G. I. Taylor

How to use this course

Geoffrey Ingram Taylor, known to everyone as G. I., spent nearly his whole working life in Cambridge, mostly without teaching duties, and published some 250 papers across fluid mechanics, solid mechanics, meteorology and oceanography. His method was distinctive. He chose a physical question, often prompted by something he had seen at sea, in the air or in a war, built a simple apparatus to look at it, and then found the mathematical model that matched what he saw. Many of his results carry his name: Taylor vortices, the Taylor–Proudman theorem, Taylor dispersion, the Taylor microscale, the Rayleigh–Taylor instability, the Saffman–Taylor fingers, the Taylor cone.

Part I traces his formation as the grandson of George Boole, a Cambridge mathematician turned meteorologist, and a wartime pilot. Part II presents his main contributions in rough chronological order. Part III covers the Second World War, including his work on the atomic bomb, and his long and productive retirement.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. Many exercises are back-of-the-envelope estimates in Taylor's own style: physical reasoning, dimensional analysis, and a number at the end.

Timeline

London and the Booles, 1886–1905

Geoffrey Ingram Taylor was born on 7 March 1886 in St John's Wood, London. His father, Edward Ingram Taylor, was an artist who painted landscapes and flowers and designed decorative interiors for ocean liners. His mother, Margaret Boole, was the second daughter of the logician George Boole and of Mary Everest Boole, a writer on the teaching of mathematics. Several of Margaret's sisters were remarkable. Alicia Boole Stott found the regular polytopes of four-dimensional space without formal training in mathematics; Ethel Lilian Voynich wrote the novel The Gadfly. Geoffrey had a younger brother, Julius.

Geoffrey went to a school in Hampstead and in 1899 entered University College School in London, where his interest in science grew. The sea and the wind would later supply many of his scientific problems.

Trinity, feeble light and the ice patrol, 1905–1914

In 1905 Taylor won a scholarship to Trinity College, Cambridge. He read mathematics, hearing lectures by Alfred North Whitehead, Edmund Whittaker and G. H. Hardy, and was twenty-second wrangler in Part I of the Mathematical Tripos in 1907. He then switched to physics and took first-class honours in Part II of the Natural Sciences Tripos in 1908.

Interference with feeble light

His first research project came from J. J. Thomson, who wondered whether interference fringes would survive if light were so weak that only a few quanta of energy were in the apparatus at once. Taylor photographed the diffraction pattern of a needle with a gas flame dimmed by smoked glass screens, using exposures of up to several months. The fringes were as sharp as with strong light. His short 1909 paper was the first experiment of its kind; with hindsight it shows that interference is a property of each photon, not of many photons interacting. Taylor did not pursue quantum physics further.

Shock waves and meteorology

His second piece of work was theoretical. Building on nineteenth-century work, he analysed the structure of a weak shock wave, showing how viscosity and heat conduction give it a definite thickness. It won him a Smith's Prize and, in 1910, a prize fellowship at Trinity. In 1911 he was appointed to the new Schuster Readership in Dynamical Meteorology.

After the Titanic sank in April 1912, the British government sent the ship Scotia to the North Atlantic in 1913 to watch for icebergs, and Taylor sailed as its meteorologist. He flew kites and balloons carrying instruments to measure temperature, humidity and wind at different heights above the cold sea. The data showed how heat and water vapour were carried up from the surface by turbulent eddies. His analysis, Eddy motion in the atmosphere (1915), estimated eddy diffusivities from measured profiles, and his essay on turbulent motion in fluids won the Adams Prize of 1915 (Lesson 04).

Farnborough and the Cavendish, 1914–1925

When war broke out in 1914, Taylor joined the group of Cambridge scientists working at the Royal Aircraft Factory in Farnborough. He studied the stresses in propeller shafts, and with A. A. Griffith turned Ludwig Prandtl's membrane analogy for torsion into a practical soap-film method of measurement. This work led to his later interest in the plasticity of metals. He studied the stability of parachutes and the pressure distribution on wings. To understand aircraft he learned to fly, gaining his pilot's certificate in 1915, and made parachute jumps himself.

In 1919 he returned to Trinity as fellow and lecturer in mathematics and was elected a Fellow of the Royal Society. Ernest Rutherford, the new Cavendish Professor, gave him space for experiments in the Cavendish Laboratory. Teaching did not suit him, and in 1923 the Royal Society appointed him its Yarrow Research Professor, a post with no teaching duties that he held at the Cavendish until 1952. The same year produced two of his finest papers, on rotating fluids and on the stability of flow between cylinders (Lesson 05), and he gave the Royal Society's Bakerian Lecture.

