Claude-Louis Navier

How to use this course

Claude-Louis Navier was a civil engineer who believed that bridges, beams and canals should be designed from mathematical principles instead of rules of thumb. In pursuing that belief he wrote down, in 1821 and 1822, the first general equations for elastic solids and for viscous fluids. The fluid equations, completed by others over the next quarter-century, carry his name together with that of George Gabriel Stokes and stand at the centre of one of the seven Millennium Prize Problems. This course follows his career in the French engineering corps and works through the mechanics he created.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. Lessons 04 to 10 follow the order of Navier’s work: strength of materials (1813–1826), plates (1820), elastic solids (1821), viscous fluids (1822), suspension bridges (1823) and machines. Lesson 08 follows his fluid equations to the present day.

Timeline

Dijon, Gauthey and the Revolution, 1785–1802

Claude-Louis-Marie-Henri Navier was born in Dijon on 10 February 1785. His father was a lawyer and a member of the National Assembly during the Revolution. He died in 1793, when the boy was eight. His mother returned to Chalon-sur-Saône, and the boy remained in Paris in the care of his granduncle Émiland Gauthey.

Gauthey was one of the most respected engineers of the Corps des Ponts et Chaussées, the state body responsible for roads, bridges and canals. He had built the Canal du Centre in Burgundy and designed bridges across the region. He took charge of Navier’s education and steered him toward engineering. The boy grew up surrounded by the practical knowledge of the eighteenth-century French engineers: rules for arches, tables of the strength of timber and stone, and long experience with water.

Polytechnique and the Ponts, 1802–1806

In 1802 Navier entered the École Polytechnique, founded eight years earlier to train engineers for the state services on a common base of mathematics and physics. He was admitted near the bottom of the list, but by the end of his first year he had risen into the top ten. His teacher of analysis was Joseph Fourier, then completing his first work on heat conduction. Fourier became Navier’s mentor and lifelong friend, and Fourier series appear throughout Navier’s later work.

In 1804 he moved to the École des Ponts et Chaussées, the corps’ own school of application, and graduated near the top of his class in 1806. That year Gauthey died, leaving an unfinished treatise on bridge construction. The Polytechnique training gave Navier something his granduncle had lacked: fluency with the calculus and with the analytical mechanics of Joseph-Louis Lagrange. The rest of his career was an attempt to bring the two traditions together.

Editor of Gauthey and Bélidor, 1807–1819

After a short posting in the provinces, Navier was recalled to Paris in 1807 to publish Gauthey’s manuscripts. The first volume of the Traité de la construction des ponts appeared in 1809, with only light notes. As he worked, his notes grew into original research in analytical mechanics and the strength of materials. The second volume (1813) and his new edition of Bernard Forest de Bélidor’s La Science des ingénieurs (1813), an eighteenth-century engineering manual, carried long notes and new tables of strength. He married Marie Charlot around this time.

Between 1814 and 1818 he studied machines and the measurement of their output, work that led him to the quantity later called mechanical work (Lesson 10). By 1819, when he began teaching applied mechanics at the École des Ponts et Chaussées, he had become known as the engineer who could put analysis behind the rules of practice. He turned the school’s course from a catalogue of empirical results into a course in mechanics.

Beams and the neutral axis

For the general reader. When a beam bends, its top is squeezed and its bottom is stretched. Somewhere in between lies a surface that is neither, the neutral axis. Earlier engineers had placed it in the wrong position. Navier put it in the right place and built from it a complete theory of how far beams bend and when they break, the theory still taught to every engineering student.

Assumptions

Consider a straight beam along the \(x\)-axis with cross-section \(A\) in the \((y,z)\)-plane, bent in the \((x,y)\)-plane. Navier assumed that plane cross-sections remain plane and perpendicular to the deformed axis, and that each fibre obeys Hooke’s law with a modulus \(E\). If the axis has curvature \(\kappa\), a fibre at height \(y\) above the neutral surface has strain \(\varepsilon=-\kappa y\) and stress \(\sigma=-E\kappa y\).

Derivation. Zero axial force means \(\int_A\sigma\,dA=-E\kappa\int_A y\,dA=0\), so the first moment of area about the neutral axis vanishes: the axis passes through the centroid. Galileo had analysed a cantilever as if it pivoted about its bottom edge; Edme Mariotte and others proposed other positions. Navier fixed the correct one in his lectures from 1819, and in the Résumé des leçons of 1826 he separated the material property \(E\) from the geometric property \(I\).

