söndagen den 29:e juli 2012

Confused Feynman on Direction of Time

                                             Feynman illustrating reversible mechanics.

Feynman thrills an audience of Cornell students by a lecture on the Distinction of Past and Future or  the Direction of Time and Irreversibility (Part 1, Part 2, Part 3, Part 4):

  • If you drop a cup and it breaks, you can sit there a long time waiting for the pieces to come together and come back into your hands.
  • We remember the past, we do not remember the future.
  • Basic laws of physics are time-reversible, without distinction between past and future.
  • Many ordinary phenomena based on reversible atomics are irreversible.  How come?
  • It comes from the characteristic that if we start with an ordered system and have the irregularities of nature of bouncing then the thing goes one way.
  • The thing goes from ordered to disordered.
  • How did it get ordered in the first place?
  • We have to add the hypothesis that the past was more organized than the present.
We hear Feynman present the standard explanation of the irreversibility of certain processes as an effect of irregularities destroying order, combined with a hypothesis that the past was more ordered than the present. 

But Feynman's explanation is not convincing, since the hypothesis of more ordered past is not explained, not even made plausible.

The interested reader finds a new approach to the irreversibility of systems based on reversible basic laws based on finite precision computation as realization of the irregularities hinted at by Feynman, developed in a setting directed to a general audience in The Clock and the Arrow: A Brief Theory of Time

Irreversibility is here explained as an effect of instability of certain processes combined with finite  precision computation: The smashed cup cannot be re-assembled because the required high degree of precision cannot be realized. Read and reflect! 



Making of the Prandtl Myth by Prandtl


The Father of Modern Aerodynamics inspecting the Ho III 1938 Rhön Contest Challenger:
Did Germany lose the war because of incorrect aerodynamics?

On the request by NACA (US National Advisory Committee for Aeronautics) in 1921, Ludwig Prandtl "prepared for the reports of the committee a detailed treatise on the present condition of those applications of hydrodynamics which lead to the calculation of the forces acting on airplane wings and airship bodies" (NACA Report 116 Applications of Modern Hydrodynamics to Aeronautics). 

Prandtl "acceded to the request all the more willingly because the theories in question have at this time reached a certain conclusion where it is worth while to show in a comprehensive manner the leading ideas and the results of these theories and to indicate what confirmation the theoretical results have received by tests".

Prandtl states in his report defining the state-of-the-art with my comments in parenthesis:

1. Friction between fluid and solid body never comes into consideration in the fields of application to be treated here, because it is established by reliable experience that fluids like water and air never slide on the surface of the body; what happens is, the final fluid layer immediately in contact with the body is attached to it (is at rest relative to it), and all the friction of fluids with solid bodies is therefore an internal friction with the fluid. (This is Prandtl's dictate of no-slip boundary condition which made 20th century fluid mechanics into uncomputable magics).

2. In this layer, which we call the boundary layer, the forces due to viscosity are of the same order of magnitude as the forces due to inertia, as may be seen without difficulty. (This is misleading: both forces due to viscosity and inertia can be small.)

3. Closer investigation concerning this shows that under certain conditions there may occur a reversal flow in the boundary layer which is set in rotation by the viscous forces, so that, further on, the whole flow is changed owing to the formation of vortices. (This is Prandtl's main thesis: The boundary layer changes the whole flow).

4. In the rear of blunt bodies vortices are formed…on the other hand, in the rear of very tapering bodies…there is no noticeable formation of vortices. The principal successful results of hydrodynamics apply to this case…the theory can be made useful exactly for those bodies which are of most technical interest. (Prandtl's theory is based on the formation of vortices in the boundary layer, which Prandtl claims are not formed in the cases of interest: Missing logic).

5. On resistance of airships: It is seen that the agreement (pressure distribution) is very complete; at the rear end, however, there appears a characteristic deviation in all cases, since the theoretical pressure distribution reaches the full dynamical pressure at the point where the flow reunites again, while actually this rise in pressure, owing to the influence of the layer of air retarded by friction, remains close to the surface. (Prandtl suggests that the lack of pressure rise at separation in real flow is du to boundary layer friction. We show that this is incorrect: It is instead caused by 3d rotational separation.)

6. As is well known there is no resistance for the theoretical flow in a nonvoscpus fluid (potential flow). The actual drag consists of two parts, one resulting from all the noral forces (pressures) acting on the surface of the body, the other from all tangential forces (friction). The pressure resistance, arises in the main from the deviation mentioned at the rear end, and is, as is known, very small. (This is incorrect: Pressure drag is the main part of drag in slightly viscous flow)

7. We shall concern ourselves in what follows only with non viscous and incompressible fluid, also called "ideal fluid". (Confusing since friction forces are supposed to change the flow.)