On 15 August 1925 he married Stephanie Ravenhill, a schoolteacher. They had no children. Together they sailed long distances in small boats, including a voyage to the Lofoten Islands of Norway, and in 1929, after a Pacific science congress, they explored the interior of Borneo. Sailing also produced an invention: in the 1930s Taylor designed the CQR plough anchor, which holds far better than traditional anchors for its weight and is still made today.

Turbulent diffusion

For the general reader. Smoke from a chimney spreads far faster than molecules alone could carry it, because turbulent eddies tear the plume apart and mix it with clean air. Taylor's first great theoretical paper, in 1921, asked how far a single particle wanders in a turbulent flow. His answer used only one statistical function, the correlation of the particle's velocity with itself at different times. It explained why spreading is fast and ballistic at first and then slows to a random walk.

Eddy diffusivity

From his Scotia data, Taylor modelled the turbulent flux of heat as if it were molecular conduction with a much larger diffusivity, \(\overline{w'T'}=-K\,\partial\bar T/\partial z\). Unlike a molecular diffusivity, \(K\) is a property of the flow, not of the fluid. The 1921 paper explained where such a diffusivity comes from.

Diffusion by continuous movements (1921)

Let a marked fluid particle start at the origin and have velocity \(v(t)\) in one direction, so that its displacement is \(X(t)=\int_0^tv(t')\,dt'\). Suppose the turbulence is statistically stationary and homogeneous, with \(\overline{v}=0\), \(\overline{v^2}=u'^2\), and Lagrangian autocorrelation

Proof. \(\frac{d}{dt}\overline{X^2}=2\overline{X\,v(t)}=2\int_0^t\overline{v(t')v(t)}\,dt'=2u'^2\int_0^tR(t-t')\,dt'\), using stationarity. Substitute \(\tau=t-t'\). Integrating once more and exchanging the order of integration gives the second formula. The limits follow from \(R\approx1\) for small \(\tau\) and from \(\int_0^tR\to T_L\).

The theorem needs no equation of motion, only stationarity. It anticipated results that later became standard in the theory of stationary random processes, and it connects to Einstein's 1905 relation between Brownian displacement and diffusion.

Rotating fluids and Taylor vortices

For the general reader. Rotation makes fluids behave strangely. In a spinning tank of water, a small object pushed slowly along the bottom drags a whole column of fluid above it, as if the water had become rigid along the axis. And when liquid fills the gap between two cylinders and the inner one spins fast enough, the smooth flow breaks into a stack of doughnut-shaped vortices. Taylor showed both effects in 1923, and in the second case he did something new: he predicted the exact speed at which the instability starts and confirmed it by experiment.

The Taylor–Proudman theorem

In a frame rotating with angular velocity \(\boldsymbol\Omega\), the momentum equation of an incompressible fluid is

where \(P\) includes the centrifugal potential. When the flow is slow compared with the rotation (small Rossby number \(U/\Omega L\)), steady, and nearly inviscid (small Ekman number \(\nu/\Omega L^2\)), only the Coriolis and pressure terms remain.

Proof. Take the curl: \(\nabla\times(\boldsymbol\Omega\times\mathbf u)=\boldsymbol\Omega(\nabla\cdot\mathbf u)-(\boldsymbol\Omega\cdot\nabla)\mathbf u=-(\boldsymbol\Omega\cdot\nabla)\mathbf u\), while the curl of a gradient vanishes. Taylor tested it by towing a small cylinder slowly across the bottom of a rotating tank. Dye showed that the fluid above the cylinder, all the way to the free surface, moved around it as if it were a solid extension of the obstacle: a Taylor column. Such columns appear above seamounts in the ocean and in laboratory models of planetary atmospheres.

Stability of Couette flow (1923)

Fluid between cylinders of radii \(r_1<r_2\) rotating at \(\Omega_1,\Omega_2\) has the exact circular solution

The reason is an exchange argument: swapping two rings of fluid conserves their angular momentum, and the kinetic energy rises if \((rv_\theta)^2\) increases outward. For Couette flow with the outer cylinder at rest, the circulation decreases outward, so the inviscid flow is unstable. Viscosity delays the instability. Taylor solved the linearized viscous problem for axisymmetric disturbances proportional to \(\cos(kz)\,e^{\sigma t}\), found the boundary \(\sigma=0\), and built an apparatus precise enough to test it.