The beam equation

For small deflection \(w(x)\), \(\kappa\approx w''\). Equilibrium of a beam element gives \(V=dM/dx\) and \(dV/dx=-q\) for a distributed load \(q\) (downward positive, \(w\) measured downward). Combining,

Navier also treated statically indeterminate beams, those with more supports than equilibrium alone can resolve, by adding the compatibility conditions of deformation. This method is the core of structural analysis.

Elastic plates

For the general reader. A floor slab or a ship’s deck bends in two directions at once. In 1820 Navier wrote down the equation for the bending of a thin elastic plate and solved it for a rectangular plate supported along its edges, by breaking the load into a sum of waves, the method his teacher Fourier had invented for heat.

The biharmonic equation had been reached earlier, by Sophie Germain with Lagrange’s correction (1811–1816), in connection with vibrating plates; Poisson also derived it. Navier’s 1820 memoir derived it from a molecular model, discussed the boundary conditions, and solved concrete static problems. The coefficient \(D\), the flexural rigidity, took its present form later in the century.

Navier’s solution

For the rectangle \(0\le x\le a\), \(0\le y\le b\), simply supported (\(w=0\) and zero bending moment on each edge), every term \(\sin\frac{m\pi x}{a}\sin\frac{n\pi y}{b}\) satisfies the boundary conditions. Expand the load:

Since \(\nabla^4\) multiplies each term by \(\pi^4\big(\frac{m^2}{a^2}+\frac{n^2}{b^2}\big)^2\),

The equations of elastic solids

For the general reader. Beams and plates are special shapes. In May 1821 Navier presented to the Académie the first general equations for the small deformations of an elastic body of any shape. He pictured the solid as a lattice of molecules pulling on each other. The picture was too simple, but the equations were the start of the modern theory of elasticity.

The molecular model

Navier assumed that molecules interact in pairs by forces along the line joining them, depending on distance, and that in a deformation each pair’s force changes in proportion to the change in its separation. Summing over neighbours in an isotropic arrangement, and using a variational (virtual work) argument, he found one material constant \(\varepsilon\).

In the modern form derived by Cauchy from his concept of stress (1822–1828) and by George Green from a strain energy function (1837),

with two independent Lamé constants \(\lambda,\mu\). Navier’s equation is the special case \(\lambda=\mu=\varepsilon\). This one-constant restriction, shared by Poisson’s molecular theory, fixes Poisson’s ratio at \(1/4\). The question of whether isotropic solids need one constant or two (more generally 15 or 21) divided the “rari-constant” and “multi-constant” schools for much of the century, until experiment settled it in favour of two.

Waves

Taking the divergence and the curl of the two-constant equation (with \(f=0\)) separates it into two wave equations: \(\nabla\cdot u\) travels at \(c_P=\sqrt{(\lambda+2\mu)/\rho}\) and \(\nabla\times u\) at \(c_S=\sqrt{\mu/\rho}\). These are the P and S waves of seismology.

The equations of viscous fluids

For the general reader. Leonhard Euler’s equations of 1757 describe a fluid without friction. In such a fluid, a ship would feel no drag and water in a pipe would need no pump. On 18 March 1822 Navier read a memoir to the Académie that added internal friction to Euler’s equations. The result is the equation that engineers now use to design aircraft and that meteorologists use to forecast the weather.

From Euler to Navier

Navier applied his molecular picture to a liquid. He supposed that between two moving molecules there is, besides the static pressure force, an extra force proportional to their relative velocity. Summing over neighbours as he had done for solids, he obtained a single new term.

The form is exactly right for an incompressible Newtonian fluid, although Navier had no notion of shear stress and his molecular hypothesis is not how viscosity arises in liquids. At a solid wall he proposed that the fluid may slip, with tangential friction proportional to the slip velocity. In modern form the Navier slip condition reads \(u_t=\ell\,\partial u_t/\partial n\), with \(n\) the normal pointing into the fluid and \(\ell\) a slip length. For ordinary flows \(\ell\) is tiny and the condition reduces to no slip, \(u=0\); in microfluidics and rarefied gases the slip length matters, and Navier’s condition is used.