8. In order that the flow may be like the actual one, the circulation must always be so chosen that the rear rest point coincides with the trailing edge….We are accordingly, by the help of such constructions, in the position of being able to calculate the velocity at every point in the neighborhood of the wing profile…. The agreement on the whole is as good as can be expected from a theory which neglects completely the viscosity. (Empty statement.)

9. That a circulatory motion is essential for the production of lift of an aerofoil is definitely established. The question is then how to reconcile this with the proposition that the circulation around a fluid line in a non viscous fluid remains constant.... There is instantly formed at the trailing edge a vortex of increasing intensity… (This is the mystery of circulation theory: If circulation generates lift, the question is how circulation is generated. The standard answer is by a non-real sharp trailing edge and starting vortex. This is however not the real physics of 3d rotational separation at a real rounded trailing edge.)

Concluding analysis: Prandtl's message of 1921 has become the text book canon, which however is non-physical and thus incorrect.

fredagen den 27:e juli 2012

Max Munk: Failure of German Flight Theory





Max Munk, student and protege of Ludwig Prandtl, was responsible for aerodynamic theory at NACA during its formative years 1921-26. In General Theory of Thin Wing Sections  Munk presents a version of the Kutta-Zhukovsy Circulation Theory, with the following plead that theory is useful:
  • It is useful to discuss this phenomenon (the action of a wing) from the theoretical point of view, however imperfect the result may be as a consequence of neglecting the viscosity of the air.
  • A theorectical investigation may at least give the limit of what to expect.
  • It enables the investigator to survey and keep in mind a great number of isolated experiences, whether the agreement between theory and experiment be more or less close.
  • It induces him to reflect on the phenomenon and thus becomes a source of progress by guiding him to new observations and experiments.
  • It has often occurred even that some relation was thought to be confirmed by experience till the progress of theory made the relation improbable. And only then the experiments confirmed the improved relation, contrary to what they were supposed to do before.
  • But is it really necessary to plead for the usefulness of theoretical work?
  • This is nothing but systematical thinking and is not useless as sometimes supposed, but the difficulty of theoretical investigations makes many people dislike it.

We read that Munk intended to educate the uneducated engineers at NACA by teaching difficult Kutta-Prandtl theory, which he exposed in a flood of articles (later collected into the book Fundamentals of Fluid Dynamics for Aircraft Designers).  This became too much for NACA and Munk was forced out of office in 1927: American NACA engineers did not embrace German theory, which in the light of the New Theory of Flight was fully rational.

John D Anderson's Fundamentals of Aerodynamics



Fundamentals of Aerodynamics by John D Anderson "offers the most readable, interesting, and up-to-date overview of aerodynamics to be found in any text".

The generation of lift of a wing is presented as follows in Chapter 4 Incompressible Flow over Airfoils:
  • The Kutta condition states that the circulation around an airfoil is just the right value to ensure that the flow smoothly leaves the trailing edge. How does nature generate this circulation?
  • Does it come from nowhere, or is circulation somehow conserved over the whole flow field?
  • As the flow over the airfoil is started, the large velocity gradients of the sharp trailing edge result in the formation of a region of intense vorticity which ... forms the starting vortex.
  • The starting vortex builds up to just the right strength such that the equal and opposite cirrculation around the airfoil leads to a smooth flow from the trailing edge.
This is the classical Kutta-Zhukovsky Circulation Theory (CT), which thus still 100 years after its conception forms the basis of aerodynamical theory of flight. 

We show in New Theory of Flight that CT does not describe the actual generation of lift of a wing. We do this by solving the Navier-Stokes equations describing the flow around the airfoil and inspecting the solutions to see that the aerodynamics described by Anderson is not actual physics, only fictitious physics invented by incorrect mathematics and physics. 


AIAA Aircraft Design Confusion



Our New Theory of Flight featured on Secret of Flight has been submitted to AIAA Journal with AIAA "the world's largest technical society dedicated to the global aerospace profession".

