneously developed in other German universities, especially Berlin. Quite naturally, Ludwig Prandtl and Richard von Mises established GAMM, the German society Gesellschaft für Angewandte Mathematik und Mechanik , in 1922 while the sister Society for Industrial and Applied Mathematics (SIAM) in the USA wasn’t founded until 1952. (The European Consortium for Mathematics in Industry (ECMI) didn’t start until 1986.)

Erich Rothe was a joint student of Erhard Schmidt and von Mises at Berlin in 1926. Far ahead of the crowd, he subsequently wrote several papers on the asymptotic solution of singularly perturbed partial differential equations and on the skin effect in electrical conductors (cf., e.g., Rothe [422]). Later, while a refugee schoolteacher in Iowa, he published a very good paper on singularly perturbed two-point boundary value problems in the obscure Iowa State College Journal of Science (cf. Rothe [423]). (Most of his career was spent at the University of Michigan, where his work included topological degree theory and his first PhD student was Jane Cronin in 1949.)

A major upheaval occurred in Germany on April 7th, 1933. The “Law for the Reorganization of the Civil Service” dismissed Jews from government positions (except those appointed prior to 1914 or who (like Courant) had served in the front lines during World War I (cf. Segal [446] and Siegmund-Schultze [462]). As of 1937, anyone married to a Jew also lost his or her university position. Hitler also targeted others, besides Jews, that he labeled Untermenschen (or subhumans). James [225] reports that from 1938 those dismissed were not only forbidden to teach, they were no longer allowed to enter university buildings, including libraries.

After Courant was placed on leave in the spring of 1933, Prandtl joined 27 others in a petition in support of Courant. Petitioners included Artin, Blaschke, Caratheódory, Hasse, Heisenberg, Herglotz, Hilbert, von Laue, Mie, Planck, Prandtl, Schrödinger, Sommerfeld, van der Waerden, and Weyl. Friedrichs later reported:

Several of the names on the list are those of people who later were considered to be Nazis or near-Nazis, and even at the time some of them were known to be in sympathy with the regime.

Recall that Nazi refers to a supporter or member of Hitler’s political party, the National Socialist German Workers’ Party or NSDAP. Courant had indeed asked Prandtl to write the Kurator (government representative at the university) on his behalf since he felt Prandtl

had acted with courage and decision, firing one of his assistants when he discovered that the man was an informer for the Nazi forces

(cf. Reid [411] and Trischler [484]. The assistant was Johann Nikuradse, whose research under Prandtl is still cited.).

From his intermediate stop at Cambridge University in 1934, Courant (according to Siegmund-Schultze [462]) wrote

Germany’s best friends such as Hardy, Flexner, Lord Rutherford, the Rockefeller Foundation get alienated while our institutions, which were unequalled in the world, are destroyed—even Cambridge cannot compare to the old Göttingen. Foreign countries take advantage of the situation and employ people, particularly physicists and chemists, who will in the long run give science and its applications there a big boost.

Ball [24] notes that Courant decided to emigrate when

My youngest son did not seem able to understand why he should not be in the Hitler Youth, too.

Prandtl protested to the Minister of the Interior

that the rigid system devised by the race theorists should be flexible enough to allow scholars who were half or a quarter Jewish, who were then logically half or three-quarters German, to be persuaded to join the people’s cause.

Mehrtens and Kingsbury [313] report that the number of math students at Göttingen decreased from 432 in 1932 to 37 in 1939. According to Cornwell [99], when Max Planck (as president of the Kaiser Wilhelm Society) expressed his concerns about the deterioration , Hitler replied:

If the dismissal of Jewish scientists means the annihilation of contemporary German science, then we will do without science for a few years.

(By 1942, the converted Hitler was quoted

$$\displaystyle{\mbox{ I'm mad on technology.}}$$

(cf. Jacobsen [223])). Beyerchen [43] reports that the aging David Hilbert (1862–1943) (who had retired in 1930) when asked at a banquet by the Nazi minister of science

And how is mathematics in Göttingen now that it has been freed of the Jewish influence?

Hilbert replied:

Mathematics in Göttingen? There is really none anymore.