Navier–Stokes after Navier

For the general reader. Navier wrote the right equations for the wrong reasons. Over the next quarter-century, Cauchy, Poisson, Barré de Saint-Venant and Stokes rederived them from sounder principles. Two centuries later, nobody has proved that the equations always have smooth solutions in three dimensions. A million-dollar prize awaits whoever settles the question.

Stress and the constitutive law

Cauchy (1822–1823) showed that the internal forces in any continuum are described by a symmetric stress tensor \(\sigma_{ij}\), and that momentum balance reads \(\rho\,Du_i/Dt=\partial_j\sigma_{ij}+f_i\). For a fluid at rest \(\sigma_{ij}=-p\,\delta_{ij}\). Stokes (1845) assumed that the extra stress is linear and isotropic in the rate of strain \(e_{ij}=\frac12(\partial_iu_j+\partial_ju_i)\):

For incompressible flow \(\nabla\cdot u=0\) and \(\partial_j(2\mu e_{ij})=\mu\nabla^2u_i\), which recovers Navier’s equation exactly. Poisson (1829) and Saint-Venant (1843) gave intermediate derivations. Stokes’s 1845 paper, which also discussed the conditions at a solid wall, made them standard; his 1850 study of pendulums in air applied them.

Similarity

Scaling lengths by \(L\), speeds by \(U\) and time by \(L/U\) gives the dimensionless form

so geometrically similar flows with equal Reynolds number, named after Osborne Reynolds, behave identically. This is the principle behind wind tunnels and ship tanks.

Jean Leray proved in 1934 that weak solutions exist for all time. In two dimensions, global smooth solutions are known (Olga Ladyzhenskaya and others). In three dimensions, smoothness is known only for small data or short times, and the set of possible singular times and points is known to be small (Caffarelli, Kohn and Nirenberg, 1982).

Suspension bridges

For the general reader. In the early 1820s British engineers were building iron chain bridges with spans no stone arch could match. The French government sent Navier to study them. His 1823 report gave the first mathematical theory of the suspension bridge: the shape of the chain, the forces in it, and how the deck responds to moving loads.

The cable

A flexible cable carries no bending moment, so its tension \(T\) is tangent to it. Let the deck weight \(w\) per unit horizontal length hang from the cable, and neglect the cable’s own weight. The horizontal component \(H\) of the tension is constant, since no horizontal loads act. Vertical equilibrium of an element gives \(H\,y''=w\).

If the cable’s own weight dominates, the load is uniform per unit arc length and the shape is a catenary, \(y=a\cosh(x/a)\). For the shallow cables of real bridges the two shapes are nearly the same. Navier also studied the deflection of the chain under a concentrated moving load, the extension of the chain under tension, and the dynamic oscillation of the deck, a problem that would become notorious a century later at Tacoma Narrows.

Work, machines and the quantity of action

For the general reader. How do you compare a horse, a waterwheel and a steam engine? Engineers needed a common measure of what machines deliver. Navier proposed one, which he called the quantity of action: force multiplied by the distance through which it acts. Gaspard-Gustave Coriolis refined it and named it work, the concept every physics student now learns.

From vis viva to work

Gottfried Wilhelm Leibniz’s vis viva was \(mv^2\). In his notes to Bélidor and his studies of machines (1814–1818), Navier measured the useful output of a machine by \(\int F\,dx\) and equated it to half the change in vis viva. Coriolis in 1829 introduced the factor \(\frac12\) into the definition of kinetic energy and the word travail. The French engineers, with Lazare Carnot and Jean-Victor Poncelet, built mechanics of machines on this balance.

Proof. \(F\cdot dx=m\,\dot v\cdot v\,dt=d\big(\tfrac12m|v|^2\big)\). Navier stated the balance for machines but did not reach a general principle of conservation of energy; that came in the 1840s with Julius Robert Mayer, James Prescott Joule and Hermann von Helmholtz.

The Invalides bridge and the Académie, 1821–1830

The early 1820s were Navier’s most productive years: the plate memoir (1820), the elasticity memoir (May 1821), the fluids memoir (March 1822) and the Rapport et mémoire sur les ponts suspendus (1823), based on his study of British chain bridges. In 1824 he was elected to the Académie des Sciences.