Let us study how AIAA expresses its world-leading expertise in the book Aircraft Design: A Conceptual Approach, by D. Raymer, 1992, published in the AIAA Education Series described as "creating a comprehensive library of the established practices in aerospace design" with the following conclusion of the Foreword: "For many years Aircraft Design will be a valuable text book for all who struggle with the fundamentals and intricacies of aircraft design".  We read on p 259:

  • Lift is created by forcing the air that travels over type top of the wing to travel faster than the air which passes under it. This is accomplished by the wing's angle of attack and/or wing camber. The resulting difference in air velocity creates a pressure differential between the upper and lower surfaces of the wing, which produces the lift that supports the aircraft. 
  • All aerodynamic lift and drag forces result from the combination of shear and pressure forces. However, the dozens of classification schemes for aerodynamic forces can create considerable confusion because of overlapping terminology. 
  • For example, the drag of a wing includes forces variously called airfoil profile drag, skin-friction drag, separation drag, parasite drag, camber drag, drag-due-to-lift, wave drag, wave-drag-due-to-lift, interference drag, scrubbing drag, trim drag, induced drag --- and so forth. 

These statements express either self-evident triviality (lift from pressure differential) or a zoo of drag forces beyond rationale. If this is representative of the science of AIAA, our New Theory of Flight will not be understood and well received.


måndagen den 23:e juli 2012

Feynman's Logical Fallacy




This clip shows that even Richard Feynman has fallen into the trap of the logical fallacy of confirming a hypothesis by observing consequences of the hypothesis. Or rather the other way around: Feynman in the role of the great scientist takes a firm grip of the audience by the trivial information that if a consequence of the hypothesis is at variance with observation, then something must be wrong with the hypothesis.  But by lifting this triviality to a deep insight by a great scientist, Feynman opens to the fallacy of confirming a hypothesis by observing consequences.

Feynman thrills the audience by revealing that a physicist starts out by simply guessing a law/hypothesis (which makes the audience laugh) and then seeks confirmation by observing consequences. Feynman does not say that a physicist starts by giving some direct rational reason why the hypothesis should hold. Simply guessing is the physicists method. No wonder that modern physics is so strange.

A longer clip is here. Feynman continues with a discussion about preferring hypotheses which are more likely, that is hypotheses with some rationality and not just wild guesses as he started out with. Feynman thus blurs a most essential aspect of science and causes confusing rather than enlightenment.

torsdagen den 19:e juli 2012

Låg Antagningspoäng till SimuleringsTeknik KTH

Antagningspoängen till det nya kandidatprogrammet i SimuleringsTeknik vid KTH finns nu publicerade:

  • 13.9 för BI, 12.1 för BII och 0.70 för Högskoleprovet
  • totalt 42 antagna och 0 reserver.

Antagningspoängen kunde inte ha varit lägre. KTH skall nu satsa stora resurser på att utbilda den lägsta kvartilen av sökande, som om KTH tillhörde lägsta divisionen.  Rektor och dekaner vid KTH måste vrida sig i vånda inför denna uppgift.

SimuleringsTeknik ST inrättades för start HT 2011 som ett program i modern beräkningsmatematik enligt Mathematical Simulation Technology (MST), men stoppades via bl a mediadrev, genom att Rektor och Dekaner helt enkelt förbjöd all användning av MST vid KTH, detta beskrivet som KTH-gate.

Men Rektor och Dekaner ville så gärna starta ST, som ett uttryck för KTHs ledande roll vad gäller reform av teknikutbildning i datorns tidevarv, och ST skall nu starta HT 2012 på en grund av traditionell analytisk matematik, istället för MST som var för reformistisk.

Att detta inte går ihop har presumptiva studenter upptäckt och söktrycket som skulle kunna ha varit högt med MST, har istället blivit så lågt att luften gått ur programmet.

Hur skall KTH hantera denna situation? Starta en reformutbildning utan studenter och reformprogram, eller helt enkelt lägga ner ST eftersom MST är nedlagt? Eller återuppväcka MST?

söndagen den 15:e juli 2012

Making of the Prandtl Myth by John D. Anderson

Continuation of Making of the Prandtl Myth, now with quotes from Fundamentals of Aerodynamics by John D. Anderson, 1984:


Anderson: The modern science of aerodynamics rests on a strong fundamental foundation, a large percentage of which was established in one place by one man, at the University of Göttingen by Ludwig Prandtl. Prandtl never received a Noble Prize, although his contributions to aerodynamics and fluid mechanics are felt to be of that caliber.

Comment: The myth is supported by a Nobel Prize which Prandtl did not receive.

Anderson: Prandtl was considered a tedious lecturer because he could hardly make a statement without qualifying it.

Comment: This can be read as a sign that Prandtl's science was unclear.

