Paradoxes and potentialities

A brief note on the paradoxes and mathematical potentialities of fluid dynamics.

Last week was the Nobel week and it has piqued my curiosity to write this short note on the richness of the field of hydrodynamics (fluid dynamics) and how it emerged on the scene of the golden age of physics as an independent field resulting from the merger of mathematics, physics and engineering sciences. I will further talk about the reluctance of the Royal Swedish Academy of Sciences to award the Nobel prize in Physics for the contributions of a man whose seminal work still continues to dominate the field. Some developments and recognition attained in the recent past will also be discussed.

The inception

Early attempts to apply the principles of dynamics to fluid flow began with Swiss physicist and mathematician Daniel Bernoulli's Hydrodynamica (1738). In this work, Bernoulli expounded on the subject from the standpoint of the principle of conservation of energy. Analogous to a moving body exchanging its potential energy for kinetic energy when it loses altitude, Bernoulli realized fluids must also obey a similar conservation law when in motion i.e. the fluid in motion may exchange its energy between the partitions of pressure, kinetic and potential energies. This unique insight led him to summarize his findings in the form of a very famous equation named after him - the Bernoulli Equation^†,

\[\frac{1}{2}\rho u^2 + p + \rho g h= constant\]

In addition to Daniel, his father Johann also published a treatise on flow called Hydraulica (1742) in which he approached the relevant, practical flow problems of that era from a completely different dynamical principle. The principle was now Newton's 2nd law. The most important insights and developments derived from Johann's Hydraulica were the concept of internal pressure and the idea of flow acceleration ($u\partial u / \partial x)$ originating from the convection of the fluid, the convective acceleration as we know it today, in addition to the local acceleration which originates by virtue of the velocity variation per unit time $(\partial u / \partial t$).

Jean le Rond d'Alembert, the French philosopher and mathematician rederived the results of Bernoullis using the principle of equilibrium named after him -- D'Alembert's principle. In layman terms, a simple explanation of this principle is that different parts/regions of the flow (which is essentially a classical system of particles in motion) must be in equilibrium with respect to the pseudo-forces obtained by subtracting from real external forces the product of their mass and acceleration. Johann Bernoulli may have introduced a rudimentary concept of convective acceleration of the fluid in addition to the local acceleration, but it was D'Alembert who is cited to have used the partial differentials in explaining the concept. Most strikingly, in his treatise on fluid resistance (1749), D'Alembert obtained particular cases of Euler's equations and the continuity equation.

Euler's emergence

At this point of time, the stage was set for Shakespeare of Math, Leonard Euler to emerge at the scene of theoretical and mathematical fluid dynamics. Influenced by the works of D'Alembert, Euler proposed the Eulerian method of deriving the general equations of fluid motion, the crux of which was to apply the Newton's 2nd law to each fluid element and invoking the idea of the pressure from the surrounding fluid, resulting in the following,

\[ \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} = -\frac{1}{\rho}\nabla p +\mathbf{g} \ \ \ \ (Euler \ equation) \]

and the continuity equation,

\[ \nabla \cdot \mathbf{u} = 0 \ \ \ \ (Continuity)\]

Paradoxes and Potentials*

From the principle of dynamical equilibrium, D'Alembert concluded (mathematically) that the curl of velocity must vanish $(\nabla \times \mathbf{u}= 0)$ for the problem of uniform flow around a symmetric body. This result derives from the issue that his differential equations admitted only one solution which gave rise to a symmetric and unique flow around the body yielding a symmetrical pressure distribution over the surface. We know that D'Alembert expressed the force exerted by the fluid on the surface of the solid as a surface integral of the pressure forces, a unique solution which is similar at the front and rear of the body. This would mean the pressure at the front would exactly balance the pressure at the rear end of the body, implying zero resistance. Bernoulli's equation yields the same solution of zero resistance for such a solution of the velocity field, as the pressures at the rear end and the front end exactly balance each other. Euler repeated the same mistake of considering the trivial, zero vorticity $(\nabla \times \mathbf{u}= 0)$ as a necessary condition for the validity of the vorticity equation. Thus, out of this illegitimacy, the velocity potential $\phi$ was born.