Despite his high opinion of Prandtl’s scientific merits, G. I. Taylor expressed some personal reservation in his obituary of von Kármán (cf. Taylor [481]):

By the time the fourth Congress (of Applied Mechanics) was held in Cambridge, England in 1934, the German Jewish members were having a bad time. Theodore was well out of it but was doing a lot for his unfortunate fellow countrymen. Prandtl, who was not Jewish, appeared to be completely taken in by the Nazi propaganda$\ldots$ In 1938, however, when the fifth Congress was organized in Cambridge, Massachusetts by Theodore and Jerome Hunsaker, conditions had changed. Prandtl and my wife and I were staying with Jerome at his home in Boston. The German delegation was strictly watched by political agents who had come as scientist members and Prandtl did not dare to be seen reading American papers. He used to ask my wife to tell him what was in them. After I returned to Cambridge from the fifth Congress I had a letter from Prandtl telling me what a benevolent man Hitler was and including a newspaper cutting showing the Führer patting children’s heads. I imagine the poor man did this under pressure from the propaganda machine, for other people told me they had similar letters from him.

Epple et al. [140] reports:

After the congress, Prandtl tried to convince Taylor in a letter dated October 1938 that Hitler did in fact “turn one million people against himself while eighty million people, however, are his most loyal and enthusiastic followers.” The battle Germany had to fight against the Jews, Prandtl continued to explain, was necessary for its self-preservation. He invited Taylor to Germany so that he could see with his own eyes “that we are, in fact, being ruled very well.”

Sreenivasan [471] reports that Prandtl and Taylor exchanged 25 letters between 1923 and 1938. Prandtl would usually write in typewritten German (in which Taylor’s wife was fluent) while Taylor would reply in handwritten English. In 1933, Prandtl complained about Taylor’s handwriting (which is reproduced in Sreenivasan [471])

Would it be possible for your letters which at the moment are extremely hard to read and cost me a large amount of time be written by someone else who writes more clearly? I hope you don’t take umbrage against this remark.

Taylor called Hitler a

$$\displaystyle{\mbox{ criminal lunatic}}$$

in his last letter and didn’t answer two letters from Prandtl after the Second World War.

Prandtl had hoped that the 1938 or 1942 Congress would be held in Germany, assuming no distinction would be made between Jews and non-Jews. According to Mehrtens and Kingsbury [313], however, the Reich’s Ministry of Education wrote Prandtl that

“Jews of foreign citizenship” who took part would not be regarded as Jews here, but that there would be no place for non-Aryans of German citizenship.

Prandtl accepted the Hermann Göring medal (named for the commander-in-chief of the Luftwaffe or German Air Force and also Hitler’s minister without portfolio) in 1939 from the Academy of Aeronautical Research and he chaired an advisory panel for the Air Ministry (Reichsluftfahrtsministerium). Prandtl’s institutes grew to have hundreds of workers (cf. Hirschel et al. [207] and Meier [315]). In a letter from 1933, Prandtl wrote

Hopes increased that, after years of ‘unjustified penny-pinching,’ the importance of research for the good of the state would fully be recognized.

Trischler [484] reported

It is then not surprising that the scientists did not regret the passing of democracy, or that they quietly aligned themselves with the new dictatorship, particularly when, with regard to their actual work, virtually no limits were set on their traditional autonomy.

As Epple et al. [140], observed

we may assume that Prandtl realized, even before the disclosure of the German air force armament plans in the summer of 1935, that the expansion of his institute, at least from the point of view of his official patrons, served to prepare for a new war $\ldots$ Prandtl thus not only anticipated the actual dynamics of the main phases of war research but also justified some of the research of the prewar arming period not immediately usable by reference to a secondary benefit in the eventuality of a war.

von Kármán reported that after the war, Prandtl said

he was not a Nazi, but had to defend his country

(cf. von Kármán and Edson [241]). In a manuscript “Reflections of an unpolitical German on the denazification” (1947), Prandtl wrote that he never played a role in politics, but had always served

$$\displaystyle{\mbox{ State and Science.}}$$

His former students, Stanford engineering professors Irmgard Flügge–Lotz and Wilhelm Flügge , to a large extent, defended him (cf. Flügge-Lotz and Flügge [150]). They wrote:

In the Applied Mechanics Institute, Prandtl was politely squeezed out by a group of flag-waving people, but the Fluid Mechanics Institute started to grow and to enjoy prosperity under the golden rain of government support. A. Betz, while never submitting to party rule, used the interest of the new German airforce to buy expensive equipment and to expand the staff. Among this staff–scientific, technical, and administrative– there was a wide variety of attitudes. Many went with flying colors into the camp of the new masters, and from them the precinct wardens were chosen who had to watch our thoughts and actions and to denounce us if they caught us in a word of doubt or criticism. They were the known enemies, and in their presence people fell silent. For those who did not approve of the regime, there was only the choice between martyrdom and compromise. We do not remember anyone who became a martyr, but the compromise was a walk on a tightrope. No one really knew where the other stood, whether he was a member of the muffled opposition, a spy, or perhaps, at times the one and then the other. This uncertainty, even with regard to former friends, fell like a blight upon the social life.