Taylor's apparatus confirmed the predicted onset speeds and cell sizes over a range of \(\Omega_2/\Omega_1\), including counter-rotating cylinders. It was the first quantitative confirmation of a hydrodynamic stability calculation and of the no-slip condition in such a setting. Taylor–Couette flow has since become a model system for the transition to turbulence.

Dislocations and plastic flow

For the general reader. A perfect crystal should be enormously strong: to shear it, whole planes of atoms would have to slide over each other at once. Real metals yield at stresses thousands of times lower. In 1934 Taylor, and independently Egon Orowan and Michael Polanyi, explained the paradox. Crystals contain line defects, called dislocations, that let one row of atoms slip at a time, like moving a rug by pushing a ruck along it. Taylor also explained why metals harden as they are deformed: the dislocations multiply and get in each other's way.

The strength paradox

Shearing two planes of atoms with spacing \(h\) and atomic period \(b\) through a displacement \(x\), the stress is periodic in \(x\). Approximating it by \(\tau=\tau_{\max}\sin(2\pi x/b)\) and matching Hooke's law \(\tau=Gx/h\) at small \(x\) gives Frenkel's estimate (1926)

For copper this is several gigapascals; single crystals yield near one megapascal.

Taylor's 1934 theory

Taylor's dislocation is a line along which the crystal on one side has slipped by one lattice vector \(\mathbf b\) (the Burgers vector) relative to the other. Moving the line across the slip plane shears the crystal by \(b\), but at any moment only atoms near the line are displaced, so a small stress suffices. Counting the dislocations crossing a region gives Orowan's relation, and Taylor's model of interacting dislocations gives the hardening law.

Sketch. The shear stress at distance \(r\) from a straight dislocation is of order \(Gb/(2\pi r)\). To force a dislocation past a neighbour at the typical spacing \(r\sim\rho^{-1/2}\) requires a stress of that order, \(\tau\sim Gb\sqrt\rho\). Since deformation multiplies dislocations, \(\rho\) and hence \(\tau\) rise with strain: work hardening. In 1938 Taylor extended the analysis to polycrystals. Requiring each grain to deform with the overall strain, using five independent slip systems, he found that the tensile yield stress is \(M\) times the critical resolved shear stress of a single crystal, with \(M\approx3.1\) for randomly oriented face-centred cubic metals (the Taylor factor; later refined to 3.06).

The statistical theory of turbulence

For the general reader. A turbulent velocity record looks random. Taylor proposed in 1935 to stop trying to follow individual eddies and to describe turbulence by averages: how strongly the velocity at one point is correlated with the velocity nearby, and how the energy is spread over eddies of different sizes. He introduced the simplest case, turbulence that looks the same in every direction, and gave the tools that experimenters still use to analyse wind-tunnel data.

Isotropic turbulence (1935)

Turbulence is homogeneous if its statistics do not depend on position, and isotropic if they are also unchanged by rotations and reflections. Grid turbulence in a wind tunnel, made by blowing air through a mesh, approximates this well. For such flows, the longitudinal correlation

is a single function that determines the whole two-point correlation tensor (as shown later by Theodore von Kármán and Leslie Howarth). Taylor defined two length scales from it: the integral scale \(L=\int_0^\infty f\,dr\), the size of the energy-containing eddies, and the microscale \(\lambda\), from the curvature of \(f\) at the origin.

The first statement is a Taylor expansion of the correlation; the factor 15 comes from expressing all nine velocity gradients \(\overline{(\partial u_i/\partial x_j)^2}\) through the single longitudinal one using isotropy. The second statement is the cornerstone of later theory. Combined, they show that \(\lambda/L\) shrinks like \(\mathrm{Re}^{-1/2}\), so the eddies that dissipate energy are much smaller than those that contain it. Andrey Kolmogorov's theory of 1941 grew from this observation.

The spectrum and frozen turbulence (1938)

A hot-wire probe at a fixed point records \(u(t)\). To get the spatial structure, Taylor proposed that if the turbulent fluctuations are small compared with the mean speed \(U\), the pattern of eddies is carried past the probe nearly unchanged:

He showed that the one-dimensional spectrum of \(u_1\), measured as a function of frequency, is the Fourier transform of the correlation \(f\), so that the spectrum and the correlation contain the same information. The frozen-turbulence hypothesis is used in nearly every experimental study of turbulence and of atmospheric wind.