Anderson: By the 1930s, Prandtl was recognized worldwide as the "elder statesman" of fluid dynamics…his "Nobel-Prize-level" contributions had all been made….He was clearly the father of modern aerodynamics - - a monumental figure in fluid dynamics. His impact will be felt for centuries to come.

Comment: Prandtl's 1904 paper with its suggestion that drag originates from a vanishingly thin boundary layer was long ignored, in particular by the leading UK aerodynamicist Lanchester, who was not at all convinced by Prandtl's sketchy speculations.

Further quotes form Prandtl's Boundary Layer by John D. Anderson:


Anderson: Despite the important work by Blasius and the sub- sequent publication of several papers on boundary-layer theory by Prandtl’s research group at Göttingen, the aero- dynamics community paid little attention, especially out- side of Germany. Finally in 1921, Theodore von Kármán, a former student of Prandtl’s and a professor at the University of Aachen, obtained a momentum-integral equa- tion through the simple expedient of integrating the boundary-layer equations across the boundary layer. That equation proved to be directly applicable to a large number of practical engineering problems, and with it, the boundary-layer theory finally began to receive more attention and acceptance in the technical community.

Anderson: The delayed acceptance of the boundary-layer concept is illustrated by the fifth and sixth editions of Horace Lamb’s classic text Hydrodynamics.6 The fifth edition, published in 1924, devoted only one paragraph to the boundary-layer concept and described Prandtl’s work as follows: “The calculations are necessarily elaborate, but the results, which are represented graphically, are interesting.” In contrast, the sixth edition, published in 1932, had an entire section on boundary-layer theory and the governing equations.

Comment: The Prandtl myth was carefully drafted by Prandtl's clever students von Karman and Schlichting. Science needs myths and heroes, and Prandtl showed to be useful.  

Anderson: Prandtl’s boundary-layer idea revolutionized how scientists conceptualized fluid dynamics. Before Prandtl, there was much confusion about the role of viscosity in a fluid flow. After Prandtl’s paper, the picture was made clear; in most cases, viscosity only played a role in the thin layer of flow immediately adjacent to a surface. What a breakthrough in the analysis and understanding of a vis- cous flow! Before Prandtl, there was no mathematically based, quantitative means to calculate the drag due to friction on a surface immersed in a fluid flow. After Prandtl’s paper, the fluid dynamicist could quantitatively calculate the skin-friction drag. Before Prandtl, there was no un- derstanding of the physical mechanism that caused a flow to separate from a surface. After Prandtl’s paper, the physics of separated flow became clear and the understanding of fluid dynamics underwent a revolutionary change.

Comment: This is a summary of the myth: Notice the shift from drag to skin-friction drag claimed to be computable by Prandtl's theory. But skin-friction drag is small compared to the pressure drag and can be set to zero in most applications. 


Making of the Prandtl Myth by Von Karman




Here is a continuation of the previous post analyzing the making of the myth of Prandtl as the Father of Modern Fluid mechanics, this time by his student Theodore von Karman in the book Aerodynamics: Selected Topics in the Light of their Historical Development from 1954: 

Von Karman: At the time of the first human flight, no theory existed that would explain the sustenation obtained by means of a curved surface at zero angle of attack. It seemed that the mathematical theory of fluid motion was unable to explain the fundamental facts revealed by experimental aerodynamics.

Comment: This is correct.

Von Karman: The Kutta-Zhukovsky condition seems to be a reasonable hypothesis, both because it is indicated by visual observation and also because the lift calculated by means of this condition seems is in fair accordance with measurements. The usefulness of the theory is restricted to a limited range of angle of attack, comprising relatively small angles. Beyond this range the the measured lift falls far below the values predicted by the theory.

Comment: Von Karman here uses the logical fallacy of confirming an assumption by observing a consequence: If there is circulation then there is lift, and since lift is observed there must be circulation.
This makes the theory fool-proof, and as such pseudo-scientific.

Von Karman: The man who gave modern wing theory its practical mathematical form was one of the most prominent representatives of the science of mechanics, and especially fluid mechanics, Ludwig Prandtl. His creates contributions to fluid mechanics were in the field of wing theory and the theory of the boundary layer. His control of mathematical methods and tricks was limited: many of his collaborators and followers surpassed him in solving difficult mathematical problems. but his ability to establish systems of simplified equations which expressed the essential physical relations and dropped the non-essentials, was unique.


Comment: This is the Prandtl myth: Prandtl reveals the mathematical secrets of fluid mechanics without knowing much math.

