The triumvirate of Cambridge University in the mid-1800’s consisted of three towering figures of mathematics and physics:  George Stokes (1819 – 1903), William Thomson (1824 – 1907) (Lord Kelvin), and James Clerk Maxwell (1831 – 1879).  Their discoveries and methodology changed the nature of natural philosophy, turning it into the subject that today we call physics.  Stokes was the elder, establishing himself as the predominant expert in British mathematical physics, setting the tone for his close friend Thomson (close in age and temperament) as well as the younger Maxwell and many other key figures of 19^th century British physics.

            George Gabriel Stokes was born in County Sligo in Ireland as the youngest son of the rector of Skreen parish of the Church of Ireland.  No miraculous stories of his intellectual acumen seem to come from his childhood, as they did for the likes of William Hamilton (1805 – 1865) or George Green (1793 – 1841).  Stokes was a good student, attending school in Skreen, then Dublin and Bristol before entering Pembroke College Cambridge in 1837.  It was towards the end of his time at Cambridge that he emerged as a top mathematics student and as a candidate for Senior Wrangler.

The Cambridge Wrangler

Since 1748, the mathematics course at Cambridge University has held a yearly contest to identify the top graduating mathematics student.  The winner of the contest is called the Senior Wrangler, and in the 1800’s the Senior Wrangler received a level of public fame and admiration for intellectual achievement that is somewhat like the fame reserved today for star athletes.  In 1824 the mathematics course was reorganized into the Mathematical Tripos, and the contest became known as the Tripos Exam.  The depth and length of the exam was legion.  For instance, in 1854 when Edward Routh (1831 – 1907) beat out Maxwell for Senior Wrangler, the Tripos consisted of 16 papers spread over 8 days, totaling over 40 hours for a total number of 211 questions.  The winner typically scored less than 50%.  Famous Senior Wranglers include George Airy, John Herschel, Arthur Cayley, Lord Rayleigh, Arthur Eddington, J. E. Littlewood, Peter Guthrie Tait and Joseph Larmor.

            In his second year at Cambridge, Stokes had begun studying under William Hopkins (1793 – 1866).  It was common for mathematics students to have private tutors to prep for the Tripos exam, and Tripos tutors were sometimes as famous as the Senior Wranglers themselves, especially if a tutor (like Hopkins) was to have several students win the exam.  George Stokes became Senior Wrangler in 1841, and the same year he won the Smith’s Prize in mathematics.  The Tripos tested primarily on bookwork, while the Smith’s Prize tested on originality.  To achieve top scores on both designated the student as the most capable and creative mathematician of his class.  Stokes was immediately offered a fellowship at Pembroke College allowing him to teach and study whatever he willed.

            After Stokes graduated, Hopkins suggested that Stokes study hydrodynamics.  This may have been in part motivated by Hopkins’ own interest is hydraulic problems in geology, but it was also a prescient suggestion, because hydrodynamics was poised for a revolution.

The Early History of Hydrodynamics

Leonardo da Vinci (1452 – 1519) believed that an artist, to capture the essence of a subject, needed to understand its fundamental nature.  Therefore, when he was captivated by the idea of portraying the flow of water, he filled his notebooks with charcoal studies of the whorls and vortices of turbulent torrents and waterfalls.  He was a budding experimental physicist, recording data on the complex phenomenon of hydrodynamics.  Yet Leonardo was no mathematician, and although his understanding of turbulent flow was deep, he did not have the theoretical tools to explain what he saw.  Two centuries later, Daniel Bernoulli (1700 – 1782) provided the first mathematical description of water flowing smoothly in his Hydrodynamica (1738).  However, the modern language of calculus was only beginning to be used at that time, preventing Daniel from providing a rigorous derivation. 

            As for nearly all nascent mathematical theories of the mid 1700’s, whether they be Newtonian dynamics or the calculus of variations or number and graph theory or population dynamics or almost anything, the person who placed the theory on firm mathematical foundations, using modern notions and notations, was Leonhard Euler (1707 – 1783).  In 1752 Euler published a treatise that described the mathematical theory of inviscid flow—meaning flow without viscosity.  Euler’s chief results is

where ρ is the density, v is the velocity, p is pressure, z is the height of the fluid and φ is a velocity potential, while f(t) is a stream function that depends only on time.  If the flow is in steady state, the time derivative vanishes, and the stream function is a constant.  The key to the inviscid approximation is the dominance of momentum in fast flow, as opposed to creeping flow in which viscosity dominates.  Euler’s equation, which expresses the well-known Bernoulli principle, works well under fast laminar conditions, but under slower flow conditions, internal friction ruins the inviscid approximation.