Blast waves and accelerated interfaces

For the general reader. During the Second World War Taylor was asked what happens when a very large amount of energy is released in a very small space. He found that the expanding fireball and shock follow a simple law: the radius grows as the two-fifths power of time, with a constant fixed by the energy. When photographs of the first atomic test were released in 1947, he used the law to estimate the yield, a figure that was still secret. In the same years he showed why the boundary between a heavy and a light fluid becomes unstable when the light fluid pushes the heavy one, a question that mattered for the implosion bomb and now matters for fusion and supernovae.

The intense point explosion

Energy \(E\) is released at a point in air of density \(\rho_0\). Once the shock is strong, the pressure behind it is so large that the ambient pressure is irrelevant. The only parameters are \(E\), \(\rho_0\), the time \(t\) and the dimensionless ratio \(\gamma\) of specific heats.

The only combination of \(E\) (kg m² s⁻²), \(\rho_0\) (kg m⁻³) and \(t\) (s) with the dimension of length is \((Et^2/\rho_0)^{1/5}\), which fixes the law up to the constant. In 1950 Taylor published both his wartime theory and a comparison with the photographs of the Trinity test. Plotting \(\log R\) against \(\log t\) gave a straight line of slope 2/5, and his yield estimates were close to the true value of about 20 kilotons.

The Rayleigh–Taylor instability (1950)

A layer of heavy fluid resting on light fluid under gravity is unstable, as Rayleigh had noted in 1883. Taylor observed that an acceleration acts in the same way: in the accelerated frame it is equivalent to gravity. An interface is unstable when it is accelerated in the direction from the light fluid toward the heavy one. D. J. Lewis confirmed it in experiments published alongside Taylor's paper, in which a liquid layer was driven downward by compressed air faster than free fall.

Derivation. Take irrotational flow on each side with potentials \(\phi_j\propto e^{\mp k z}\cos kx\,e^{\sigma t}\), decaying away from the interface \(z=\eta=\eta_0\cos kx\,e^{\sigma t}\). Kinematic conditions give \(\partial_z\phi_j=\sigma\eta\). The linearized Bernoulli equation on each side, \(\rho_j(\partial_t\phi_j+g\eta)+p_j=\text{const}\), and the pressure jump \(p_1-p_2=-\gamma\eta_{xx}=\gamma k^2\eta\) combine to give the dispersion relation. With \(\rho_1>\rho_2\) (stable layering) the same formula gives oscillating gravity–capillary waves.

Slow viscous flow

For the general reader. After he retired, Taylor turned to flows so slow or so small that inertia does not matter: bacteria and sperm swimming, dye spreading along a narrow tube, oil being pushed out of rock by water. In this world motion is reversible and intuition from everyday life fails. Taylor made films and demonstrations of it that are still shown to students, and found three results that each started a field.

Swimming (1951)

At very small Reynolds number the Stokes equations \(\mu\nabla^2\mathbf u=\nabla p\), \(\nabla\cdot\mathbf u=0\) hold. They have no time derivative, so a swimmer that runs through a sequence of shapes and then reverses it returns to its starting point. Taylor asked how a microorganism can swim at all, and modelled it as an infinite sheet carrying a travelling wave, \(y=b\sin(kx-\omega t)\).

The speed is second order in \(b\) because reversing the sign of \(b\) gives the same motion shifted by half a wavelength, so \(U\) must be even in \(b\).

Taylor dispersion (1953)

Solute injected into a liquid flowing slowly through a narrow tube spreads along it far faster than molecular diffusion allows, yet far more slowly than the velocity differences across the tube would suggest. Taylor showed that the two effects combine.

The striking feature is that \(D_{\text{eff}}\) grows as \(D\) decreases. Shear stretches the solute along the tube; radial diffusion samples fast and slow streamlines and so limits the stretching. The faster the radial diffusion, the less the spreading. Taylor used the result to measure molecular diffusivities, and it underlies the analysis of chromatography and of solute transport in blood vessels and rivers.

Viscous fingering (1958)

In a Hele-Shaw cell, two glass plates a small distance \(b\) apart, the gap-averaged velocity obeys Darcy's law \(\mathbf u=-\frac{b^2}{12\mu}\nabla p\), the same law as flow in porous rock. Taylor and Philip Saffman studied what happens when one fluid displaces another.