Prandtl had little understanding for politics and was at times as helpless as a child. He knew that some of the people were like mad dogs, but he could not understand how results of clear logical argument could be rejected furiously if they went against the new doctrine. Standing at the top of the pyramid, he could not avoid giving once in a while a public address, and this was always a nervous strain for the scientific community of the institute. Usually someone had had an advance look at Prandtl’s draft of a speech, but who could be sure that he would not be carried away and make some extempore remark that could lead us all into trouble?

The author recalls an earlier seminar by Flügge-Lotz, given in a darkened Stanford classroom with a spotlighted portrait of Prandtl, entitled “The Ludwig Prandtl I Knew.”

The biography Vogel-Prandtl [509], by Prandtl’s younger daughter Johanna, portrayed him as hostile to the regime, noting that he refused to have a picture of Hitler in his office. Her book also makes public a 1941 letter from Prandtl to Göring which states

they [the antagonists of “Jewish physics” ] have poisoned the air with $\ldots$ disdain for the past.

This followed efforts by Prandtl to defend Werner Heisenberg and theoretical physics. He summarized by writing:

In short, it boils down to one thing, namely that a group of physicists, to whom the Führer listens, is raging against theoretical physics and defaming the most meritorious theoretical physicists.

The Nazi physicists and Nobel laureates J. Stark and P. Leonard had attacked “Jewish Aryans” or “white Jews,” particularly Heisenberg, the 1933 Nobel prizewinner in physics, because they decided he

$$\displaystyle{\mbox{ thinks like a Jew}}$$

(cf. Reeves [410]). The term Jude in Geiste (Jewish in spirit) was used. Some mathematicians wrote similar diatribes, especially in the journal Deutsch Mathematik, edited by L. Bieberbach and T. Vahlen. Also note Rowe [424].

The Kreisleiter (local Nazi coordinator) in Göttingen wrote in 1937:

Prof. Prandtl is a typical scientist in an ivory tower. He is only interested in his scientific research which has made him world famous. Politically, he poses no threat whatsoever$\ldots$. Prandtl may be considered one of those honourable, conscientous scholars of a bygone era, conscious of his integrity and respectability, whom we certainly cannot afford to do without, nor should we wish to, in light of his immensely valuable contributions to the development of the air force

(cf. Ball [24]).

Bodenschatz and Eckert [49] report, based on Prandtl’s correspondence,

By July 1945 the institute was under British administration and had “many British and American visitors.” Prandtl was allowed to work on some problems that were not finished during the war and from which also reports were expected. Starting any new work was forbidden.

They also quote a British Intelligence report on a visit to the Kaiser-Wilhelm-Institut (KWI) 26–30 April 1946:

Much of the equipment of the A.V.A. has been or is in the process of being shipped to the U.K. under M.A.P. direction, but the present proposals for the future of the K.W.I. Göttingen appear to be that it shall be reconstituted as an institute of fundamental research in Germany under allied control, in all branches of physics, not solely in fluid motion as hitherto. Scientific celebrities now at the K.W.I. include Professors Planck, Heisenberg, Hahn and Prandtl among others. In the view of this policy, it is only with difficulty that equipment can be removed from the K.W.I. The K.W.I. records and library have already been reconstituted.

When Courant visited Göttingen in 1947, about the time of Prandtl’s retirement, he reported that the aeronautics institute had become a

$$\displaystyle{\mbox{ veritable fortress.}}$$

Although ill and depressed, Prandtl was mentally alive. He had given much thought to analog computing machines with a view toward meteorological computations, a longtime interest. As we continue to describe further progress in boundary layer theory, we encourage readers to try to understand Prandtl’s behavior in the context of Nazi Germany (cf. Goldhagen [174], Medawar and Pyke [312], and Barrow-Green et al. [30]).

As Flügge-Lotz and Flügge [150] put it:

The seeds sown by Prandtl have sprouted in many places, and there are now many ‘second growth’ Göttingens who do not know that they are.