The same year he began building his own showpiece: a single-span iron suspension bridge across the Seine opposite the Hôtel des Invalides, about 155 metres between the towers, far longer than any bridge in Paris. Construction started in August 1824 under a private concession. In July 1826 a crack appeared in one of the anchorage masses, which engineers judged ordinary settling. In September 1826 a water main burst near the anchorage on the Champs-Élysées side and soaked the ground. The anchorage moved, and the city council pressed for the bridge’s removal. In 1827 the project was cancelled and the structure dismantled; its materials went into other bridges. A government committee criticised Navier for relying too much on calculation. A different suspension bridge opened on a nearby site in 1829.

The scientific debates of these years were sharper than the political ones. Poisson criticised the way Navier summed the molecular forces in his theory of elasticity. Their exchange ran in the Annales de chimie et de physique in 1828 and 1829. Cauchy, meanwhile, was building elasticity on the concept of stress, which made the molecular hypothesis unnecessary.

Last years and legacy, 1830–1836

The setback did not slow his career. In 1830 Navier became titular professor at the École des Ponts et Chaussées. In 1831, after Cauchy left France following the July Revolution, Navier succeeded him as professor of analysis and mechanics at the École Polytechnique, and was made a chevalier of the Légion d’honneur. He wrote a preface for Fourier’s posthumous Analyse des équations déterminées (1831), and he clashed with Poisson over how Fourier’s theory of heat should be taught. From 1830 he advised the government on roads and on the new railways. Auguste Comte, the founder of positivism, worked as one of his assistants at the Polytechnique, and Navier shared the Saint-Simonian belief in a society organised by science and industry. He died in Paris on 21 August 1836, aged 51.

Legacy

• Fluid mechanics: the Navier–Stokes equations govern aerodynamics, oceanography, weather and climate models, blood flow and combustion. Their global regularity in three dimensions is a Millennium Prize Problem.
• Elasticity: Navier’s 1821 memoir was the first general field theory of elastic solids. The equation \(\mu\nabla^2u+(\lambda+\mu)\nabla(\nabla\cdot u)=0\) is often called the Navier (or Navier–Cauchy, or Lamé–Navier) equation.
• Structural engineering: the neutral axis, \(EI\), statically indeterminate structures, and Navier’s double-series solution for plates. Barré de Saint-Venant’s annotated 1864 edition of the Résumé des leçons made it a foundation text of the subject.
• Honours: his name is one of the 72 inscribed on the Eiffel Tower, and the Rue Navier in the 17th arrondissement of Paris is named after him.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Claude Louis Marie Henri Navier, MacTutor History of Mathematics.
2. Claude-Louis Navier, Wikipedia.
3. Navier, Claude-Louis-Marie-Henri, Dictionary of Scientific Biography.
4. K.-E. Kurrer, The History of the Theory of Structures, Ernst und Sohn, 2008; biographical note reproduced here.
5. E. Kranakis, Constructing a Bridge: An Exploration of Engineering Culture, Design, and Research in Nineteenth-Century France and America, MIT Press, 1997.
6. The Paris bridge that never was, Parisian Fields, 2014.

Mechanics

1. C.-L. Navier, Mémoire sur les lois de l’équilibre et du mouvement des corps solides élastiques, Mém. Acad. Sci. Inst. France 7 (1827).
2. C.-L. Navier, Mémoire sur les lois du mouvement des fluides, Mém. Acad. Sci. Inst. France 6 (1827).
3. C.-L. Navier, Rapport à M. Becquey et mémoire sur les ponts suspendus, Paris, 1823.
4. C.-L. Navier, Résumé des leçons données à l’École des Ponts et Chaussées sur l’application de la mécanique, Paris, 1826; 3rd ed. with notes by Barré de Saint-Venant, 1864.
5. G. G. Stokes, On the theories of the internal friction of fluids in motion, Trans. Cambridge Phil. Soc. 8 (1845).
6. O. Darrigol, Worlds of Flow: A History of Hydrodynamics from the Bernoullis to Prandtl, Oxford University Press, 2005.
7. C. L. Fefferman, Existence and smoothness of the Navier–Stokes equation, Clay Mathematics Institute problem description.
8. F. W. Hehl and Y. Itin, The Cauchy relations in linear elasticity theory, arXiv:cond-mat/0206175.
9. S. Timoshenko, History of Strength of Materials, McGraw-Hill, 1953.