A less viscous fluid pushing a more viscous one forms fingers, which compete until, in a channel, one finger of about half the channel width remains. The instability limits the recovery of oil by water flooding, and the selection of the finger width became a celebrated problem of pattern formation in the 1980s.

Electrified drops and the Taylor cone

For the general reader. In his late seventies Taylor took up a question from meteorology: what happens to raindrops in the strong electric fields of a thunderstorm? He found that a drop of conducting liquid stretches in a field, and that above a critical field its ends become sharp cones that shoot out fine jets. He showed that the cone has a universal angle, the same for every liquid. The Taylor cone is now the working heart of electrospray, the technique that lets mass spectrometers weigh large proteins, and of electrospinning and some spacecraft thrusters.

Balancing stresses

The surface of a perfect conductor is an equipotential, and the electric field \(E\) just outside it pulls the surface outward with stress \(\tfrac12\varepsilon_0E^2\). Surface tension \(\gamma\) pulls inward with pressure \(\gamma\kappa\), where \(\kappa\) is the mean curvature (sum of principal curvatures). For a cone of half-angle \(\alpha\), one principal curvature vanishes and the other, at distance \(r\) from the apex, is \(\cot\alpha/r\). Balance at every \(r\) requires \(E^2\propto1/r\), so \(E\propto r^{-1/2}\) and the potential varies like \(r^{1/2}\).

Sketch. Laplace's equation in spherical coordinates with axial symmetry admits separated solutions \(r^\nu P_\nu(\cos\theta)\). The stress balance forces \(\nu=\tfrac12\). The function \(P_{1/2}(\cos\theta)\) is regular on the axis \(\theta=0\) and has exactly one zero in \((0,\pi)\), at \(\theta_0\approx130.71^\circ\). Taylor verified the angle by photographing cones formed on drops and soap films between electrodes. Real cones emit a thin jet from their tips, which breaks into charged droplets: this is the cone-jet mode of electrospray. John Fenn's development of electrospray ionization for large biomolecules earned a share of the 2002 Nobel Prize in Chemistry.

War and Los Alamos, 1939–1945

Through the 1930s Taylor's work had turned increasingly to solids and to turbulence. His correspondence with Ludwig Prandtl, mostly on turbulence, ran from 1923 to 1938. In letters of 1938 and 1939 to Taylor and his wife, Prandtl defended Nazi policy toward the Jews (see Prandtl's course, Lesson 11).

From 1939 Taylor worked as a consultant on the physics of explosions for several British ministries and committees. He analysed the propagation of blast waves in air and of shock waves and gas bubbles in underwater explosions, the response of plates and structures to blast, and the penetration of armour. Early in 1941 he wrote the analysis of the very intense point explosion summarized in Lesson 08; it circulated to the Civil Defence Research Committee of the Ministry of Home Security in June of that year. He was knighted and received the Copley Medal of the Royal Society in 1944.

In 1944 and 1945 he visited Los Alamos as a member of the British mission to the Manhattan Project. His expertise on the instability of accelerated interfaces bore directly on the implosion design of the plutonium bomb, in which a heavy shell is driven inward by explosives. He was among the observers of the Trinity test in New Mexico on 16 July 1945. He was elected to the US National Academy of Sciences in 1945 and received the US Medal for Merit in 1946. In 1950, after the Trinity photographs had been released, he published his blast-wave analysis with the yield estimate that, in effect, revealed a classified number from public data.

The long afternoon, 1945–1975

Taylor retired from the Yarrow professorship in 1952 but kept his room and workshop at the Cavendish and worked for another twenty years. In 1959 George Batchelor founded the Department of Applied Mathematics and Theoretical Physics at Cambridge, and Taylor's presence helped make Cambridge a world centre of fluid mechanics. His style of work became famous: simple apparatus, often built by his own hands, followed by an elegant analysis. In the 1960s he appeared in the American series of educational films on fluid mechanics, demonstrating low-Reynolds-number flows himself.

His doctoral students and younger collaborators included George Batchelor, Albert Green, Rosa Morris, Philip Drazin, Francis Bretherton, Stewart Turner and Philip Saffman. Batchelor edited his collected scientific papers in four volumes (1958–71) and later wrote his biography. Among Taylor's many honours were the Royal Medal (1933), the Copley Medal (1944), the De Morgan Medal of the London Mathematical Society (1956), the Timoshenko Medal (1958), the Franklin Medal (1962) and the Order of Merit (1969). In 1972 he received the SIAM Theodore von Kármán Prize.