In 1749, D'Alembert ended his fluid dynamics on a paradox, concluding,

"Thus I do not see, I admit, how one can satisfactorily explain by theory the resistance of fluids. On the contrary, it seems to me that the theory, developed in all possible rigor, gives, at least in several cases, a strictly vanishing resistance; a singular paradox which I leave to future geometers for elucidation."

-- J. R. D'Alembert, 1749

Vorticity evolution.
Source**: ICFD UCLA.

The solutions obtained by Euler and De'Alembert contradicted with the common observations. For example, these solutions forbade the flight of birds! As a result of the abstraction of these solutions, fluid dynamics branched off into a more practical and less rigorous Hydraulics whose proponents considered theoretical fluid dynamics merely a complex structure of mathematical truths. Charles Bossut, the prominent hydraulician commented,

"These great geometers [d'Alembert and Euler] seem to have exhausted the resources
that can be drawn from analysis to determine the motion of fluids: their formulas are
so complex, by the nature of things, that we may only regard them as geometric
truths, and not as symbols fit to paint the sensible image of the actual and physical
motion of a fluid.
"

-- Charles Bossut, 1786

For a very long time, the mathematical fluid dynamics took a back seat because of its unreal, abstract solutions to practical flow problems. Two worlds of flow were born; the real one and the one in the minds of theoretical physicists and mathematicians working on fluid dynamics.

Escaping the paradox

In the 19th century, Anglo-Irish physicist and mathematician George Gabriel Stokes attempted to arrive at solutions to D'Alembert's paradox by considering fluid friction, finite slip between the adjacent fluid layers and instability leading to a turbulent wake behind the immersed solid. He introduced the concept of viscosity to bridge the gap between the ideal flow solutions derived from the theory and the real flows existing in nature and engineering systems. As a result of his attempts, the Navier-Stokes^‡ equations (NSE) were discovered,

\[ \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} = -\frac{1}{\rho}\nabla p + \nu \nabla^2 \mathbf{u} +\mathbf{g} \ \ \ \ (NSE) \]

19th century mathematicians like Lagrange, Laplace, Poisson, Rayleigh, Boussinesq, Helmholtz, Cauchy, Poisson and Saint Venant devoted their work to develop the models of flow for a variety of practical problems relevant to natural and engineering flows. The idea of unstable vortex sheets (analogous to Stokes' finite slip surfaces) introduced by Helmholtz (1858) proved to be an effective solution to bridging the gap between theory and observation. This paved the way for future physicists and mathematicians to resolve the D'Alembert's paradox.

Prandtl: The man, and Swedish reluctance

By the end of 19th century, a number of approaches to reconcile the theory of fluid dynamics with the observations gave rise to new mathematical concepts like the viscous stress tensor, vortex motion, and hydrodynamic instability etc. Even though these approaches closed the gap between the ideal and real worlds of flow somewhat, they lacked robustness and as a result proved to be ineffective in some real flows.

In 1904, the German physicist and fluid dynamicist Ludwig Prandtl introduced the revolutionary idea of a boundary layer to tackle the problem of vanishing resistance. He suggested that the effects of a thin viscous boundary layer could possibly be the source of substantial resistance experienced by a solid body immersed within a flow. He proposed that at high Reynolds numbers, a no-slip boundary condition at the wall causes a strong variation of flow velocity over a thin layer near the body. The presence of shear generates vorticity and dissipates the kinetic energy of fluid in the boundary layer. As he pointed out, this dissipation of energy results in separation of flow in the case of bluff bodies and acts as the origin of substantial form drag. Here is a link to his groundbreaking paper: On Motion of Fluids Flow with Very Little Viscosity in which he described the boundary layer and its importance, ultimately resolving a 150 year old paradox by highlighting the unphysicality of the 'zero resistance potential solution'^⸸. Prandtl was the saviour who saved fluid dynamics from imminent collapse. The impact of his 1905 paper is often compared to the famous papers of Albert Einstein published in the same year. In 2005, celebrating 100 years of Einstein's work and Prandtl's seminal paper, aerodynamicist John D. Anderson says[1],

"In 2005, concurrent with the World Year of Physics celebration of, among other things, Albert Einstein and his famous papers of 1905, we should also celebrate the 100th anniversary of Prandtl’s seminal paper. The modern world of aerodynamics and fluid dynamics is still dominated by Prandtl’s idea. By every right, his boundary-layer concept was worthy of the Nobel Prize. He never received it, however; some say the Nobel Committee was reluctant to award the prize for accomplishments in classical physics."