Von Karman: To be sure, Prandtl's theory has limitations, as does every theory. Its first limitation is caused by the phenomenon of stall.

Comment: By admitting limitation the theory is strengthened. The fact that stall is not predicted, should alone be sufficient to eliminate the theory as unphysical, and thus very dangerous to use for the design of real airplanes.

Von Karman: Our knowledge of the reasons "why we can fly" and "how we fly" has increased both in scope and depth in a rather impressive way.

Comment: The qualification "rather" means that the knowledge is not convincing.

Von Karman: We aerodynamicists were always more modest (than physicists) and did not attempt to change basic beliefs of the human mind or to interfere with the business of the good Lord or divine Providence.

Comment: The theory is strengthened by showing a humble attitude, different from that of physicists.

lördagen den 14:e juli 2012

Making of the Prandtl Myth by Schlichting




Listen to Herrmann Schlichting forming the Myth of Prandtl as Father of Modern Fluid Mechanics


The myth of Prandtl as the Father of Modern Fluid Mechanics was shaped by his former students Theodor von Karman and Hermann Schlichting (photo) serving as aeronautics expert scientists in the US and Germany during the 2nd World War. Upon request from the Allied forces Schlichting documented Prandtl's expertise in the book Boundary Layer Theory, viewed as the bible of modern fluid mechanics. It is possible to argue that the outcome of the war was influenced by incorrect German aeronautics.

Let us analyze how Schlichting builds the Prandtl myth in the Introduction to his book in a sequence of quotes followed by comments:

Schlichting: The present book is concerned with the branch known as boundary-layer theory. This is the oldest branch of modern fluid dynamics; it was founded by Prandtl in 1904 when he succeeded in showing how flows involving fluids of very small viscosity, in particular  water and air, the most important ones from the point of view of applications, can be made amenable to mathematical analysis.

Comment: This sets the scence with boundary-layer theory opening to technological progress by mathematical analysis in the hands of Prandtl.

Schlichting: This was achieved by taking the effects of friction into account only in regions where they are essential, namely in the thin boundary layer which exists in the immediate neighbourhood of a solid body.

Comment: This is a clever circular formulation with effects of viscosity taken into account where they are essential and should be taken into account.

Schlichting: This concept made it possible to clarify many phenomena which occur in flows and which
had previously been incomprehensible.

Comment: Vague. Nothing was clarified, only further mystified.

Schlichting: Most important of all, it has become possible to subject problems connected with the occurrence of drag to a theoretical analysis.

Comment: This is the central dogma of Prandtl: Drag originates from boundary layer effects. We show that thus is incorrect by obtaining correct drag without boundary layers.

Schlichting: The science of aeronautical engineering was making rapid progress and was soon
able to utilize these theoretical results in practical applications. It did, furthermore,  pose many problems which could be solved with the aid of the new boundary layer theory. Aeronautical engineers have long since made the concept of a boundary layer one of everyday use and it is now unthinkable to do without it.

Comment: Vague. 

Schlichting: In other fields of machine design in which problems of flow occur, in particular in the design of turbomachinry, the theory of boundary layers made much slower progress, but
in modern times these new concepts have come to the fore in such applications as well.

Comment: This in an admittance that the boundary layer theory is not useful in applications.

Schlichting: Towards the end of the 18th century the science of fluid mechanics began to
develop in two directions which had practically no points in common. On the one
side there was the science of theoretical hydrodynamics which was evolved from
Euler's equations of motion for a frictionless, non-viscous fluid and which achieved a
high degree of completeness. Since, however, the results of this so-called classical
science of hydrodynamics stood in glaring contradiction to experimental results — in
particular as regards the very important problem of pressure losses in pipes and
channels, as well as with regard to the drag of a body which moves through a mass
of fluid — it had little practical importance. For this reason, practical engineers,
prompted by the need to solve the important problems arising from the rapid
progress in technology, developed their own highly empirical science of hydraulics.
The science of hydraulics was based on a large number of experimental data and
differed greatly in its methods and in its objects from the science of theoretical
hydrodynamics.

Comment: This is an admittance that theory and practice do not come together.