Stephanie Taylor died in 1967. Taylor continued to publish into his eighties, including six papers on electrohydrodynamics between 1964 and 1969. A severe stroke in 1972 ended his research. He died at his home, Farmfield, on Huntingdon Road in Cambridge, on 27 June 1975, aged 89.

Legacy

• Fluid mechanics: Taylor–Couette flow and Taylor columns in rotating fluids; turbulent diffusion, the statistical theory of turbulence, the microscale and the frozen-turbulence hypothesis; the Rayleigh–Taylor and Saffman–Taylor instabilities; the intense blast wave; Taylor dispersion; swimming at low Reynolds number.
• Solid mechanics: dislocations, Taylor hardening and the Taylor factor for polycrystals; the Taylor impact test for dynamic yield strength.
• Geophysics and engineering: eddy diffusion in the atmosphere and ocean, tidal friction in shallow seas, the CQR anchor, parachute design.
• Method: a model for how to combine simple experiments, scaling arguments and exact solutions. The G. I. Taylor Medal of the Society of Engineering Science honours this approach.

Sources and further reading

Biography and history

1. J. J. O'Connor and E. F. Robertson, Geoffrey Ingram Taylor, MacTutor History of Mathematics.
2. G. K. Batchelor, Taylor, Sir Geoffrey Ingram, Dictionary of National Biography (via MacTutor).
3. G. I. Taylor, Wikipedia (honours, doctoral students).
4. Trinity College Cambridge, Papers of Sir Geoffrey Taylor; correspondence with L. Prandtl.
5. G. K. Batchelor, The Life and Legacy of G. I. Taylor, Cambridge University Press, 1996.
6. G. K. Batchelor, Geoffrey Ingram Taylor, Biographical Memoirs of Fellows of the Royal Society 22 (1976).
7. M. Eckert, Ludwig Prandtl: A Life for Fluid Mechanics and Aeronautical Research, Springer, 2019 (the Prandtl–Taylor letters).

Physics

1. G. I. Taylor, Interference fringes with feeble light, Proc. Camb. Phil. Soc. 15 (1909).
2. G. I. Taylor, Eddy motion in the atmosphere, Phil. Trans. R. Soc. A 215 (1915).
3. G. I. Taylor, Diffusion by continuous movements, Proc. London Math. Soc. 20 (1921).
4. G. I. Taylor, Stability of a viscous liquid contained between two rotating cylinders, Phil. Trans. R. Soc. A 223 (1923).
5. G. I. Taylor, Experiments on the motion of solid bodies in rotating fluids, Proc. R. Soc. A 104 (1923).
6. G. I. Taylor, The mechanism of plastic deformation of crystals, Proc. R. Soc. A 145 (1934).
7. G. I. Taylor, Statistical theory of turbulence, parts I–IV, Proc. R. Soc. A 151 (1935); The spectrum of turbulence, Proc. R. Soc. A 164 (1938).
8. G. I. Taylor, The formation of a blast wave by a very intense explosion, I and II, Proc. R. Soc. A 201 (1950).
9. G. I. Taylor, The instability of liquid surfaces when accelerated in a direction perpendicular to their planes, Proc. R. Soc. A 201 (1950).
10. G. I. Taylor, Analysis of the swimming of microscopic organisms, Proc. R. Soc. A 209 (1951).
11. G. I. Taylor, Dispersion of soluble matter in solvent flowing slowly through a tube, Proc. R. Soc. A 219 (1953).
12. P. G. Saffman and G. I. Taylor, The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid, Proc. R. Soc. A 245 (1958).
13. G. I. Taylor, Disintegration of water drops in an electric field, Proc. R. Soc. A 280 (1964).
14. G. K. Batchelor (ed.), The Scientific Papers of Sir Geoffrey Ingram Taylor, 4 vols., Cambridge University Press, 1958–71.
15. G. K. Batchelor, The Theory of Homogeneous Turbulence, Cambridge University Press, 1953.
16. P. G. Drazin and W. H. Reid, Hydrodynamic Stability, 2nd ed., Cambridge University Press, 2004 (Rayleigh criterion, Taylor–Couette and Rayleigh–Taylor problems).