The surviving victims of Hitler’s Third Reich (including applied mathematicians) spread worldwide as an intellectual diaspora . Courant’s transition from Göttingen to founding what become the Courant Institute at New York University is described in Reid [411]. As in Göttingen, Courant was ambitious to develop as international center at NYU for basic science, ultimately gaining support from the Rockefeller Foundation, the Office of Naval Research, the Atomic Energy Commission, and rich and well-connected individuals. He started anew, with practically no resources. His institute educated many outstanding students and later employed some of them, including the brothers J. B. and H. B. Keller (from Paterson, N. J.), the Hungarian Peter Lax, and the Canadians Cathleen Synge Morawetz and Louis Nirenberg. Among those later involved in asymptotics, Wiktor Eckhaus survived the war in Poland before finishing high school in Holland and a Ph.D. at MIT (cf. Eckhaus [137]); the Hungarian Arthur Erdélyi emigrated from Czechoslovakia to Edinburgh in 1939 with the help of E.T. Whittaker (cf. Colton [94] and Jones [228]), and subsequently split his career between Caltech and Edinburgh; the German/Palestinian Abraham Robinson fled from France to England where he did aeronautical research during the war (cf. Dauben [111]); and Richard von Mises went to Istanbul and then to Harvard (initially, without salary, but with a girl friend!). James [225] reports that of the 2600 Jews assisted by the British Academic Assistance Council, twenty become Nobel laureates, fifty-four were selected Fellows of the Royal Society, thirty-four become Fellows of the British Academy, and ten received knighthoods.

A substantial effort successfully found university positions for most of the émigré mathematicians (cf. Siegmund-Schultze [462]), although there was a counter point of view that the big migration of established academics from Europe to the USA would force some young American mathematicians to become

hewers of wood and drawers of water

(cf. Birkhoff [44] and Joshua 9:23). Einstein (who did not return from America to Germany in 1933) labeled G. D. Birkhoff (1884–1944), the most prominent American-trained mathematician at the time,

one of the world’s great anti-Semites

while Courant said

I don’t think he was any more antisemitic than good society in Cambridge, Massachusetts.

In any case, of the 148 mathematical émigrés after 1933 listed in Siegmund-Schultze [462], 82 came to the United States and all but 7 obtained positions by 1945. As Courant anticipated, many contributed to the allied war effort, including developing the atomic bomb. As Medawar and Pyke [312] conclude:

The great majority of the scientific emigrants were young and unknown people. Those who later made worthwhile contributions were able to do so because their host countries generally gave them the chance that Germany denied them.

The subsequent prominence of Jewish mathematicians in America is considered in Hersh [203].

K.-O. Friedrichs (1901–1982) left a professorship in Braunschweig in 1937 to come to New York University in order to marry a non-Aryan (cf. Reid [412]). He had been Courant’s student at Göttingen, learned fluid mechanics as a postdoc with von Kármán in Aachen, and he encountered boundary layers in plate theory (cf. Friedrichs and Stoker [162]). His lecture notes from the Brown University summer school (cf. von Mises and Friedrichs [322]), his NYU lectures on special topics in fluid mechanics (Friedrichs [160]), and his Gibbs lecture (Friedrichs [161]) all demonstrate his mastery of asymptotics, as part of much broader contributions to analysis and differential equations (cf. Morawetz [326]). In studying nonlinear oscillations, like those described by the van der Pol equation , Friedrichs and Wasow [163] introduced the now universally accepted term singular perturbations to distinguish them from the more common situation of a regular perturbation for which uniform convergence implies that a single asymptotic power series suffices. Wasow [516] observed:

In later years, neither of the two authors could remember to which of them belongs the credit for coining the terminology, but I believe it was Friedrichs.

Wolfgang Wasow (1909–1993) had studied mathematics in Göttingen, seeking to pass his Staatsexamen to become a teacher (cf. Wasow [516] and O’Malley [369]). He passed his orals in 1933 and applied for practice teaching, but was not employed. After some wandering, including teaching at a boarding school in Florence and at Choate School and Goddard College, he took a $600 fellowship at NYU in 1940, arranged by Courant. His 133-page thesis was nearly completed the following summer, under the direction of Friedrichs. It described many singular perturbation examples and made Prandtl’s boundary layer theory into a mathematical topic. The work was not immediately well received, however. The first papers he submitted were rejected, but a ten-page paper ultimately appeared in the MIT journal (cf. Wasow [511]),

with some behind the scenes support from Courant

(according to Wasow [516]). Wasow’s autobiography modestly states that his post-thesis research

was soon overshadowed by two articles by MIT’s Norman Levinson who obtained more general results by different methods.

(Levinson’s critical work on singular perturbations is summarized in Nohel [353]). Wasow continued to do important work on asymptotics, as summarized in Asymptotic Expansions for Ordinary Differential Equations [513] and Linear Turning Point Theory [515], as well as much other mathematical and numerical analysis (while working mostly at UCLA and the University of Wisconsin, Madison).