            The violation of the inviscid flow approximation became one of the important outstanding problems in theoretical physics in the early 1800’s.  For instance, the flow of water around ship’s hulls was a key technological problem in the strategic need for speed under sail.  In addition, understanding the creation and propagation of water waves was critical for the safety of ships at sea.  For the growing empire of the British islands, built on the power of their navy, the physics of hydrodynamics was more than an academic pursuit, and their archenemy, the French, were leading the way.

The French Analysts

In 1713 when Newton won his priority dispute with Leibniz over the invention of calculus, it had the unintended consequence of setting back British mathematics and physics for over a hundred years.  Perhaps lulled into complacency by their perceived superiority, Cambridge and Oxford continued teaching classical mathematics, and natural philosophy became dogmatic as Newton’s in Principia became canon.  Meanwhile Continental mathematical analysis went through a fundamental transformation.  Inspired by Newton’s Principia rather than revering it, mathematicians such as the Swiss-German Leonhard Euler, the Frenchwoman Emile du Chatelet and the Italian Joseph Lagrange pushed mathematical physics far beyond Newton by developing Leibniz’ methods and notations for calculus.

            By the early 1800’s, the leading mathematicians of Europe were in the French school led by Pierre-Simon Laplace along with Joseph Fourier, Siméon Denis Poisson and Augustin-Louis Cauchy.  In their hands, functional analysis was going through rapid development, both theoretically and applied, far surpassing British mathematics.  It was by reading the French analysts in the 1820’s that the Englishman George Green finally helped bring British mathematics back up to speed.

            One member of the French school was the French engineer Claude-Louis Navier (1785 – 1836).  He was educated at the Ecole Polytechnique and the School for Roads and Bridges where he became one of the leading architects for bridges in France.  In addition to his engineering tasks, he also was involved in developing principles of work and kinetic energy that aided the later work of Coriolis, who was one of the first physicists to recognize the explicit interplay between kinetic energy and potential energy.  One of Navier’s specialties was hydraulic engineering, and he edited a new edition of a classic work on hydraulics.  In the process, he became aware of serious deficiencies in the theoretical treatment of creeping flow, especially with regards to dissipation.  By adopting a molecular approach championed by Poisson, including appropriate boundary conditions, he derived a correction to the Euler flow equations that included a new term with a new material property of viscosity

Navier published his new flow equation in 1823, but the publication was followed by years of nasty in-fighting as his assumptions were assaulted by Poisson and others.  This acrimony is partly to blame for why Navier was not hailed alone as the discoverer of this equation, which today bears the name “Navier-Stokes Equation”.

Stokes’ Hydrodynamics

Despite the lead of the French mathematicians over the British in mathematical rigor, they were also bogged down by their insistence on mechanistic models that operated on the microscale action-reaction forces.  This was true for their theories of elasticity, hydrodynamics as well as the luminiferous ether.  George Green in England would change this.  While Green was inspired by French mathematics, he made an important shift in thinking in which the fields became the quantities of interest rather than molecular forces.  Differential equations describing macroscale phenomena could be “true” independently of any microscale mechanics.  His theories on elasticity and light propagation relied on no underlying structure of matter or ether.  Underlying models could change, but the differential equations remained true.  Maxwell’s equations, a pinnacle of 19^th-century British mathematical physics, were field equations that required no microscopic models, although Maxwell and others later tried to devise models of the ether.

            George Stokes admired Green and adopted his mathematics and outlook on natural philosophy.  When he turned his attention to hydrodynamic flow, he adopted a continuum approach that initially did not rely on molecular interactions to explain viscosity and drag.  He replicated Navier’s results, but this time without relying on any underlying microscale physics.  Yet this only took him so far.  To explain some of the essential features of fluid pressures he had to revert to microscopic arguments of isotropy to explain why displacements were linear and why flow at a boundary ceased.  However, once these functional dependences were set, the remainder of the problem was pure continuum mechanics, establishing the Navier-Stokes equation for incompressible flow.  Stokes went on to apply no-slip boundary conditions for fluids flowing through pipes of different geometric cross sections to calculate flow rates as well as pressure drops along the pipe caused by viscous drag.

