6

Triumph and Failure of Mathematics

No part of the aim of normal science is to call forth new sorts of phenomena;
indeed those that will not fit the box are often not seen at all. Nor do
scientists normally aim to invent new theories, and they are often intolerant
of those invented by others. Instead, normal-scientific research is directed to
the articulation of those phenomena and theories that the paradigm already
supports. (Thomas Kuhn in The Structure of Scientific Revolutions, 1962)

The field of hydrodynamic phenomena which can be explored with exact
analysis is more and more increasing. (Zhukovsky, 1911)

6.1 Triumph: Celestial Mechanics

In the famous treatise Celestial Mechanics in five volumes published during
1799–1825, Laplace formulated Newton’s theory of gravitation in the form
−∆φ = ρ, where φ is the gravitational potential and ρ the mass distribu-
tion. Knowing the mass distribution, e.g. one heavy point mass representing
the Sun, surrounded by lighter point masses representing the planets, that
is, knowing ρ(x) at a given time instant, one can solve for φ(x) and obtain
the gravitational force field F (x) =

∇φ(x), from which the acceleration of

the masses can be determined using Newton’s Law F = ma, where m is the
mass and a the acceleration. From the acceleration, the velocity and motion
of the masses can then be determined. Laplace could thus summarize celes-
tial mechanics in the differential equation

−∆φ = ρ, and in particular this

way prove Newton’s inverse square law, which Newton just assumed to be
true. Laplace could thus, and also did, predict the positions of the planets
many years ahead from knowing their present positions and velocities. This
is probably the most important triumph of mathematics all times, and gave
mathematics and science an enormous boost.

50

6 Triumph and Failure of Mathematics

Fig. 6.1. Isaac Newton (1643–1727), Pierre-Simon Laplace (1749–1827), and Jean
Le Rond d’Alembert (1717–1783).

6.2 Failure: Potential Flow

The triumph for mathematics in celestial mechanics starting with Newton,
stimulated mathematicians of the 18th and 19th centuries (Fig. 6.1) to try the
same approach for fluid mechanics, with the hope of summarizing also this
scientific discipline in the form ∆φ = 0, with φ now a velocity potential with
∇φ representing the flow velocity. The prospects seemed really good: In this
form one could represent a variety of ideal, stationary (time-independent),
incompressible, irrotational flows as potential flows. We recall that a flow
velocity u is irrotational if

∇ × u = 0, which holds if u = ∇φ.

We know that potential flows had been studied already by d’Alembert in

the mid 18th century and d’Alembert had published his paradox in 1752: A
body of any shape can move through a lightly viscous fluid like water without
any drag! Of course nobody could believe this, which from start gave mathe-
matical fluid mechanics a strange reputation among the many practitioners of
hydraulic engineering, which probably has lasted into our days. The challenge
is to change this unfortunate situation.

The same mathematical equation, Laplace’s equation, which was so amaz-

ingly successful in celestial mechanics, thus was a complete failure in fluid
mechanics, and evidently mathematics and fluid mechanics lived a long time
with a very disturbing paradox. How could that be?

As indicated, we present below a new resolution of d’Alembert’s Mystery

to illustrate basics aspects of fluid flow, including stability and transition to
turbulence.