By 1950, singular perturbations were being studied and developed by a variety of mathematicians and engineers worldwide, although hardly in Göttingen. (Curiously, the U.S. Office of Strategic Services recruited over 1600 German scientists (including Wernher von Braun) to come to America after the war (as Operation Paperclip ), bleaching any ties to Nazi service (cf. Jacobsen [223]).) Substantial activity was taking place in the Guggenheim Aeronautical Laboratory at Caltech (GALCIT ), which von Kármán headed since 1930 (cf. Cole [93]), while many promising efforts underway elsewhere were largely unconnected to the original motivation from fluid dynamical boundary layers.

von Kármán became emeritus at Caltech in 1949, largely due to his expanding duties (since 1944) with the scientific advisory group for the Air Force and (since 1951) with NATO’s Advisory Group for Aeronautical Research and Development. In 1962, he received the first National Medal of Science from President Kennedy.

Sears [445] reports:

He also loved parties, drinks, girls, jokes, the bon mot. All his life, he played the part of the dangerous Hungarian bachelor. He succeeded in shocking some of the young wives (and their husbands), but charmed many more. He told us: ‘I have decided how I want to die. At the age of 85, I want to be shot by a jealous husband’.

More seriously, he added

Ulam saw von Kármán and asked John von Neumann who that little old guy was. ‘What, you don’t know Theodore von Kármán?’ said von Newmann, ‘Why, he invented consulting’.

More generally, the arrival of applied mathematics in academic America is considered in Richardson [417] and Siegmund-Schultze [461]. It spread from New York, Boston, Providence, and Pasadena, among other centers (cf. Rowe [426] for a colorful (Truesdellian) description of early applied mathematics at Brown, including comments on its émigré faculty in the summer of 1942).

Exercises

Although we’ve not yet accomplished much technically, readers can do the exercises that follow based on their first course in differential equations (cf. O’Malley [370]).

• 1. Find the solution of the initial value problem

  $$\displaystyle{\epsilon ^{3}\ddot{y} +\epsilon \dot{ y} + y = 1}$$

  on $t \geq 0$ with y(0) = 2 and $\epsilon \dot{y}(0) = 3$ as $\epsilon \rightarrow 0^{+}$.

  The answer is an asymptotic solution of the form

  $$\displaystyle{y(t,\epsilon ) \sim 1 + A(t,\epsilon )e^{-t/\epsilon } +\epsilon B(t,\epsilon )e^{-t(1-\epsilon )/\epsilon ^{2} }}$$

  for power series A and B. (The tilde indicates an asymptotic limit.)

• 2. Solve the two-point problem

  $$\displaystyle{\epsilon y^{{\prime\prime}} + y^{{\prime}}- (y^{{\prime}})^{2} = 0,\ \ \ 0 \leq x \leq 1}$$

  with y(0) = 1 and y(1) = 0 and describe its boundary layer behavior.

  An exact solution is

  $$\displaystyle{y(x,\epsilon ) = -\epsilon \ln \left (1 + e^{-1/\epsilon } - e^{-x/\epsilon }\right ).}$$
• 3.

  • (a) Find the exact solution to the problem

    $$\displaystyle{\epsilon y^{{\prime\prime}}- y^{{\prime}} + 2x = 0,\ \ 0 \leq x \leq 1}$$

    with y(0) = 0 and y(1) = 1.

  • (b) For small positive values of ε, show that

    $$\displaystyle{y(x,\epsilon ) = x^{2} + 2\epsilon \left (x + 1 + e^{\frac{x-1} {\epsilon } }\right )}$$

    up to exponentially negligible terms like e ^−1∕ε.

    Note that y converges uniformly to x ², that y ^′ converges nonuniformly near x = 1, and that y ^{′ ′}(1) is approximately 2∕ε.

• 4. Show that the two-point problem for the constant coefficient equation

  $$\displaystyle{\epsilon ^{2}(y^{{\prime\prime}} + ay^{{\prime}}) - b^{2}y = c,\ \ 0 \leq x \leq 1}$$

  with b > 0 has an asymptotic solution of the form

  $$\displaystyle{y(x,\epsilon ) = -\frac{c} {b^{2}} + A(x,\epsilon )e^{-bx/\epsilon } + B(x,\epsilon )e^{-\frac{b} {\epsilon } (1-x)}}$$

  where

  $$\displaystyle{A(x,\epsilon ) \sim e^{-\frac{ax} {2} }\left (y(0) + \frac{c} {b^{2}}\right )}$$

  and

  $$\displaystyle{B(x,\epsilon ) \sim e^{\frac{a} {2} (1-x)}\left (y(1) + \frac{c} {b^{2}}\right ).}$$