Sclichting: At the beginning of the present century L. Prandtl distinguished himself by
showing how to unify these two divergent branches of fluid dynamics. He achieved
a high degree of correlation between theory and experiment and paved the way
to the remarkably successful development of fluid mechanics which has taken place
over the past seventy years. It had been realized even before Prandtl that the
discrepancies between the results of classical hydrodynamics and experiment were, in
very many cases, due to the fact that the theory neglected fluid friction. Moreover,
the complete equations of motion for flows with friction (the Navier-Stokes
equations) had been known for a long time. However, owing to the great mathematical
difficulties connected with the solution of these equations (with the exception of a
small number of particular cases), the way to a theoretical treatment of viscous
fluid motion was barred. Furthermore, in the case of the two most important fluids,
namely water and air, the viscosity is very small and, consequently, the forces
due to viscous friction are, generally speaking, very small compared with the
remaining forces (gravity and pressure forces). For this reason it was very difficult
to comprehend that the frictionall forces omitted from the classical theory influenced
the motion of a fluid to so large an extent.

Comment: The claim that Prandtl unified mathematical theory and practice is without substance. Prandtl was mathematically naive and his dictate that Navier-Stokes equations cannot be combined  with force boundary conditions, is incorrect. 

Schlichting: In a paper on "Fluid Motion with Very Small Friction", read before the
Mathematical Congress in Heidelberg in 1004, L. Prandtl | showed how it was possible to
analyze viscous flows precisely in cases which had great practical importance.

Comment: The paper is very short (8 sparsely typed pages) and contains no mathematical analysis, only vague speculations, which have showed to be misleading.

Schlichting: With the aid of theoretical considerations and several simple experiments, he proved that
the flow about a solid body can be divided into two regions: a very thin layer in the
neighbourhood of the body (boundary layer) where friction plays an essential part,
and the remaining region outside this layer, where friction may be neglected.

Comment: This subdivision is mathematically and physically impossible.

Schlichting: On the basis of this hypothesis Prandfl succeeded in giving a physically penetrating
explanation of the importance of viscous flows, achieving at the same time a maximum
degree of simplification of the attendant mathematical difficulties. The theoretical
considerations were even (then supported by simple experiments performed in a
small water tunnel which Prandtl built with his own hands. He thus took the first
step towards a reunification of theory and practice. This boundary-layer theory proved
extremely fruitful in that it provided an effective tool for the development of fluid
dynamics.

Comment: The claim that Prandtl unifies theory and practice with a maximum of mathematical simplification lacks rationale.

Schlichting: Since the beginning of the current century the new theory has been
developed at a very fast rate under the additional stimulus obtained from the recently
founded science of aerodynamics. In a very short time it became one of the foundation
stones of modern Ihiid dynamics together with the other very important
developments — the aerofoil theory and the science of gas dynamics. 

Comment: This summarizes the myth of Prandtl as the Father of Modern Fluid Mechanics. The truth is that Prandtl misled a whole century of fluid dynamicists in searching for drag and lift in a vanishingly thin boundary layer.

Schlichting: The existence of tangential (shearing) stresses and the condition of no slip near  solid walls constitute the essential differences between a perfect and a real fluid. Certain fluids which are of great practical importance, such as water and air, have very small coefficients of viscosity. In many instances, the motion of such fluids of small viscosity agrees very well with that of a perfect fluid, because in most eases the shearing stresses are very small. 

Comment: This is an admittance that the skin friction forces are small in the boundary layer and thus can be approximated by a slip boundary condition, in direct violation of Prandtl's dictate of no-slip.

Schlichting: For this reason the existence of viscosity is completely neglected in the theory of perfect fluids, mainly because this introduces a far-reaching simplification of the. equations of motion, as a result of which an extensive mathematical theory becomes possible.  It is, however, important to stress the fact that even in fluids with very small viscosities, unlike in perfect fluids, the condition of no slip near a solid boundary prevails. This condition of no slip introduces in many cases very large discrepancies in the laws of motion of perfect and real fluids. In particular, the. very large discrepancy between the value of drag in a real and a perfect fluid has its physical origin in the condition of no slip near a wall. 

Comment: This is Prandtl's dictate of no-slip which has made 20th century fluid mechanics both uncomputable and impossible to rationalize.

Foreword by Dryden: Boundary-layer theory is the cornerstone of our knowledge of the flow of air and other fluids of small viscosity under circumstances of interest in many engineering applications. Thus many complex problems in aerodynamics have been clarified by a study of the flow within the boundary layer and its effect on the general flow around the body. Such problems include the variations of minimum drag and maximum lift of airplane wings with Reynolds number, wind-tunnel turbulence, and other parameters. Even in those cases where a complete mathematical analysis is at present impracticable, the boundary-layer concept has been extraordinarily fruitfull and useful.


Comment:  Big words without real substance.