            Stokes then turned to experimental results to explain why a pendulum slowly oscillating in air lost amplitude due to dissipation.  He reasoned that when the flow of air around the pendulum bob and stiff rod was slow enough the inertial effects would be negligible, simplifying the Navier-Stokes equation.  He calculated the drag force on a spherical object moving slowly through a viscous liquid and obtained the now famous law known as Stokes’ Law of Drag

in which the drag force increases linearly with speed and is proportional to viscosity.  With dramatic flair, he used his new law to explain why water droplets in clouds float buoyantly until they become big enough to fall as rain.

The Lucasian Chair of Mathematics

There are rare individuals who become especially respected for the breadth and depth of their knowledge.  In our time, already somewhat past, Steven Hawking embodied the ideal of the eminent (almost clairvoyant) scientist pushing his field to the extremes with the deepest understanding, while also being one of the most famous science popularizers of his day as well as an important chronicler of the history of physics.  In his own time, Stokes was held in virtually the same level of esteem. 

            Just as Steven Hawking and Isaac Newton held the Lucasian Chair of Mathematics at Cambridge, Stokes became the Lucasian chair in 1849 and held the chair until his death in 1903.  He was offered the chair in part because of the prestige he held as first wrangler and Smith’s prize winner, but also because of his imposing grasp of the central fields of his time. The Lucasian Chair of Mathematics at Cambridge is one of the most famous academic chairs in the world.  It was established by Charles II in 1664, and the first Lucasian professor was Isaac Barrow followed by Isaac Newton who held the post for 33 years.  Other famous Lucasian professors were Airy, Babbage, Larmor, Dirac as well as Hawking.  During his tenure, Stokes made central contributions to hydrodynamics (as we have seen), but also the elasticity of solids, the behavior of waves in elastic solids, the diffraction of light, problems in light, gravity, sound, heat, meteorology, solar physics, and chemistry.  Perhaps his most famous contribution was his explanation of fluorescence, for which he won the Rumford Medal.  Certainly, if the Nobel Prize had existed in his time, he would have been a Nobel Laureate.

Derivation of Stokes’ Law

The flow field of an incompressible fluid around a smooth spherical object has zero divergence and satisfies Laplace’s equation.  This allows the stream velocities to take the form in spherical coordinates

where the velocity components are defined in terms of the stream function ψ.   The partial derivatives of pressure satisfy the equations

where the second-order operator is

The vanishing of the Laplacian of the stream function

allows the function to take the form

The no-slip boundary condition on the surface of the sphere, as well as the asymptotic velocity field far from the sphere taking the form v•cosθ  gives the solution

Using this expression in the first equations yields the velocities, pressure and shear

The force on the sphere is obtained by integrating the pressure and the shear stress over the surface of the sphere.  The two contributions are

Adding these together gives the final expression for Stokes’ Law

where two thirds of the force is caused by the shear stress and one third by the pressure.

Stokes Timeline

• 1819 – Born County Sligo Parish of Skreen
• 1837 – Entered Pembroke College Cambridge
• 1841 – Graduation, Senior Wrangler, Smith’s Prize, Fellow of Pembroke
• 1845 – Viscosity
• 1845 – Viscoelastic solid and the luminiferous ether
• 1845 – Ether drag
• 1846 – Review of hydrodynamics (including French references)
• 1847 – Water waves
• 1847 – Expansion in periodic series (Fourier)
• 1848 – Jelly theory of the ether
• 1849 – Lucasian Professorship Cambridge
• 1849 – Geodesy and Clairaut’s theorem
• 1849 – Dynamical theory of diffraction
• 1850 – Damped pendulum, explanation of clouds (water droplets)
• 1850 – Haidinger’s brushes
• 1850 – Letter from Kelvin (Thomson) to Stokes on a theorem in vector calculus
• 1852 – Stokes’ 4 polarization parameters
• 1852 – Fluorescence and Rumford medal
• 1854 – Stokes sets “Stokes theorem” for the Smith’s Prize Exam
• 1857 – Marries
• 1857 – Effect of wind on sound intensity
• 1861 – Hankel publishes “Stokes theorem”
• 1880 – Form of highest waves
• 1885 – President of Royal Society
• 1887 – Member of Parliament
• 1889 – Knighted as baronet by Queen Victoria
• 1893 – Copley Medal
• 1903 – Dies
• 1945 – Cartan establishes modern form of Stokes’ theorem using differential forms

Further Reading

Darrigol, O., Worlds of flow : A history of hydrodynamics from the Bernoullis to Prandtl. (Oxford University Press: Oxford 2005.) This is an excellent technical history of hydrodynamics.