He did this through his deep analysis of particular problems, such as the similarities in movements of sea breezes, avalanches and long air pockets in tubes; or why thin vortices on aircraft wings can behave like shock-waves and suddenly become thicker and change the whole flow; how tiny bubbles collapse, and thence cause the damaging erosion found on propellers; how the flexible skins of dolphins and, nowadays, "smart" submarines can damp out fluctuations in the turbulent flow over their skins to increase their speed and reduce the noise heard by nearby prey or predators. It is perhaps a surprising list of accomplishments for the Sedleian Professor at Oxford University, but not when you consider Brooke Benjamin's remarkable life and career.
Coming from a Merseyside family, he studied electrical engineering at Liverpool University and then took an MEng at Yale University in 1952 in the rather ageing subject of thermionic valves (he saw transistors around the corner). Although he came up to King's College, Cambridge, in the same year to continue this research in the Engineering Department, the story goes that he popped his head round the door of the hydraulics laboratory, with its experiments on swirling flow in tubes, the unusual square shaped "G.I. Taylor" wave tank, and A.M. Binnie FRS clambering around the huge pipes and tanks. Benjamin was hooked and stayed to work with Binnie, on a new subject for the laboratory, the formation of the tiny "cavitation" bubbles that form in high-speed water flow and their sudden collapse; they used electronics to trigger the bubbles and to take remarkable high-speed photographs of individual bubbles.
A characteristic feature of the elegant theories Benjamin built on these experiments, and on his other studies of "avalanche-like" gravity currents, steep sub-surface "internal" waves in the ocean, and swirling flows, was the use of appropriate averages (in fact integrals of mathematical functions of the velocity and pressure over spatial volumes), to explain the underlying principles of a hydrodynamic problem.
His research led him to pose questions about flows in ever more general and abstract terms, beginning with engineering questions such as the total energy or impulse of the flow, and later moving to questions about whether a particular type of flow can exist or not. One might trace over the years the movement of his mathematics towards more abstractness in relation to the growing capacity of electronic computers to calculate flows with ever greater detail. Benjamin pioneered the way, despite significant scepticism at Cambridge and elsewhere, for applied mathema- ticians to ask and answer more general questions, now that computers can provide more of the numerical answers.
On the basis of his thesis he was elected to a Fellowship at King's College in 1955. He had already published the first of his papers, on the emerging topic of non-linear effects on steep water waves, with James Lighthill, which led to his appointment to the specially created post of Assistant Director of Research, held jointly between the Engineering Department and the embryonic Department of Applied Mathematics and Theoretical Physics. He helped set up a new fluid mechanics laboratory there in the old Press Building in 1964 and was appointed a Reader in 1967.
During this period he widened his interests in hydrodynamical problems as an active member of the illustrious group of researchers drawn together by G.K. Batchelor and G.I. Taylor, and winnowed by the ferocious arguments of famous Friday-afternoon seminars.
However, despite his brilliance, Benjamin did not find his research easy; he was always focusing on really tough, but well-defined, problems. In the warm summer of 1964 he and his student Jim Feir were struggling to make perfect sinusoidal water waves travel down the new 10m-long wave tank set alongside the brassy old steam engines in the Thermo Lab.
Despite ever finer electronic and mechanical control of the harmonic motion of the wave maker, the initially regular wave train would keep spawning higher harmonics along the tank. The penny dropped - this was the discovery of the fundamental Benjamin-Feir "side-band" instability of all steep waves.
Benjamin's teaching of undergraduates at Cambridge was rather minimal, but no one who attended can forget his slow entry into a lecture-room, the precise wiping of the board, the ceremonial breaking of chalk, the elegant thin writing with serifs on the letters, and his quiet, economical explanations as he led his class through the analysis of water waves.
In 1970 he left for a chair in Mathematics at Essex University. There he set up the Fluid Mechanics Research Institute, with a particular aim of showing how more abstract mathematical analysis could really contribute to understanding well-defined but difficult experimental problems; such as why, as concentric cylinders rotate, the flow between them sets up slow eddying motions that have so many and such sensitive flow patterns. He took an active interest in the turbulent politics of the Essex campus, as well as conducting the university choir.
On his election to the Sedleian chair at the Mathematical Institute in Oxford in 1979, and a Fellowship at Queen's College, Benjamin continued the fruitful interaction with those experimenting on very sensitive flows, especially his close collaborator Dr T. Mullin. He also returned to a number of the hydrodynamic problems he had studied earlier in his career, on vortex breakdown, bubbles and others, but now approaching the problems with more general and powerful mathematical apparatus.
He had a number of visiting appointments abroad, but he returned most often to Pennsylvania State University, where he celebrated his 60th birthday at a symposium in his honour. Benjamin was a great supporter of scientific and academic communities; he supported the International Centre for Theoretical Physics in Trieste for its work for Third World young scientists; as President of the UK National Conference of University Professors he stimulated public debate in the hope that the government would recognise publicly and materially the value to Britain of its universities; he persuaded those running the annual British Theoretical Mechanics Colloquia to broaden their mathematical appeal and rename them the British Applied Mathematics Colloquia, the first being held in Oxford in 1991; and finally as a Fellow of the Royal Society, he was twice a member of Council and gave the Bakerian Lecture in 1992.
Brooke Benjamin had a most attractive aura - wise, sensitive and having a touch of mystery - in fact a bit of a Cheshire cat. He was about 6ft tall, moving slowly, smiling slightly, often with a pipe in his mouth, and speaking so that every word counted, with surprisingly strong views on science in all its aspects. He and his second wife Natalia were a joyous couple with many friends from all over the world. He was an exceptional man in another way, being the longest surviving diabetic in Britain, who had regularly injected himself since he was a teenager; indeed he was something of a medical specimen. Most people never knew.
As a finale, we should remember one of the best "Brooke" stories. He was invited to a dinner at 10 Downing Street in the 1980s. Circulating after dinner he met Denis Thatcher, who asked him who he was. Brooke replied that he was a professor at Oxford and Denis said, ''I should keep that quiet around here, if I were you.''
Thomas Brooke Benjamin, mathematician: born 15 April 1929; Fellow of King's College Cambridge 1955-64; Assistant Director of Research, Cambridge University 1958-67, Reader in Hydrodynamics 1967-70; FRS 1966; Professor of Mathematics and Director, Fluid Mechanics Research Institute, Essex University 1970-78; Sedleian Professor of Natural Philosophy, Oxford University 1979-95; Fellow of Queen's College, Oxford 1979-95; married 1956 Helen Ginsburg (one son, two daughters; marriage dissolved 1974), 1978 Natalia Court (one daughter); died Oxford 16 August 1995.