-- John D. Anderson

In reply to G. I. Taylor's (another modern giant) letter in early 1930s, Prandtl responded to Taylor's strong conviction of initiating a Nobel prize (Physics) nomination for him,

“If the Swedes classify the sciences in a similar way as here, then I will be taken under consideration as little as the mathematicians, and for the rest I will know how to console myself just like the mathematicians.”

-- Ludwig Prandtl

Prandtl had been nominated for the Nobel prize already in 1928 by German colleagues, and again in 1936 by the British Nobel prize laureate W. L. Bragg, whose letter of nomination hints at Taylor's initiative in the background. Unfortunately, the Swedish were evidently reluctant to honor the fundamental advances in classical physics at that point of time[2].

The golden age

After the huge strides in the fundamental physical discoveries of fluid dynamics in the early part of 20th century, fluid dynamics carved a niche of its own as an immensely rich and successful mathematical discipline. Solutions to old paradoxes resulted in new discoveries and the resulting advances posed new challenges and questions. Werner Heisenberg started out his illustrious academic career with a Ph.D. expounding stability of laminar flows and nature of turbulence. He returned to this problem after World War II. Prior to Heisenberg, another famous physicist Arnold Sommerfeld also developed a theory of hydrodynamic stability. In fact, Heisenberg's doctoral thesis topic was suggested by Sommerfeld himself. At the same time, Russian mathematician Andrey Kolmogorov developed a groundbreaking theory of isotropic turbulence -- a topic so rigorous that it demands an article of its own.

Theoretical research into the mathematical nature of turbulence is considered as one of the unsolved mysteries in classical physics. The Navier–Stokes existence and smoothness problem that concerns the mathematical properties of solutions to the Navier–Stokes equations is listed as one of the seven Millennium Prize problems in mathematics by the Clay Mathematics Institute. It offers a US $1,000,000 prize to the first person providing a solution for a specific statement of the problem listed here.

Overall, two Nobel prizes have been awarded in the field of fluid dynamics -- the 1970 prize to Hannes Olof Gösta Alfvén "for fundamental work and discoveries in magnetohydrodynamics with fruitful applications in different parts of plasma physics", and the 1996 prize to David M. Lee, Douglas D. Osheroff and Robert C. Richardson for the discovery of superfluidity in helium-3.

Summing up

From the dynamical principles of equilibrium and energy conservation to the novel concepts of plasma physics and quantum turbulence, fluid dynamics has evolved into an enterprise of immense scientific and mathematical potential. It is not unreasonable to say that it lies at the core of many fields of applied sciences. Fundamental advances in fluid dynamics have divulged the secrets of the microscopic and the cosmic realms - all alike; from the pulsating flow of blood in our hearts to the dynamos of of plasma inside stars!

Footnotes

† This equation served as the theoretical underpinning of the first invasive blood pressure measuring technique in which the patient's arteries were punctured to stick point-ended glass tubes to measure the pressure at a specific point in the artery by recording the height of the blood column in the glass tube.

* Please note that the 'Potentials' is capitalized here unlike the article title where it is small case 'potential'. Capitalized 'P' stresses that the potential talked about here is the velocity potential $\phi$ while as in the article title 'potentialities' signifies the general adjective to highlight the immense potential of the field of fluid dynamics as a fundamental mathematical science.

** https://www.math.ucla.edu/~yanovsky/Incompressible.htm

‡ In 1827, French engineer and physicist Claude-Louis Navier also carried out studies to explore the mathematical models of flow.

⸸ Advances since Prandtl have offered alternative and more plausible explanations to resolve the D'Alembert's paradox.

Cite as

Bader, Shujaut H., “Paradoxes and potentialities: A brief note on the paradoxes and mathematical potentialities of fluid dynamics."

References

Ludwig Prandtl's Boundary Layer Theory, John D. Anderson Jr. http://www.math.lsa.umich.edu/~krasny/math654_prandtl.pdf
Ludwig Prandtl and the growth of fluid mechanics in Germany, Michael Eckert. https://www.sciencedirect.com/science/article/pii/S1631072117300815
Worlds of Flow, Olivier Darrigol, ISBN-13: 978-0199559114.

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